Algebra Chapters 1 2 3 4 5 1.1 SOME BASICS OF ALGEBRA The following figure shows the relationships among various kinds of numbers, along with examples of how real numbers can be sorted. -19,-i,-7,0, 3,i,T7.8.39 Positivc intcg~:rs real numhers: 1, 39 page 50 135.4 cm + 50.28 cm = 185.68 cm round off to 185.7 cm 2.1 cm x 6.45 cm = 13.545 cm2. Multiplication. 4.371 x 10 7 can become = 43,710,000 Instructor's Solutions Manual isbn 0-201-73489-3 Answer book isbn 0-201-73727-2 mymathlab.com Exercise Set 1.1 When all members of one set are found in a second set, the first set is a sub-set of the second set. if A = {2, 4, 6} and B = {1, 2, 4, 5, 6}, we write A S B to indicate that A is a subset of B. Similarly, if f~ represents the set of all natural numbers and 71 the set of all integers, we can write ~l S 7L. Additional state-ments can be made using other sets in the diagram above. FOR EXTRA HELP Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MyMathtab com Tn thP ctm%Irm~! ~~~W the i~~~rrr~~t~~w Throughout this text, Use mathematical symbols to translate each phrase. selected exercises are marked with the icon Ana~' . These CHAPTER 7 ALGEBRA AND PROBLEM SOLVING `Aha!" exercises can be answered quite easily if the stu- dent pauses to inspect the exercise rather than proceed mechanically. This is done to discourage rote memoriza- tion. Some `~iha!" exercises are left unrnarked to encour- age students to always pause before working a problem. Exercise Set 1.1 1. Three less than some number 3-6 2. Four more than some number 1+4=5 3, Twelve times a number. 12 x 4. Twice a number 2 5. Sixty-five percent of some number 65% ' 6. Thirty-nine percent of some number 39% 7. Ten more than twice a number 10+10=20 8. Six less than half of a number 6-12 9. Eight more than ten percent of some number. 10. Five less than six percent of some number 6-5=1 11. One less than the difference of two numbers 12. Two more than the product of two numbers 13. Ninety miles per every four gallons of gas 14. One hundred words per every sixty seconds Evaluate each expression using the values provided. 15.7x+y,for x=3and y=4 16. 6a - b, for a = 5 and b = 3 17. 2c divide by = 3b, for b = 4 and c = 6 3 x 2 =6 18. 3z divide = 2y, for y = 1 and z = 6 3 x 2 = 6 19. 25 + r2 - s, for r = 3 and s = 7 20. n3 + 2 - p, for n = 2 and p = 5 A~`~ 21. 3n2p - 3pn2, for n = 5 and p = 9 22. 2a3b - 2b2 for a = 3 and b = 7 23. 5x divide by (2 + x - y) for x = 6 and y = 2 24. 3(m + 2n) divide by m for m = 7 and n = 0 25. 10 - a - b 2 for a = 7 and b = 2 26. [ 17 - (x + y)]2, for x = 4 and y = 1 27. m + [n(3 + n)]Z, for m = 9 and n = 2 28. a2 - [3(a - b)]2, for a = 7 and b = 5 29. Base = 5 ft, height = 7 ft 30. Base = 2.9 m, height = 2.1 m 31. Base = 7 m, height = 3.2 m 32. Base = 3.6 ft, height = 4 ft 9. Eight more than ten percent of some number Use roster notation to write each set. 33. The set of all vowels in the alphabet 34. The set of all days of the week 01 02 03 04 05 06 07 35. The set of all odd natural numbers 36. The set of all even natural numbers 37, The set of all natural numhers that are multiples of 5 10 15 20 25 38. The set of all natural numbers that are multiples of10 Use set-builder notation to write each set. 39. The set of all odd numbers between 10 and 20 11 13 40. The set of all multiples of 4 between 22 and 35 41. {0, 1, 2, 3, 4} 42. {-3, -2, -1, 0, 1, 2} 43. The set of all multiples of 5 between 7 and 79 22. 2a3b - 2b2, for a = 3 and b = 7 44. The set of all even numbers between 9 and 99 9 12 15 18 Classify each statement as true or false. The following sets are used. N = the set of natural numbers; W = the set of whole numbers; Z = the set of integers; O = the set of rational numbers; H= the set of irrational numbers; R= the set of real numbers. 45.9E~1 46.5.1Ef~ 47. ~1SMU 48. MUS7L _ 49. ~ E ~ 50. 3 E ail 51. ~{I S I~ 52. 10 E Il8 53. 4.3 ~ 71 54. 7L ~ ~l 55. Q~S~ 56. ~ S 71 , In Exercises 29-32, find the area of a triangular window with the given base and height.  Use roster notation to write eacla set. 7~r !i means "is greater than." The symbol <_ means "is less than or equal to" and the symbol >_ means "is greater than or equal to." These symbols are used to form inequalities. In the figure below, we have -6 < -1 (since -6 is to the left of -1) and |-6| > |-1| (since 6 is to the right of 1). -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 |-1| |-6| Examp1e 2 Write out the meaning of each inequality and determine whether it is a true statement. a) -7<-2 b) 4>-1 c) -3>_-2 d)5 < 6 e) 6 < 6 Solution Inequality Meaning a) -7 < -2 "-7 is less than -2" is true because -7 is to the left of -2. b) 4 > -1 "4 is greater than -1" is true because 4 is to the right of -1. c) -3 >_ -2 "-3 is greater than or equal to -2" is false because -3 is to the left of -2. d) 5 <_ 6 "5 is less than or equal to 6" is true because 5 < 6 is true. e) 6 < 6 "6 is less than or equal to 6" is true because 6 = 6 is true. Addition, Subtraction, and Opposites We are now ready to review the addition of real numbers. 1.2 OPERATIONS AND PROPERTIES OF REAL NUMBERS 13 Addition of Two Real Numbers 1. Positive numbers: Add the numbers. The result is positive. 2. Negative numbers: Add absolute values. Make the answer negative. 3. A negative and a positive number: If the numbers have the same absolute value, the answer is 0. Otherwise, subtract the smaller absolute value from the larger one: a) If the positive number is further from 0, make the answer positive. b) If the negative number is further from 0, make the answer negative. 4. One number is zero: The sum is the other number. 3 1 E x a m p 1 e 3 Add: (a) -9 + (-5); (b) -3.2 + 9.7; (c) -4 + 3. Solution a) -9 + (-5) We add the absolute values, getting 14. The answer is negative, -14. b) -3.2 + 9.7 The absolute values are 3.2 and 9.7. Subtract 3.2 from 9.7 to get 6.5. The positive number is further ' from 0, so the answer is positive, 6.5. - 3 + 1 = - 9 + 4 4 3 12 12 9 4 5 The absolute values are l2 and 12. Subtract to get 12. The negative number is further from 0, so the an-swer is negative, -5 12 When numbers like 7 and -7 are added, the result is 0. Such numbers are called opposites, or additive inverses, of one another. The Law of Opposites For any two numbers a and -a, a+(-a)=o. (When opposites are added, their sum is 0.) Example 4 4 Find the opposite: (a) -17.5; (b) 5; (c) 0. Solution a) The opposite of -17.5 is 17.5 because -17.5 + 17.5 = 0. 4 4 4 4 b) The opposite of 5 is -5 because 5 + (-5) = 0. c) The opposite of 0 is 0 because 0 + 0 = 0. 14 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING To name the opposite, we use the symbol"-" and read the symbolism -a as "the opposite of a." : Caution! -a does not necessarily denote a negative number. In par-ticular, when a is negative, -a. is positive. Example 5 Find -x for the following: (a) x = -2; (b) x = 3 4. technology connection Solution a) If x = -2, then -x = -(-2) = 2. The opposite of -Z is 2. b) If x = 4 , then -x = - 4 . The opposite of 4 is - 4. Technology Connections high-light situations in which calcula-tors (primarily graphing calculators) or computers can be used to enrich the learning experience. Most Technology Connections present informa-tion in a generic form-consult an outside reference for specific keystrokes. (Henceforth in the text we will refer to all graphing utilities as graphers.) Graphers have different keys for subtracting and writing negatives. The key labeled (-) is used to create a negative sign, whereas - is used for subtraction. 1. Use a grapher to check Example 6. Using the notation of opposites, we can formally define absolute value. Absolute Value |x| = x if x >_ 0, -x ifx<0 (When x is nonnegative, the absolute value of x is x. When x is negative, the absolute value of x is the opposite of x. |x| is never negative.) A negative number is said to have a negative "sign" and a positive number a positive "sign." To subtract, we can add an opposite. we sometimes say that we "change the sign of the number being subtracted and then add." Example 6 Subtract: (a) 5 - 9; (b) -1.2 - (-3.7); (c) -4 - 2 . 5 3 Solution a) 5-9=5+(-9) Change the sign and add. = -4 b) -1.2 - (-3.7) = -1.2 + 3.7 Instead of subtracting -3.7, we add 3.7. =2.5 c)- 4 - 2 = - 4 + (-2) 5 3 5 3 =-12 + (-10) Finding a common denominator 15 15 =-22 15 14page OPERATIONS AND PROPERTIES OF REAL NUMBERS Multiplication, Division, and Reciprocals Multiplication of real numbers can be regarded as repeated addition or as repeated subtraction that begins at 0. For example, 3 . (_2) = 0 + (-2) + (-2) + (_2) - _6 and (-2)(-5)=0-(-5)-(-5)=0+5+5=10. Multiplication of Two Real Numbers 1. To multiply two numbers with unlike signs, multiply their absolute values. The answer is negative. 2. To multiply two numbers having the same sign, multiply their absolute values. The answer is positive. we have (-4)9 = -36 and _2 _3 =2 3 7 7 To divide, recall that the quotient a = b (also denoted a/b) is that number c for which c ' b = a. For example, 10 divide (-2) _ -5 since (-5) (-2) = 10; (-12) divide 3 = -4 since (-4)3 = -12; and -18 divide (-6) = 3 since 3(-6) _ -18. The rules for division are just like those for multiplication. Division of Two Rea! Numbers 1. To divide two numbers with unlike signs, divide their absolute values. The answer is negative. 2. To divide two numbers having the same sign, divide their absolute values. The answer is positive. we have -45 20 -15=3 and -4=-5. Note that since -8 8 _8 2=-2 2 =-4 we have the following generalization. The 5ign of a Fraction For any number a and any nonzero number b, - a = a = -a b -b b 16 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Recall that _a __a _1 _ _1 b l~b-a~b~ That is, if we prefer, we can multiply by 1/b rather than divide by b. Provided that b is not 0, the numbers b and 1/b are called reciprocals, or multiplicative inverses, of each other. The Law of Reciprocals For any two numbers a and 1/a (a ~ 0), 1 a.-=1. a (When reciprocals are multiplied, their product is 1.) 2 3 E x a m p I e 7 Find the reciprocal: (a) 8; (b) -4; (c) -8. Solution 7 8 7 8 a) The reciprocal of 8 is 7 because 8. 7 = 1. b) The reciprocal of -3 is - 4 4 3 underneath 3 4. c) The reciprocal of -8 is 1 or -1 -8 8 To divide, we can multiply by a reciprocal. We sometimes say that we "invert and multiply." E x a m p 1 e 8 Divide: (a) -1 divide into 3 (b) -6 divide (-10). 4 5 underneath 1 and 3. 7 under 6. Solution a) 1 3 1 5 -4 divide 5 = -4 . 3 5 Inverting 3 and changing division to multiplication. = -12 b) -6 divide -10 = -6 . - 1 = 6 or 3 7 7 10 70 35 we have never divided by 0 or, equivalently, had a denominator of 0. There is a reason for this. Suppose 5 were divided by 0. The answer would have to be a number that, when multiplied by 0, gave 5. But any number times 0 is 0. we cannot divide 5 or any other nonzero number by 0. What if we divide 0 by 0? In this case, our solution would need to be some number that, when multiplied by 0, gave 0. But then any number would work as a solution to 0 = 0. This could lead to contradictions so we agree to exclude division of 0 by 0 also. 'I .2 OPERATIONS AND PROPERTIES OF REAL NUMBERS 17 Division by Zero We never divide by 0. If asked to divide a nonzero number by 0, we say that the answer is undefined. If asked to divide 0 by 0, we say that the answer is indeterminate. The rules for order of operations discussed in Section 1.1 apply to all real numbers, regardless of their signs. a fraction bar, absolute-value symbol, or radical sign (~). Take the time to include all the steps when working your homework problems. Doing so will help you organize your thinking and avoid computational errors. It will also give you complete, step-by-step solutions of the exercises that will make sense whenstudyingforquizzes The COmmutative, ASSOCIatIVe, and andtests. DIStYIFJUtIVe LdWS Study Tip E x a m p 1 e 9 " Simplify: 7 - 5 2 + 6 divide by 2(-5)2. Solution 7 - 5 2 + 6 divide by 2(-5)2 = 7 - 25 + 6 divide 2 . 25 Simplifying 5Z and (-5) E x a m p 1 e 1 0 ' Calculate: 12|7 - 9| + 4 - 5 , (-3)4 + 2 3 = 7 - 25 + 3 divide 25 Dividing = 7 - 25 + 75 Multiplying _ -18 +. 75 Subtracting = 57 Adding Besides parentheses, brackets, and braces, groupings may be indicated by Solution We simplify the numerator and the denominator and divide the 12~7-9~+4~5_12~-2~+20 (-3)4 + 23 81 + 8 12(2) + 20 = 89 _44 = Multiplying and adding 89~ When a pair of real numbers are added or multiplied, the order in which the numbers are written does not affect the result. 1H CHAPTER 1 ALGEBRAAND PROBLEM SOLVING The Commutative Laws For any real numbers a and b,  a+b=b+a; a~b=b~a. (for Addition) (for Multiplication) The commutative laws provide one way of writing equivalent expressions. Equivalent Expressions  Two expressions that have the same value for all possible replace-ments are called equivalent expressions. Much of this text is devoted to finding equivalent expressions. E x a m p 1 e 1 1 Use a commutative law to write an expression equivalent to 7x + 9. Solution Using the commutative law of addition, we have 7x+9=9+7x. We can also use the commutative law of multiplication to write 7x +9= x7+9. The expressions 7x + 9, 9 + 7x, and x7 + 9 are all equivalent. They name the same number for any replacement of x. The associative laws also enable us to form equivalent expressions. The Associative Laws For any real numbers a, b, and c, a+(b+c)=(a+b)+c; a~(b*c)=(a~b)~c. (for Addition) (for Multiplication) E x a m p 1 e 1 2 ' Write an expression equivalent to (3x + 7y) + 9z, using the associative law of addition. Solution We have (3x + 7y) + 9z = 3x + (7y + 9z). The expressions (3x + 7y) + 9z and 3x + (7y + 9z) are equivalent. They name the same number for any replacements of x, y, and z. 1.2 OPERATIONS AND PROPERTIES OF REAL NUMBERS 19 2O CHAPTER 1 ALGEBRA AND PROBLEM SOLVING FOR EXTRA HELP Exercise Set '/,Z ' ~~ Cid MI Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MvMath! ab.com Find each absolute value. 1. |-9| 2. |-7| 3. |6| 4. |47| 5. |-6.2| 6. |-7.9| 7. |0| 8. |3 3| 4 9. |1 7| 8 10. |7.24| 11. |-4.21| 12. |-5.309| 13. -6 < -2 14. -1 < -5 15. -9 >1 16. 7 > -2 17. 3 > -5 18. 9 < 9 19. -8 < -3 20. 7 > -8 21. -4 > -4 22. 2 < 2 23. -5 < -5 24. -2 > -12 25. 4 + 7 26. 8+3 27. -4+ (-7) 28. -8 + (-3) 29. -3.9 + 2.7 30. -1.9 + 7.3 31. 2 3 7 + (-5) 32. 3 2 8 + (-5) 33. -4.9 + (-3.6) 34. -2.1 + (-7.5) 35. 1 2 -9 + 3 36. 1 4 -2 + 5 37. 0 + (-4.5) 38. -3.19 + 0 39. -7.24 + 7.24 40. -9.46 + 9.46 41. 15.9 + (-22.3) 42. 21.7 + (-28.3) 3 43. 3.14 44. 5.43 , 45. -4 1 3 Divide. 46. 2 3 5 47. 0 .~ 48. - 2 3 4 Find -x for each of the following. 49. x = 7 50. x = 3 51. x = -2.7 52. x = -1.9 53. x = 1.79 54. x = 3.14 55. x=0 56. x=-1 Subtract. 57.9-2 58.10-3 59.2-9 60.3-10 61. -6 - (-10) 62. -3 - (-9) 63. -5 -14 64. -7 -8 65. 2.7 - 5.8 66. 3.7 - 4.2 67. 3 1 - 5 - 2 68. 2 1 - 3 - 5 69. -3.9 - (-6.8) 70. -5.4 - (-4.3) 71. 0-(-7.9) 72. 0-5.3 Multiply. 73. (-5)6 74. (-4)7 75. (-3) (-8) 76. (-7) (-8) 77. (4.2) (-5) 78. (3.5) (-8) Add. 79. 3 ( -1) 7 80. -1 - 2 5 81. (-17.45) -0 82. 15.2 X 0 83. (-3.2) X (-1.7) 84. (1.9) x (4.3) Divide. 85.-10 -2 86. -15 -3 87. -100 20 88. -50 -5 89. 73 -1 90. -62 1 91. 0 -7 92. 0 -11 Find the reciprocal, or multiplicative inverse. 93. 4 94. 3 95. -9 96. -5 97. 2 3 98. 4 7 Write the meaning of each inequality, and determine whether it is a true statement. 7 Find the opposite, or additive inverse. 99. _3 11 100. 7 -3 Divide 101. 2 divide by 4 3 5 102. 2 divide by 6 7 5 Find -x for each of the following. 103. 3 1 -5 divide by 2 104. (- 4) 1 7 divide by 3 105. 2 (-9) divide by (-8) 106. (-2) divide by (-6) 11 107. -12 divide by -12 7 (- 7 ) 108. 2 (-7) divide by (-1) Calculate using the rules for order of operations. 109. 7-(8-3-2 3) 110. 19-(4+2- 3 2) 1.2 OPERATIONS AND PROPERTIES OF REAL NUMBERS page 21 111. 5 * 2- 4 2is upper right corner. 27 - 2 4 112. 7 * 3 - 5 2 9 + 4 * 2 113. 3 4 - (5 - 3)4 8 - 2 3 114. 4 3 - (7 - 4)2 3 2 - 7 115. (2-3)3-5|2-4| 7 - 2 * 5 2 116. 8 divide by 4 * 6|4 2 - 5 2| 9 - 4 + 11 - 4 2 117. |2 2 - 7|3 + 1 118. |-2-3|*4 2 -1 119. 28 - (-5)2 + 15 divide (-3) * 2 120. 43 -(-9+2)2 + 18 divide 6 * (-2) 121. 12 -V11 - (3 + 4) divide [-5 - (-6)]2 122. 15 - 1 +V5 2 - (3 + 1)2(-1) Evaluate. pears once as a verb and once as a noun. Write an equivalent expression using an associative law. ~ Insert one pair of parentheses to convert each false state- ment into a true statement. SYNTHESIS Answers may vary. 123. 4a + 7b 124. 6 + xy 125. (7x)y 126. -9(ab) Write an equivalent expression using a commutative law. 127. (3x)y 128. -7(ab) 129. x+(2y+5) 130. (3y+4)+10 Write an equivalent expression using the distributive law. , 131. 3(a + 7) 132. 8(x + 1) 133. 4(x - y) 134. 9(a - b) 135. -5(2a + 3b) 136. -2(3c + 5d) 137. 9a(b - c + d) 138. 5x( y - z + w) 139. 5x + 25 Find an equivalent expression by factoring. 140. 7a + 7b 141. 3p - 9 142. 15x - 3 143. 7x - 21y + 14z 144. 6y - 9x - 3w 145. 255 - 34b 146. 132a + 33 Why or why not? 147. Describe in your own words a method for determining the sign of the sum of a positive number and a negative number. 148. What is the difference between the associative law of multiplication and the distributive law? Skill Maintenance review skills previously studied in the text. Usually these exercises provide preparation for the next section of the text. The answers to all Skill Mainte nance exercises appear at the back of the book, along with a section number indicating where that type of problem first appeared. SKILL MAINTENANCE Evaluate 149. 2(x + 5) and 2x + 10, for x = 3 150.2a-3 and a -3+a,for a = 7 151. Explain in your own words why 7/0 is undefined. 152. Write a sentence in which the word "factor" ap- pears once as a verb and once as a noun. Insert one pair of parentheses to convert each false state- ment into a true statement. 153. 8-5 3+9=36 154. 2*7+3 2 * 5=104 155. 5*2 3 divide 3 - 4 4 =40 156. 2 - 7 * 2 2 + 9= -11 157. Find the greatest value of a for which |a| > 6.2 and a < 0 158. Use the commutative, associative, and distribu-tive laws to show that 5(a + bc) is equivalent to c(b5) + a5. Use only one law in each step of your work. 159. Are subtraction and division commutative? Why or why not? 160. Are subtraction and multiplication associative? ZZ CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Solving Equations Equivalent Equations • The Addition and Multiplication Principles • Combining Like Terms • Types of Equations Solving equations is an essential part of problem solving in algebra. In this section, we review and practice solving simple equations. Equivalent Equations In Section 1.1, we saw that the solution of 41 + x = 65 is 24. That is, when x is replaced with 24, the equation 41 + x = 65 is a true statement. Although this solution may seem obvious, it is important to know how to find such a solution using the principles of algebra. These principles can produce equivalent equa-tions from wtlich solutions are easily found. Equivalent Equations  Two equations are equivalent if they have the same solution(s). . . .. . Example 1 Determine whether 4x = 12 and lOx = 30 are equivalent equations. Solution The equation 4x = 12 is true only when x is 3. Similarly, lOx = 30 is true only when x is 3. Since both equations have the same solution, they are equivalent. . E x a m p 1 e 2 "' Determine whether x + 4 = 7 and x = 3 are equivalent equations. Sol ution Each equation has only one solution, the number 3. Thus the equa-tions are equivalent. Example 3 Determine whether 3x = 4x and 3/x = 4/x are equivalent equations. Solution Note that 0 is a solution of 3x = 4x. Since neither 3/x nor 4/x is defmed for x = 0, the equations 3x = 4x and 3/x = 4/x are not equivalent. The Addition and Multiplication Principles Suppose that a and b represent the same number and that some number c is added io a. If c is also added to b, we will get two equal sums, since a and b are 1.3 SOLVING EQUATIONS 23 the same number. The same is true if we multiply both a and b by c. In this manner, we can produce equivalent equations. The Addition and Multiplication Principles for Equations For any real numbers a, b, and c: a) a = b is equivalent to a + c = b + c; b) a = b is equivalent to a * c = b * c, provided c 0 0. Solution y - 4.7 = 13.9 y - 4.7 + 4.7 = 13.9 + 4.7 Using the addition principle; adding 4.7 y + 0 = 13.9 +4.7 The law of opposites y = 18.6 18.6 - 4.7 ? 13.9 Substituting 18.6 for y 13.9 ~ 13.9 TRUE In Example 4, why did we add 4.7 to both sides? Because 4.7 is the opposite of -4.7 and we wanted y alone on one side of the equation. Adding 4.7 gave us y + 0, or just y, on the left side. This led to the equivalent equation, y = 18.6, from which the solution, 18.6, is immediately apparent. Example 5 Solve: 2 9 5x =10. Solution We have 2 x = 9 5 10 5 * 2 x = 5 * 9 2 5 2 10 Using the multiplication principle, we multiply bY5 , - 1 x = 45 the reciprocal of 5. 20 The law of reciprocals x = 3 Simplifying 4 The check is left to the student. The solution is 4. In Example 5, why did we multiply by 5 ? Because 5 is the reciprocalof2 2 2 2 5 and we wanted x alone on one side of the equation. When we multiplied by 5, we got Ix, or just x, on the left side. 2 This led to the equivalent equation x = 9 9 4, from which the solution, 4, is clear 24 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING There is no need for a subtraction or division principle because subtraction can be regarded as adding opposites and division can be regarded as mul-tiplying by reciprocals. Combining Like Terms In an expression like 8a5 + 17 + 4/b + (-6a3b), the parts that are separated by addition signs are called terms. A term is a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. 8a5, 17, 4/b, and -6a3b are terms in 8a5 + 17 + 4/b + (-6a3b). When terms have variable factors that are exactly the same, we refer to those terms as like, or similar, terms. 3xZy and -7xZy are similar terms, but 3xZy and 4xyZ are not. We can often simplify expressions by combining, or collecting, like terms. Example Combine like terms: 3a + 5a2 + 4a + a2. 3a + 5a2 + 4a + a2 = 3a + 4a + 5a2 + a2 Using the commutative law =(3 + 4)a + (5 + 1)a2 Using the distributive law. Note that a2 = l a2. = 7a + 6a2 Sometimes we must use the distributive law to remove grouping symbols before combining like terms. Remember to remove the innermost grouping symbols first. E x a m p 1 e 7 Simplify: 3x + 2[4 + 5(x + 2y)]. Solution 3x + 2[4 + 5(x + 2y)] = 3x + 2[4 + 5x + l0y] Using the distributive law = 3x + 8 + lOx + 20y Using the distribu-tive law = 13x + 8 + 20y Combining like terms The product of a number and -1 is its opposite, or additive inverse. For example, -1 * 8 = -8 (the opposite of 8). we have -8 = -1 * 8, and in general, -x = -1 * x. We can use this fact along with the distributive law when parentheses are preceded by a negative sign or subtraction. 1.3 SOLVING EQUATIONS 25 Example 8 Simplify -(a - b), using multiplication by -1. =(a - b) _ -1 * (a - b) Replacing - with multiplication by -1 = -1 * a - (-1) * b Using the distributive law = -a - (-b) Replacing -1 * a with -a and (-1) * b with -b = -a + b, or b - a. Try to go directly to this step. The expressions -(a - b) and b - a are equivalent. They name the same num-ber for all replacements of a and b. Example 8 illustrates a useful shortcut worth remembering: The opposite of a - b is -a + b, or b - a. Example 9 Simplify: 9x - 5y - (5x + y - 7). Solution 9x-5y-(5x+y-7)=9x-5y-5x-y + 7 Using the distributive law = 4x - 6y + 7 Combining like terms CONNECTING THE CONCEPTS It is important to distinguish between forming equivalent expressions and writing equivalent equations. In Examples 4 and 5, we used the addition and multiplication principles to write a series of equiv-alent equations that led to an equation for which the solution is clear. Because the equations were equivalent, the solution of the last equation was also a solution of the original equation. In Examples 6-9, we used the commutative, associative, and distributive laws to write equivalent expressions that take on the same value when the variables are replaced with numbers. This is how we "simplify" an expression. Equivalent Equations Equivalent Expressions y-4.7=13.9 3x+2[4+5(x+2y)] y-4.7+4.7=13.9 + 4.7 =3x+2[4+5x+10y] y+0=13.9+4.7 =3x+8+ 10x+20y y=18.6 =13x+8+20y Often, as in Example 10, we merge these ideas by forming an equivalent equation by replacing part of an equation with an equivalent expression. 26 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Example 10 Solve: 5x - 2(x - 5) = 7x - 2. Solution 5x-2(x-5)=7x-2 5x-2x+10=7x-2 3x+ 10=7x-2 3x + 10-3x=7x-2-3x Using the distributive law Combining like terms Using the addition principle; adding -3x, the opposite of 3x, to both sides Combining like terms Using the addition principle 12 = 4x Simplifying 10=4x-2 10+2=4x-2+2 12 = 4x 1 * 12 = 1 * 4x 4 4 3 = x Using the multiplication principle; multiplying both sides by 4, the reciprocal of 4 The law of reciprocals; simplifying 5x -2(x-5) = 7x-2 5 * 3 - 2 (3 - 5) ? 7 * 3 - 2 15 -2 (-2) 21 - 2 15 + 4 19 19 19 TRUE The solution is 3. Types of Equations In Examples 4, 5, and 10, we solved linear equations. A linear equation in one variable-say, x-is an equation equivalent to one of the form ax = b with a and b constants and a =A 0. Don't rush to solve equations in your head. Work neatly, keeping in mind that the number of steps in a solution is less important than producing a sim-pler, yet equivalent, equation in each step. Every equation falls into one of three categories. An identity is an equation that is true for all replacements that can be used on both sides of the equation (for example, x + 3 = 2 + x + 1). A contradiction is an equation, like n + 5 = n + 7, that is never true. A conditional equation, like 2x + 5 = 17, is sometimes true and sometimes false, depending on what the replacement of x is. Most of the equations examined in this text are conditional. Example 11 Solve each of the following equations and classify the equation as an identity, a contradiction, or a conditional equation. a) 2x+7=7(x+l)-5x b) 3x-5=3(x-2)+4 c} 3-8x=5-7x 13 SOLVING EQUATIONS 27 2x + 7 = 7(x + 1) - 5x 2x + 7 = 7x + 7 - 5x 2x + 7 = 2x + 7 Using the distributive law The equation 2x + 7 = 2x + 7 is true regardless of what x is replaced with, so all real numbers are solutions. Note that 2x + 7 = 2x + 7 is equivalent to 2x = 2x, 7 = 7, or 0 = 0. All real numbers are solutions and the equation is an identity. b) 3x-5=3(x-2)+4 3x - 5 = 3x - 6 + 4 Using the distributive law 3x - 5 = 3x - 2 Combining like terms -3x + 3x - 5 = -3x + 3x - 2 Using the addition principle -5 = -2 Since our original equation is equivalent to -5 = -2, which is false for any choice of x, there is no solution to this problem. There is no choice of x that will solve the original equation. The equation is a contradiction. c) 3-8x=5-7x 3 - 8x + 7x = 5 - 7x + 7x Using the addition principle 3 - x = 5 Simplifying -3+3-x=-3+5 Using the addition principle -x = 2 Simplifying x = 2, or -2 Dividing both sides by -1 or multiplying -1 both sides by 1 -1 There is one solution, -2. For other choices of x, the equation is false. This equation is conditional since it can be true or false, depending on the replacement for x. We will sometimes refer to the set of solutions, or solution set, of a partic-ular equation. The solution set for Example 11(c) is {-2}. The solution set for Example 11(a) is simply R, the set of all real numbers, and the solution set for Example 11(b) is the empty set, denoted 0 or { }. As its name suggests, the empty set is the set containing no elements. 28 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING FOR EXTRA HELP Exercise Set 1.3 Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 1 Chapter 1 page 28 Exercise set 1.3 Determine whether the two equations in each pair are 1. 3x=15 and 5x=25 2. 6x = 24 and 15x = 60 3. t+5=11 and 3t=18 4. t-3=7 and 3t=24 5. 12 - x = 3 and 2x = 20 2 x 10=20 6. 3x-4=8 and 3x=12 7. 5x = 2x and 4 = 3 x 8. 6 = 2x and 5 = 2 3-x Solve. Be sure to check. 9. x-2.9=13.4 10. y+4.3=11.2 11. 8t = 72 12. 9t = 63 13. 4x - 12 = 60 14. 4x - 6 = 70 15. 3n + 5 = 29 16. 7t + 11 = 74 17. 2y - 11 = 37 18. 3x - 13 = 29 19. 3x + 7x 20. 9x + 3x 21. 7rt - 9rt 22. 3ab + 7ab 23. 9t2 + t2 24. 7a2 + a2 25. 12a - a 26. 15x - x 27. n-8n 28. x-6x 29. 5x - 3x + 8x 30. 3x - 11x + 2x 31. 4x-2x2+3x 32. 9a-5a2+4a 33. 6a+7a2-a+4a2 34. 9x + 2x3 + 5x - 6x2 35. 4x-7+ 18x + 25 36. 13p+5-4p+7 37. -7t2+3t+5t3-t3+2t2-t 38. -9n + 8n2 + n3 - 2n2 - 3n + 4n3 equivalent. 39. 7a - (2a + 5) 40. x-(5x+9) 2. 6x = 24 and 15x = 60 41. m-(3m-1) 42. 5a-(4a-3) 43. 3d-7-(5-2d) 5. 12-x=3 and 2x=20 44. 8x - 9 - (7 - 5x) 45. -2(x+3)-5(x-4) 46. -9(y + 7) - 6(Y - 3) 47. 4x - 7(2x - 3) 48. 9y-4(5y-6) 49. 9a - [7 - 5(7a - 3)] 50. 12b - [9 - 7(5b - 6)] 51. 5[-2a + 3[4 - 2(3a + 5)]} 52. 7[-7x + 8[5 - 3(4x + 6)]} 53. 2y + (7[3(2y - 5) - (8y + 7)] + 9) 17. 2y-11=37 18.3x-13=29 54. 7b - {6[4(3b - 7) - (9b + 10)] + 11} Simplify by combining like terms. Use the distributive Solve. Be sure to check. law as needed. 55. 6x + 2x = 56 56. 3x + 7x = 120 57. 9y - 7y = 42 58. 8t - 3t = 65 59. 5t - 13t = -32 60. -9y - 5y = 28 61. 2(x + 6) = 8x 62. 3(y + 5) = 8y 63. 70 = 10(3t - 2) 64. 27 = 9(5y - 2) 65. 180(n-2)=900 66. 210(x-3)=840 67. 5y - (2y - 10) = 25 68. 8x - (3x - 5) = 40 69. 7y-1=23-5y 70. 14t+ 20 =8t-22 71. 1 3 4 5+ 10 x=5 72. 5 1 -2x+2=-18 73. 0.9y - 0.7 = 4.2 74. 0.8t - 0.3t = 6.5 75. 7r-2+5r=6r+6-4r 76. 9m-15-2m=6m-1-m 77. 1 1 4(16y+8)-17=-2(8y-16) 1.4 INTRODUCTION TO PROBLEM SOLVING 29 78. 1 1 3 (6t + 48) - 20 = - 4 (12t - 72) 79. 5 + 2(x - 3) = 2[5 - 4(x + 2)] Find each solution set. Then classify each equation as a conditional equation, an identity, or a contradiction. 80. 3[2 - 4(x - 1)] = 3 - 4(x + 2) 81. 5x + 7 - 3x = 2x laws can be used to rewrite 3x + 6y + 4x + 2y as 82.3t+5+t=5+4t 7x+8y. 83. 1 + 9x = 3(4x + 1) - 2 Solve and check. The symbol `-', indicates an exercise de- 84. 4 + 7x = 7(x + 1) signed to be solved with a calculator. ® Pha~~ 85. -9t + 2 = -9t - 7(6 divide 2(49) + 8) ® gg, -0,00458y + 1.7787 = 13.002y - 1.005 86. -9t + 2 = 2 - 9t - 5(8 divide 4(1 + 3 4 is upper corner)) 87. 2{9 - 3[-2x - 4]} = 12x + 42 88. 3{7 - 2[7x - 4]}= -40x + 45 89. As the first step in solving 2x+5=-3 90. Explain how an identity can be easily altered so that it becomes a contradiction. 91. Nine more than twice a number. Introduction to Problem Solving SYNTHESIS 92. Forty-two percent of half of a number 93. Explain the difference between equivalent expres- sions and equivalent equations 94. Explain how the distributive and commutative laws can be used to rewrite 3x + 6y + 4x + 2y as 7x + 8y. 95. 4.23x - 17.898 = -1.65x - 42.454 96. -0.00458y + 1.7787 = 13.002y - 1.005 97. 4x - {3x -[2x(5x -(7x - 1)}] = 4x + 7 98.3x-{5x-[7x-(4x-(3x+1))]}=3x+5 99. 17 - 3{5 + 2[x - 2]} + 4{x-3(x+7)} =9{x + 3[2 + 3[4-x)]} 100. 23 - 2{4 +3[x - 1]}+ 5 {x - 2{x + 3)} =7{x-2[5- (2x + 3)]} 101. Create an equation for which it is preferable to use the multiplication principle before using the addition principle. Explain why it is best to solve the equation in this manner. SKILL MAINTENANCE Translate to an algebraic expression. that it becomes a The Five-Step Strategy • Problem Solving We now begin to study and practice the "art" of problem solving. Although we are interested mainly in using algebra to solve problems, much of what we say here applies to solving all kinds of problems. What do we mean by a problem? Perhaps you've already used algebra to solve some "real-world" problems. What procedure did you use? Is there an approach that can be used to solve problems of a more general nature? These are some questions that we will answer in this section. In this text, we do not restrict the use of the word "problem" to computa-tional situations involving arithmetic or algebra, such as 589 + 437 = a or 3O CHAPTER 1 ALGEBRA AND PROBLEM SOLVING 3x + 5x = 9. We mean instead some question to which we wish to find an an-swer. Perhaps this can best be illustrated with some sample problems: 1. Can I afford to rent a bigger apartment? 2. If I exercise twice a week and eat 3000 calories a day, will I lose weight? 3. Do I have enough time to take 4 courses while working 20 hours a week? 4. My fishing boat travels 12 km/h in still water. How long will it take me to cruise 25 km upstream if the river's current is 3 km/h? Although these problems are all different, there is a general strategy that can be applied to all of them. The Five-Step Strategy Since you have already studied some algebra, you have had some experience with problem solving. The following steps constitute a strategy that you may al-ready have used and a good strategy for problem solving in general. Five 5teps for Problem Solving with Algebra 1. Familiarize yourself with the problem. 2. Translate to mathematical language. 3. Carry out some mathematical manipulation. 4. Check your possible answer in the original problem. 5. State the answer clearly. Of the five steps, probably the most important is the first: becoming famil-iar with the problem situation. Here are some ways in which this can be done. The First Step in Problem Solving with Algebra To familiarize yourself with the problem: 1. If the problem is written, read it carefully. 2. Reread the problem, perhaps aloud. Verbalize the problem to yourself. 3. List the information given and restate the question being asked. Select a variable or variables to represent any unknown(s) and clearly state what each variable represents. Be descriptive! For example, let t = the flight time, in hours; let p = Paul's weight, in kilograms; and so on. 4. Find additional information. Look up formulas or definitions with which you are not familiar. Geometric formulas appear on the inside back cover of this text; important words appear in the index. Consult an expert in the field or a reference librarian. 5. Create a table, using variables, in which both known and un-known information is listed. Look for possible patterns. 6. Make and label a drawing. 7. Estimate or guess an answer and check to see whether it is correct. page 31 Example 1 How might you familiarize yourself with the situation of Problem 1: "Can I afford to rent a bigger apartment?" Solution Clearly more information is needed to solve this problem. You might: a) Estimate the rent of some apartments in which you are interested. b) Examine what your savings are and how your income is budgeted. c) Determine how much rent you can afford. When enough information is known, it might be wise to make a chart or table to help you reach an answer. Example 2 How might you familiarize yourself with Problem 4: "How long will it take the boat to cruise 25 km upstream?" Solution First read the question very carefully. This may even involve speak-ing aloud. You may need to reread the problem several times to fully under-stand what information is given and what information is required. A sketch or table is often helpful. Distance to be Traveled 25 km Speed of Boat in Still Water 12 km/h Speed of Current 3 km/h Speed of Boat Upstream ? Time Required ? To continue the familiarization process, we should determine, possibly with the aid of outside references, what relationships exist among the various quantities in the problem. With some effort it can be learned that the current's speed should be subtracted from the boat's speed in still water to determine the boat's speed going upstream. We also need to either find or recall an extremely important formula: Distance = Speed x Time. We rewrite part of the table, letting t = the number of hours required for the boat to cruise 25 km upstream. 1.4 INTRODUCTION TO PROBLEM SOLVING 31 Distance to be Traveled 25 km. Speed of Boat Upstream 12 - 3 = 9 km/h Time Required , 32 CHAPTER'I ALGEBRAAND PROBLEM SOLVING At this point we might try a guess. Suppose the boat traveled upstream for 2 hr. The boat would have then traveled 9 km x 2 hr = 18 km. Note that km * hr = km. hr hr ^ ^ ^ Speed x Time x =Distance Speed x Time = Distance Since 18 =/ 25, our guess is wrong. Still, examining how we checked our guess sheds light on how to translate the problem to an equation. Note that a better guess, when multiplied by 9, would yield a number closer to 25. The second step in problem solving is to translate the situation to mathe-matical language. In algebra, this often means forming an equation. The Second Step in Problem Solving with Algebra Translate the problem to mathematical language. In some cases, translation can be done by writing an algebraic expression, but most problems in this text are best solved by translating to an equation. In the third step of our process, we work with the results of the first two steps. Often this will require us to use the algebra that we have studied. The Third Step in Problem Solving with Algebra Carry out some mathematical manipulation. If you have translated to an equation, this means to solve the equation. To complete the problem-solving process, we should always check our so-lution and then state the solution in a clear and precise manner. To check, we make sure that our answer is reasonable and that all the conditions of the origi nal problem have been satisfied. If our answer checks, we write a complete English sentence stating the solution. The five steps are listed again below Try to apply them regularly in your work. Five Steps for Problem Solving with Algebra 1. Familiarize yourself with the problem. 2. Translate to mathematical language. 3. Carry out some mathematical manipulation. 4. Check your possible answer in the original problem. 5. State the answer clearly. page 33 1.4 INTRODUCTION TO PROBLEM Solving Problem Solving At this point, our study of algebra has just begun. we have few algebraic tools with which to work problems. As the number of tools in our algebraic "toolbox' increases, so will the difficulty of the problems we can solve. For now our problems may seem simple; however, to gain practice with the problem-solving process, you should try to use all five steps. Later some steps may be shortened or combined. E X a m p l e 3 Purchasing. 'Elka pays $1187.20 for a computer. If the price paid includes a 6% sales tax, what is the price of the computer itself? Solution 1. Familiarize. First, we familiarize ourselves with the problem. Note that tax is calculated from, and then added to, the computer's price. Let's guess that the computer's price is $1000. To check the guess, we calculate the amount of tax, (0.06) ($1000) = $60, and add it to $1000: $looo + (0.06) ($l000) = $l000 + $60 = $1060. $1060 =/ $1187.20 Our guess was too low, but the manner in which we checked the guess will guide us in the next step. We let C = the computer's price, in dollars. 2. Translate. Our guess leads us to the following translation: Rewording: The 6% sales the price with computer's price plus tax is sales tax. Translating: C + (0.06)C = $1187.20 3. . Carry out. Next, we carry out some mathematical manipulation: C + (0.06)C = 1187.20 1.06C = 1187.20 Combining like terms 1 1 1 06 * 1.06C = 1.06 ~ 1187.20 Using the multiplication principle C = 1120. 4. Check. To check the answer in the original problem, note that the tax on a computer costing $1120 would be (0.06)($1120)= $67.20. When this is added to $1120, we have $1120 + $67.20, or $1187.20. $1120 checks in the original problem. 5. State. We clearly state the answer: The computer itself costs $1120. 34 ChAPTER 1 ALGEBRA AND PROBLEM SOLVING Example 4 Home maintenante. In an effort to make their home more energy-efficient, Alma and Drew purchased 200 in. of 3M Press-In-Place window glazing. This will be just enough to outline their two square skylights. If the length of the sides of the larger skylight is 1 1 2 times the length of the sides of the smaller one, how should the glazing be cut? 1. Familiarize. Note that the perimeter of (distance around) each square is four times the length of a side. Furthermore, if s is used to represent the length of a side of the smaller square, then (1 1s will represent the length of 2 a side of the larger square. We make a drawing and note that the two perimeters must add up to 200 in. Perimeter of a square = 4 * length of a side. 2. Translate. Rewording the problem can help us translate: Rewording: The perimeter the perimeter of one square plus of the other is 200 in. Translating: 4s + 4 (1 1s) = 200 2 3. Carry out. We solve the equation: 4s + 4(1 1s) = 200 Simplifying 2 4s + 6s =200 10s = 200 Combining like terms s = 1 * 200 Multiplying both sides by io 10 s = 20. Simplifying 4. Check. If 20 is the length of the smaller side, then (1 1 2 (20) = 30 is the length of the larger side. The two perimeters would then be 4 * 20 in. = 80 in. and 4 * 30 in. = 120 in. Since 80 in. + 120 in. = 200 in., our answer checks. 5. State. The glazing should be cut into two pieces, one 80 in. long and the other 120 in. long. Example 5 We cannot stress too much the importance of labeling the variables in your problem. In Example 4, solving for s was not enough: We needed to find 4s and 4(1 1s) to determine the numbers we were after. 2 Three numbers are such that the second is 6 1ess than three times the first and the third is 2 more than two-thirds the first. The sum of the three numbers is 150. Find the largest of the three numbers. Solution We proceed according to the five-step process. 1. Familiarize. We need to find the largest of three numbers. We list the information given in a table in which x represents the first number. page 35 1.4 INTRODUCTION TO PROBLEM SOLVING First Number x Second Number 6 1ess than 3 times the first Third Number 2 more than 2 the first 3 First + Second + Third = 150 Try to check a guess at this point. We will proceed to the next step. 2. Translate. Because we wish to write an equation in just one variable, we need to express the second and third numbers using x ("in terms of x"). To do so, we expand the table: First Number x x Second Number 6 1ess than 3 times the first 3x - 6 Third Number 2 more than 2 the first 2x + 2 3 3 Study Tip Consider forming a study group with some of your fellow students. Exchange telephone numbers, schedules, and any e-mail addresses so that you can coordinate study time for homework and tests. We know that the sum is 150. Substituting, we obtain an equation: First + second + third = 150. x + (3x - 6) + (2x + 2) = 150 3 3. Carry out. We solve the equation: x + 3x - 6 + 2 x + 2 = 150 Leaving off unnecessary parentheses 3 (4 + 2)x - 4 = 150 Combining like terms 3 14 x - 4 = 150 3 14x = 154 Adding 4 to both sides 3 3 3 x = 14 * 154 Multiplying both sides by 14 x = 33. Remember, x represents the first number. Going back to the table, we can find the other two numbers: Second: 3x - 6 = 3 * 33 - 6 = 93; 2 2 Third: 3x+2=3 33 + 2 = 24. 4. Check. We return to the original problem. There are three numbers: 33, 93, and 24. Is the second number 6 1ess than three times the first? 3 x 33 - 6 = 99 - 6 = 93 The answer is yes. Is the third number 2 more than two-thirds the first? 2 3 x 33 + 2 = 22 + 2 = 24 The answer is yes. 36 CHAPTER 1 ALGEBRAt1t1lQ PROBLEM SOLVING Is the sum of the three numbers 150? 33 + 93 + 24= 150 The answer is yes. The numbers do check. 5. State. The problem asks us to find the largest number, so the answer is: "The largest of the three numbers is 93." Caution! In Example 5, although the equation x = 33 enabled us to find the largest number, 93, the number 33 was not the solution to the problem. By carefully labeling our variable in the first step of problem solving, we may avoid the temptation of thinking that our variable al-ways represents the solution of the problem. Exercise Set 1.4 FOR EXTRA HELP Digital Video Tutor CD 1 InterAct Math Math Tutor Center Videotape 2 MathXL MyMathLab.com For each problem, familiarize yourself with the situa- tion. Then translate to mathematical language. You need not actually solve the problem; just carry out the first two steps of the five-step strategy. You will be asked to complete some of the solutions as Exercises 31-38. 1. The sum of two numbers is 65. One of the numbers is 7 more than the other. What are the numbers? 2. The sum of two numbers is 83. One of the numbers is 11 more than the other. What are the numbers? 3. Boating. The Delta Queen is a paddleboat that tours the Mississippi River near New Orleans, Louisiana. It is not uncommon for the Delta Queen to run 7 mph in still water and for the Mississippi to flow at a rate of 3 mph (Source: Delta Queen in-formation). At these rates, how long will it take the boat to cruise 2 mi upstream? 4. Smimming. Fran swims at a rate of 5 km/h in still water. The Lazy River flows at a rate of 2.3 km/h. How long will it take Fran to swim 1.8 km upstream? 5. Moving Sidewalks. The moving sidewalk in O'Hare Airport is 300 ft long and moves at a rate of 5 ft/sec. If Alida walks at a rate of 4 ft/sec, how long it take her to walk the length of the moving sidewalk? 6. Aviation. A Cessna airplane traveling 390 km/h in still air encounters a 65-km/h headwind. How long will it take the plane to trave1 725 km into the wind? 725 -65 =660 7. Angles in a triangle. The degree measures of the angles in a triangle are three consecutive integers. 1.4 INTRODUCfION TO PROBLEM SOLVING 37 Find the measures of the angles. x + 2 x + 1 x 8. Pricing The Sound Connection prices TDK 100-min blank audiotapes by raising the wholesale price 50% and adding 25 cents. What must a tape's wholesale price be if the tape is to sell for $1.99? 9. Pricing. Becker Lumber gives contractors a 10% discount on all orders. After the discount, a con-tractor's order cost $279. What was the original cost of the order? 10. Pricing Miller Oil offers a 5% discount to cus-tomers who pay promptly for an oil delivery. The Blancos promptly paid $142.50 for their December oil bill. What would the cost have been had they not promptly paid? 11. Cruising Altitude: A Boeing 747 has been instructed to climb from its present altitude of 8000 ft to a cruising altitude of 29,000 ft. If the plane ascends at a rate of 3500 ft/min, how long will it take to reach the cruising altitude? 12. A piece of wire 10 m long is to be cut into two pieces, one of them 2 3 as long as the other. How should the wire be cut? 13. Angles in a triangle. One angle of a triangle is three times as great as a second angle. The third angle measures 12° less than twice the second angle. Find the measures of the angles. 14. Angles in a triangle. One angle of a triangle is four times as great as a second angle. The third angle measures 5° more than twice the second angle. Find the measures of the angles. 15. Find two consecutive even integers such that two times the first plus three times the second is 76. 16. Find three consecutive odd integers such that the sum of the first, twice the second, and three times the third is 70. 17. A steel rod 90 cm long is to be cut into two pieces, each to be bent to make an equilateral triangle. The length of a side of one triangle is to be twice the length of a side of the other. How should the rod be cut? 18. A piece of wire 100 cm long is to be cut into two pieces, and those pieces are each to be bent to make a square. The area of one square is to be 144 cm2 greater than that of the other. How should the wire be cut? (Remember: Do not solve.) 19. Test scores. Deirdre's scores on five tests are 93, 89, 72, 80, and 96. What must the score be on her next test so that the average will be 88? 20. Pricing Whitneys Appliances is having a sale on 13 TV sets. They are displayed in order of increas- ing price from left to right. The price of each set differs by $20 from either set next to it. For the price of the set at the extreme right, a customer can buy both the second and seventh sets. What is the price of the least expensive set? Solve each problem. Use all five problem-solving steps. 21. The number 38.2 is less than some number by 12.1. What is the number? 22. The number 173.5 is greater than a certain number by 16.8. What is the number? 23. The number 128 is 0.4 of what number? 1 24. The number 456 is 3 of what number? 25. One number exceeds another by 12. The sum of the numbers is 114. What is the larger number? 26. One number is less than another by 65. The sum of the numbers is 92. What is the smaller number? Find the measures of the angles. 38 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING 27. A rectangle's length is three times its width and its perimeter is 120 cm. Find the dimensions of the rectangle. 28. A rectangle's length is twice its width and its perimeter is 21 cm. Find the dimensions of the rectangle. 29. A rectangle's width is one-fourth its length and its perimeter is 130 m. Find the dimensions of the rectangle. 30. A rectangle's width is one-third its length and its perimeter is 32 m. Find the dimensions of the rectangle. 31. Solve the problem of Exercise 4. 32. Solve the problem of Exercise 3. 33. Solve the problem of Exercise 14. 34. Solve the problem of Exercise 13. 35. Solve the problem of Exercise 10. 36. Solve the problem of Exercise 9. 37. Solve the problem of Exercise 8. 38. Solve the problem of Exercise 15. 39. Write a problem for a classmate to solve. Devise the problem so that the solution is "The first angle is 40°, the second angle is 50°, and the third angle is 90°." 40. Write a problem for a classmate to solve. Devise the problem so that the solution is "The material should be cut into two pieces, one 30 cm long and the other 45 cm long." SKILL MAINTENANCE Solve 41. 7 = 2 (x + 6) 3 42. 9 = x 4 43. 8 = 5 + t 3 44. 6t - 8 = 0 SYNTHESIS 45. How can a guess or estimate help prepare you for the Translate step when solving problems? 46. Why is it important to check the solution from step 3 (Curry out) in the original wording of the problem being solved? 47. Test Scores Tico's scores on four tests are 83, 91, 78, and 81. How many points above his current average must Tico score on the next test in order to raise his average 2 points? 48. Geometry The height and sides of a triangle are four consecutive integers. The height is the first integer, and the base is the third integer. The perimeter of the triangle is 42 in. Find the area of the triangle. 49. Home prices. Panduski's real estate prices in-creased 6% from 1998 to 1999 and 2% from 1999 to 2000. From 2000 to 2001, prices dropped 1%. If a house sold for $117,743 in 2001, what was its worth in 1998? (Round to the nearest dollar.) 50. Adjusted wages Blanche's salary is reduced n% during a period of financial difficulty. By what number should her salary be multiplied in order to bring it back to where it was before the reduction? 1.5. FORMULAS, MODELS, AND GEOMETRY 43 Examp1e 5 Density. A collector suspects that a silver coin is not solid silver. The density of silver is 10.5 grams per cubic centimeter (g/cm3) and the coin is 0.2 cm thick with a radius of 2 cm. If the coin is really silver, how much should it weigh? Solution 1. Familiarize. From an outside reference, we find that density depends on mass and volume and that, in this setting, mass means weight. A for-mula for the volume of a right circular cylinder appears at the very end of this text. 2. Translate. We need to use two formulas: m D = V and V = ttr2h where D represents the densiry m the mass, V the volume, r the length of the radius, and h the height of a right circular cylinder. Since we need a model relating mass to the measurements of the coin, we solve for m and then substitute for V m D= V M V * D = V * V Multiplying by V V * D = m Simplifying ttr2h * D = m. Substituting 3. Carry out. The model m = ttr2hD can be used to find the mass of any right circular cylinder for which the dimensions and the density are known: m = ttr2hD =tt(2)2(0.2)(10.5) Substituting =26.3894. Using a calculator with a ar key 4. Check. To check, we could repeat the calculations. We might also check the model by examining the units: ttr2h * D = cm2 * cm * g = cm3 * g = g. tt has no unit. cm3 cm3 Since g (grams) is the unit in which the mass, m, is given, we have at least a partial check. 5. State. The coin, if it is indeed silver, should weigh about 26 grams. 44 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Exercise Set 1.5 FOR EXTRA HELP Solve. 1. d = rt, for r (a distance formula) 2. d = rt, for t 3. F = ma, for a (a physics formula) 4. A = lw, for w (an area formula) dz _ di 5. W = EI, for I (an electricity formula) 6. W=EI,for E 7. V = lwh, for h (a volume formula) 8. I = Prt, for r (a formula for interest) k 9. L = d2,for k (a formula for intensity of sound or light) Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 10. mv2 F = r , for m (a physics formula) 11. G = w + 150n, for n (a formula for the gross weight of a bus) 12. P = b + 0.5t, for t (a formula for parking prices) • 13. 2w + 2h + l = p, for l a formula used when shipping boxes 14. 2w + 2h + l = p, for w 15. 5x + 2y = 8, for y (a formula for a line) 16. 2x + 3y = 12, for y 17. Ax + By = C, for y (a formula for graphing lines) 18. P = 21 + 2w, for l (a perimeter formula) 5 19. C = 9 (F - 32), for F (a temperature formula) 20. T= 3 (I- 12.000), for I (a tax formula) 10 21. V = 4 ttr3, for r3 (a volume formula) 3 22. V = 4 ttr3, for tt 3 23. A = h 2 (bl + b2), for b2 (an area formula) 24. A = h 2 (bl + b2), for h (an area formula) 25. A = ql + q2 + q3 , for n (a formula for averaging) n (Hint: Multiply by n to "clear" fractions.) 26. g = km1m2, for d2 (Newton's law of gravitation) d2 27. v = d2 - d1 , for t (a physics formula) t 28. v = S2 - S1 , for m m 29. v = d2 - d1 ,for d1 t 30. v = s2 - sl , for sl m 31. r = m + mnp, for m 32. P = x - xyz, for x 33. y = ab - ac2, for a 34. d = mn - mp3, for m 35• Investing Janos has $2600 to invest for 6 months. If he needs the money to earn $156 in that time, at what rate of simple interest must Janos invest? 36. Banking. Yvonne plans to buy a one-year certifi- cate of deposit (CD) that earns 7% simple interest. If she needs the CD to earn $110, how much should Yvonne invest? 37. Geometry. The area of a parallelogram is 78 cm2. The base of the figure is 13 cm. What is the height? 38. Geometry The area of a parallelogram is 72 cm2. The height of the figure is 6 cm. How long is the base? 39• Weight of a coin. The density of gold is 19.3 g/cm3. If the coin in Example 5 were made of gold instead of silver, how much more would it weigh? 1.5 FORMULAS, MODELS, AND GEOMETRY 45 40. Weight of salt The density of salt is 2.16 g/cm3 (grams per cubic centimeter). An empty cardboard salt canister weighs 28 g, is 13.6 cm tall, and has a 4-cm radius. How much will a filled canister weigh? Use the model developed in Example 5. Projected birth weight. Ultrasonic images of 29-week- old fetuses can be used to predict weight. One model, de- veloped by Thurnau,* is P = 9.337da - 299; a second model, developed by Weiner,t is P = 94.593c + 34.227a - 2134.616. For both formulas, P represents the estimated fetal weight in grams, d the diameter of the fetal head in centimeters, c the circumference of the fetal head in centimeters, and a the circumference of the fetal abdomen in centimeters. 41. Use Thurnau's model to estimate the diameter of a fetus' head at 29 weeks when the estimated weight is 1614 g and the circumference of the fetal ab- domen is 24.1 cm. 42. Use Weiner's model to estimate the circumference of a fetus' head at 29 weeks when the estimated weight is 1277 g and the circumference of the fetal abdomen is 23.4 cm. 43. Gardening A garden is being constructed in the shape of a trapezoid. The dimensions are as shown in the figure. The unknown dimension is to be such that the area of the garden is 90 ft2. Find that un- known dimension. 44. Fencing A rectangular garden is being con- structed, and 76 ft of fencing is available. The width of the garden is to be 13 ft. What should the length be, in order to use just 76 ft of fence? Aha'~ 45. Investing. Bok Lum Chan is going to invest $1000 at simple interest at 9%. How long will it take for the investment to be worth $1090? 46. Rik is going to invest $950 at simple interest at 7%. How long will it take for his investment to be worth $1349? Waiting time In an effort to minimize waiting time for patients at a doctor's office without increasing a physi- cian's idle time, Michael Goiten of Massachusetts Gen- eral Hospital has developed a model. Goiten suggests that the interval time I, in minutes, between scheduled appointments be related to the total number of minutes T that a physician spends with patients in a day and the number of scheduled appointments N according to the formulal= 1.08(T/N).4. 47. Dr. Cruz determines that she has a total of 8 hr a day to see patients. If she insists on an interval time of 15 min, according to Goiteris model, how many appointments should she make in one day? 48. A doctor insists on an interval time of 20 min and must be able to schedule 25 appointments a day. According to Goiten's model, how many hours a day should the doctor be prepared to spend with patients? 49. Is every rectangle a trapezoid? Why or why not? 50. Predictions made using the models of Exercises 41 and 42 are often off by as much as 10%. Does this mean the models should be discarded? Why or why not? SKILL MAINTENANCE Use the associative and commutative laws to write two equivalent expressions for each of the following. Answers may vary. 51. (7a) (3a) 52. (4y) (xy) SYNTHESIS 53. Which would you expect to have the greater den- sity, and why: Cork Or Steel? *Thurnau, G. R., R. K. Tamura, R. E. Sabbagha, et al. Am. J. Obstet Gynecol 1983; 145:557. tWeiner, C. P, R. E. Sabbagha, N. Vaisrub, et al. Obstet Gynecol 1985; 65:812. ~New England Journal of Medicine, 30 August 1990, pp. 604-608. 46 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING 54. Both of the models used in Exercises 41 and 42 have P alone on one side of the equation. Why? 55. The density of platinum is 21.5 g/cm3. If the ring shown in the figure below is crafted out of platinum, how much will it weigh? 0.5 cm 2 cm 0.15cm 56. The density of a penny is 8.93 g/cm3. The mass of a roll of pennies is 177.6 g. If the diameter of a penny is 1.85 cm, how tall is a roll of pennies? 57. The density of copper is 8.93 g/cm3. How long must a copper wire be if it is 1 cm thick and has a mass of 4280 g? Solve. 58. s = v1t + 1 at2, for a 2 59. A = 4lw + w2, for l 60. P1 V1 = P2 V2 for T2 T1 T2 61. P1 V1 = P2V2 Tl T2 , for Tl 62• b a - b = c for b 63. (d/e) m = , for d (e/f) 64. a a+b =c,for a 65. s+ s+t =1 + s + t s-t t s - t,for t 66. To derive the formula for the area of a trapezoid, consider the area of two trapezoids, one of which is upside down. b1 b2 b b b2 b1 Explain why the total area of the two trapezoids is given by h(bl + b2). Then explain why the area of a trapezoid is given by h 2 (bl + b2). Properties of 1•6 Exponents The Product and Quotient Rules • The Zero Exponent • Negative Integers as Exponents • Raising Powers to Powers • Raising a Product or a Quotient to a Power. In Section 1.1, we discussed how whole-number exponents are used. We now develop rules for manipulating exponents and determine what zero and nega-tive integers will mean as exponents. The Product and Quotient Rules Note that the expression x3 . x4 can be rewritten as follows: x3. x4=x* x* x* x*x* x* x 3 factors 4 factors =x*x*x*x*x*x*x 7 Factors. =x7. 1.6 PROPERTIES OF EXPONENTS 47 This result is generalized in the product rule. Multiplying with Like Bases: The Product Rule For any number a and any positive integers m and n, am * an = am+n (When multiplying powers, if the bases are the same, keep the base and add the exponents.) (When dividing powers, if the bases are the same, keep the base and subtract the exponent of the denominator from the exponent of the numerator.) Example 1 Multiply and simplify: (a) m5 * m7; (b) (5a2b3) (3a4b5). Solution a) m5 * m7 = m5+7 = m12 Multiplying powers by adding exponents b) (5a2b3) (3a4b5) = 5 * 3 * a2 * a4 * b3 * b5 Using the associative and commutative laws = 15a2+4b3+5 = 15a6b8 Caution! 5 8 * 5 6 = 5 14; 5 8 * 5 6 =/ 25 14. Next, we simplify a quotient: x8 x*x*x*x*x*x*x*x x3 x*x*x =x*x*x *x*x*x*x*x x*x*x = x * x * x * x * x Removing a factor equal to 1: x*x*x =1 x*x*x =x5 The generalization of this result is the quotient rule. Multiplying; using the product rule Using the definition of exponential notation Dividing with Like Bases: The Quotient Rule For any nonzero number a and any positive integers m and n, m > n, am = am-n an When dividing powers if the bases are the same keep the base and subtract the exponent of the denominator from the exponent of the numerator. 48 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Example 2 r9 lOxllys Divide and simplify: r9 10x 11y5 (a) r3; (b) 2x 4Y3 Solution r9 a) r3 = r9-3 = r6 Using the quotient rule b) 10x11y5 = 5 * x11-4*y5-3 Dividing; using the quotient rule 2x4y3 = 5x7y2 Caution! 7 8 = 7 6; 7 8 =/ 7 4. 7 2 7 2 The Zero Exponent Suppose now that the bases in the numerator and the denominator are both raised to the same power. On the one hand, any (nonzero) expression divided by itself is equal to 1. For example, t5 6 4 t5 = 1 and 6 4 = 1. On the other hand, if we continue subtracting exponents when dividing powers with the same base, we have t5 6 4 t5 = t5- 5 = t° and 6 4 = 6 4-4 = 6°. This suggests that t5/t5 equals both 1 and t°. It also suggests that 6 4/6 4 equals both 1 and 6°. This leads to the following definition. The Zero Exponent For any nonzero real number a, a°=1. (Any nonzero number raised to the zero power is 1. 0° is undefined.) Example 3 Evaluate each of the following for x = 2.9: (a) x°; (b) -x°; (c) (-x)°. a) x° = 2.9° = 1 Using the definition of 0 as an exponent b) -x° = -2.9° = -1 The exponent 0 pertains only to the 2.9. c) (-x)° = (-2.9)° = 1 The base here is -2.9. 1.6 PROPERTIES OF EXPONENTS 53 There is a similar rule for raising a quotient to a power. Raising a Quotient to a Power For any integer n, and any real numbers a and b for which a/b, a n, and b n exist, a n a n b l' = bn . (To raise a quotient to a power, raise both the numerator and the denominator to that power.) Examp1e 9 x2 4 Y2z3 -3 Simplify: (a) 2 b 5 Solution a x2 4 = (x2)4 x8 < 2 * 4 = 8 b y2 z3 -3 = y2z3 -3 5 5 -3 = 5 3 Moving factors to the other side of the fraction bar ( y2Z3)3 and changing each -3 to 3 125 y6z9 The rule for raising a quotient to a power allows us to derive a useful result for manipulating negative exponents: a -n = a-n = b n = b n b b-n a n a Using this result, we can simplify Example 9(b) as follows: y2z3 -3 = 5 3 Taking the reciprocal of the base and 5 ) ( y2z3) changing the exponent's sign = 5 3 125 (Y2Z3)3 y6z9. 54 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Definitions and Properties of Exponents The following summary assumes that no denominators are 0 and that 0° is not considered. For any integers m and n, 1 as an exponent: al = a 0 as an exponent: a° = 1 Negative exponents: a-n = 1 a n a-n = bm b-m a n a -n = b n b a am * an = a m+n The Product Rule: The Quotient Rule: a m a n a m-n The Power Rule: (a m)n = a mn Raising a product to a power: (ab)n = anbn Raising a quotient to a power: a n = a n b b n Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MyMathLab.com Multiply and simplify. Leave the answer in exponential 17. m7n9 18. m12n9 notation. m2n5 m4n6 Exercise Set 1.6 1. 2 7 * 2 4 2. 7 3 * 7 4 19. 45x8y5 20. 35x7y8 3. 5 6 * 5 3 4. 6 3 * 6 5 5x2y 7xy2 5. t° * t8 6. x° * x5 21. 28x10y9z8 22. 18x8y6z7 7. 6x5 * 3x2 8. 4a3 * 2a7 -7x2Y3z2 -3x2y3z 9. (-3m4) (-7m9) 10. (-2a5) (7a4) Evaluate each of the following for x = -2. 11. (x3y4) (x7Y6z°) 12. (m6n5) (m4n7P°) 23. -x° 24. (-x)° 25. (4x)° 26. 4x° Divide and simplify. a9 xl2 Simplify. 13. a3 14. x3 27. (-3)4 28. (-2)6 29. -3 4 15. 12t7 16. 20a2° 30. -2 6 31. (-4)-2 32. (-5)-2 4t2 5a4 33. -4 -2 34. -5 -2 35. -2-4 page 55 36. -5 -3 37. -2 -6 38. -1 -8 Write an equivalent expression without negative expo-nents and, if possible, simplify. 39. a-3 40. n-6 41. 1 5-3 42. 1 43. 4x-3 2 -6 44. 7x-3 45. 2a3b-6 46. 5a-7b4 47. z-4 3x5 48. y-5 49. x -2y7 50• y4z -3 x-3 z-4 x-2 51. 1 3 4 52. 1 9 2 53. 1 (-16)2 54. 1 (-8)6 55. x5 56. n3 57. 6x2 58. -4y5 59. 1 (5y)3 60. 1 (5x)5 61. 1 3y4 62. 1 4b3 63. 8-2 • 8-4 64. 9-1 • 9-6 65. b2 • b-5 66. a4 • a-3 67. a-3 • a4 • a2 68. x-8 • x5 • x3 69. (9mn3) (-2m3n2) 70. (6x5y -2) (-3x2y3) 71. (-2x -3) (7x -8) 72. (6x -4y3) (-4x-8y-2) 73. (5a -2b-3) (2a-4b) 74. (3a-5b-7) (2ab-2) 75. l0 -3 10 6 76. 12 -4 12 8 77. 2-7 2-5 78. 9-4 9-5 79. y4 y-5 80. a3 a-2 81. 24a5b3 -8a4b 82. 9a2 3ab3 83. 14a4 b -3 -8a8b -5 84. -24x6y7 18x -3y9 85. -5x -2y4z7 30x-5y6z-3 86. 9a6b-4c7 27a-4b5c9 87. (x4)3 88. (a3)2 89. (9 3)-4 90. (8 4)-3 91. (t-8)-5 92. (x-4)-3 93. (6xy)2 94. (5ab)3 95. (a3b)4 96. (x3Y)5 97. 5(x2y2)-7 98. 7(a3b4)-5 99. (a-5b2)3(a4b-1)2 100. (x2 y-3)3(x -4y)2 Write an equivalent expression with negative exponents. 101. (5x -3y2)-4(5x-3y2)4 102. (2a-lb3)-2(2a -lb3)-2 103. (3x3y4)3 6xy3 104. (5a3b)2 10a2b 105. -4x4y -2 -4 5x -1y4 106. 2x3y-2 3 3y-3 107. 3a-2b5 -2 9a-4b6 108. 30x5y-7 0 6x-2y-6 109. 4a3b -9 0 2a -2b5 110. 5x0y-7 -2 2x-2y4 111. Explain why (-1)n = 1 for any even number n. 112. Explain why (-17)-8 is positive. Simplify. Should negative exponents appear in the answer, write a second answer using only positive exponents. SKILL MAINTENANCE 113. 4.9t2 + 3t, for t = -3 114. 16t2 + lOt, for t = -2 115. Explain in your own words why a° is defined to be l. Assume that a =/ 0. 116. Is the following true or false, and why? 5 -6 > 4 -9 Simplify. Assume that all variables represent nonzero integers. 117. 12ax-2 3a2x+2 118. -12xa+1 4x2 -a 1.6 PROPERTIES OF EXPONENTS 55 14a4b-3 83. -8asb-s -24xsy~ -12xa+1 56 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING ll9. [7y(7 - 8)-2 - 8y(8 - 7)-2](-2)2 120. {[(8-a)-2]b}-c . [(8o)a]c 121. (3a+2)a 122. (12 3-a)2b 123. 4x2a+3y2b-1 2xa+1yb+1 124. 25xa+byb-a -5xa-byb+a 125. (2-2 a). (2b) -a (2 -2)-b(2b)-2a 126. -28xb+5y4+c 7xb-5yc-4 127. 3q+3 - 3 2 (3q) 3(3q+4) 128. (a-2c -3 a4c 2 -a b7c b-3c Scientific Notation 1.7 Conversions • Significant Digits and Rounding • Scientific Notation in Problem Solving There is a variety of symbolism, or notation, for numbers. You are already fa-miliar with fraction notation, decimal notation, and percent notation. We now study scientific notation, so named because of its usefulness in work with the very large and very small numbers that occur in science. The following are examples of scientific notation: 7.2 x 10 5 means 720,000; 3.4 x 10-6 means 0.0000034; 4.89 x 10-3 means 0.00489. Scientific Notation Scientific notation for a number is an expression of the form N x lOm, where N is in decimal notation, 1 <_ N < 10, and m is an integer. Conversions Note that lOb/ lOb = lOb * 10-b = 1. To convert to scientiflC notation, we can multiply by 1, writing 1 in the form lOb/lOb or lOb * 10-b. page 59 To find scientific notation for the result, we convert 31 to scientific notation and simplify: 31 x 10 14=(3.1 x 101)x 10 14 = 3.1 x 10 15. Example 6 Divide and write scientific notation for the answer: 3.48 x 10-7 4.64 x 10 6 Solution 3.48 x 10-7=3.48 10-7 Separating factors. Our answer 4.64 x 10 6 4.64 x 10 6 must have 3 significant digits. =0.75 x 10 -13 =(7.5 x 10 -1) x 10 -13 =7.50 x 10 -14 Scientific Notation in Problem Solving Scientific notation can be useful in problem solving. Example 7 Astronomy. The largest known star, Betelgeuse, is about 700 times the size of the sun and about 3.06 x 10 15 mi from Earth (Sources: Guinness Book of Records 2000 and The Cambridge Factfinder, 4th ed.). How many light years is it from Earth to Betelgeuse? Solution 1. Familiarize. From an astronomy text, we learn that light travels about ( 5.88 x 10 12 mi in one year. 1 light year = 5.88 x l0 12 mi. We will let y represent the number of light years from Earth to Betelgeuse. Let's guess that the answer is 6 light years. Then the distance in miles would be (5.88 X lO 12) * 6 = 3.528 x 10 13. 1 light year = 5.88 x 10 12 mi Technology connection Both graphing and scientific cal- culators allow expressions to be entered using scientific nota- tion. To do so, a key normally la- beled EE or EXP is used. Often this is a secondary func- tion and a key labeled SHIFT or 2nd must be pressed first. To check Example 5, we press 7.2 EE 5 [x] 4.3 EE 9. When we then press ENTER or = the result 3.096E15 or 3.096 15 appears. We must interpret this result as 3.096 x l0 15. 3.06 x 10 15 mi Our guess is incorrect, but it tells us that the distance is more than 6 light years. We are also better able to translate to an equation. 4.64 x 106 * = 0.75 x 10-13 _ (7.5 x 10-1) x 10-13 Converting 0.75 to scientific notation = 7.50 x 10-14 Adding exponents. We write 7.50 to indicate 3 significant digits. Subtracting exponents; simplifying 1.7 SCIENTIFIC NOTATION 59 page 60 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING 2. Translate. Note that the distance to Betelgeuse is y light years, or (5.88 x 10 12)y mi. Also, note that the distance is 3.06 x lO l5 mi. Since the quantities (5.88 x lO 12)y and 3.06 x lO l5 both represent the number of miles to Betelgeuse, we form the equation (5.88 x 10 12)y = 3.06 x lO l5. 3. Carry out. We solve the equation: (5.88 x 10 12)y = 3.06 x lO 15 1 1 5.88 x 10 12 (5.88 x 10 12)y = 5.88 x lO l2 (3.06 X lO l5) Multiplying both sides by 1/(5.88 x lO l2) 3.06 x lO l5 y = 5.88 x 10 12 Simplifying 3.06 lO l5 = 5.88 x 10 12 Factoring ~ 0.5204 x 10 3 ~ 5.20 x 10 2 or 520. yr. Our answer has 3 significant digits because of the 3.06 and 5.88. 4. Check. Since light travels 5.88 x 10 12 mi in one year, in 520. it will travel 520 x 5.88 x 10 12 = 3.06 x lO 15 mi, which is approximately the distance from Earth to Betelgeuse. The decimal point in 520. indicates that we did not round to the nearest ten. A number like 527 would not need a decimal point 5. State. The distance from Earth to Betelgeuse is about 520. light years. Example 8 Teleommunicptions. A fiber-optic wire will be used for 375 km of transmission line. The wire has a diameter of 1.2 cm. What is the volume of wire needed for the line? Solution 1. Familiarize. Making a drawing, we see that we have a cylinder (a very long one). Its length is 375 km and the base has a diameter of 1.2 cm. 1.7 SCIENTIFIC NOTATION Page 61 Recall that the formula for the volume of a cylinder is V = ttr2h, where r is the radius and h is the height (in this case, the length of the wire). 2. Translate. We will use the volume formula, but it is important to make the units consistent. Let's express everything in meters: Length: 375 km = 375,000 m, or 3.75 x 10 5 m; Diameter: 1.2 cm = 0.012 m, or 1.2 x 10-2 m. The radius, which we will need in the formula, is half the diameter: Radius: 0.6 x 10-2 m, or 6 x 10-3 m. We now substitute into the above formula: V = tt(6 x 10-3)2(3.75 x 10 5). 3. Carry out. We do the calculations: v= tt x (6 x l0-3)2(3.75 x l0 5) =tt x 6 2 x 10-6 x 3.75 X 10 5 Using the properties of exponents =(tt x 6 2 x 3.75) x (10-6 x 10 5) = 423.9 x 10-1 Using 3.14 for tt and rounding = 42. Rounding 42.39 to 2 significant digits because of the 1.2 4. Check. About all we can do here is recheck the translation and calculations. 5. State. The volume of the wire is about 42 m3 (cubic meters). Exercise Set 1.7 FOR EXTRA HELP . Digital Video Tutor CD 1 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape Z Convert to scientific notation. Convert to decimal notation. Simplify and write scientific notation for the answer. Use the correct number of significant digits. 1. 83,000,000,000 is 83. 2. 2,600,000,000,000 is 2.6 3. 863,000,000,000,000,000 answer is 8.63 4. 572,000,000,000,000,000 answer is 5. 0.000000016 answer is 6. 0.000000263 answer is 7. 0.00000000007 answer is 8. 0.00000000009 answer is 9. 803,000,000,000 answer is 10. 3,090,000,000,000 11. 0.000000904 12. 0.00000000802 13. 431,700,000,000 14. 953,400,000,000 15. 5 x 10-4 16. 5 x 10-5 17. 9.73 x 10 8 18. 9.24 x 10 7 19. 4.923 x 10 -10 20. 7.034 x 10 -2 21. 9.03 x 10 10 22. 1.01 x 10 12 23. 4.037 X 10 -8 24. 3.007 x 10 -9 25. 7.01 x 10 12 26. 9.001 x lO lo 27. (2.3 x 10 6) (4.2 x 10-11) 62 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING 28. (6.5 x 10 3) (5.2 x 10-8) 29. (2.34 x lo-8) (5.7 x l0 -4) 30. (4.26 x l0-6) (8.2 x l0 -6) 31. (5.2 x l0 6) (2.6 x l0 4) 32. (6.ll x 10 3)(1.01 x 10 13) 33. (7.01 x l0 -5) (6.5 x l0 7) 34. (4.08 x 10 -10) (7.7 x l0 5) ha` 35. (2.o x 10 6) (3.02 x l0 -6) 36. (7.04 x 10 -9)(9.01 x lo -7) 5.1 x 10 6 37. 3.4 x 10 3 38. 8.5 x 10 8 3.4 x 10 5 39. 7.5 x 10 -9 2.5 x 10 -3 40. 4.0 x 10 -6 8.0 x 10 -3 41. 3.2 x 10 -7 8.0 x 10 8 42. 12.6 x 10 8 4.2 x 10 -3 43. 9.36 x 10 -11 3.12 x 10 11 44. 2.42 x 10 5 1.21 x 10 -5 45. 6.12 x 10 19 3.06 x 10 -7 46. 4.7 x 10 -9 2.0 x 10 -9 47. 780,000,000 x 0.00071 0.000005 48. 830,000,000 x 0.12 3,100,000 49. 5.9 x 10 23 + 2.4 x l0 23 50. 1.8 X 10 -34 + 5.4 x 10 -34 Solve 51. Astronomy Venus has a nearly circular orbit of the sun. If the average distance from the sun to Venus is 1.08 x 10 8 km, how far does Venus travel in one orbit? 52. Office supplies. A ream of copier paper weighs 2.25 kg. How much does a sheet of copier paper weigh? 53. Printing and engineering - A ton of five-dollar bills is worth $4,540,000. How many pounds does a five-dollar bill weigh? 54. Astronomy The average distance of the earth from the sun is about 9.3 x 10 7 mi. About how far does the earth travel in a yearly orbit about the sun? (Assume a circular orbit.) 55. Astronomy The brightest star in the night sky, Sirius, is about 4.704 x 10 13 mi from the earth. How many light years is it from the earth to Sirius? 56. Astronomy The diameter of the Milky Way galaxy is approximately 5.88 x 10 17 mi. How many light years is it from one end of the galaxy to the other? Named in tribute to Anders Angstrom, a Swedish physicist who measured light waves, 1 A (read `one Angstrom') equals 10 -10 meters. One parsec is about 3.26 light years, and one light year equals 9.46 x 10 15 meters. 57. How many Angstroms are in one parsec? 58. How many kilometers are in one parsec? For Exercises 59 and 60 approximate the average distance from the earth to the sun by 1.50 x 10 11 meters. 59. Determine the volume of a cylindrical sunbeam that is 3 A in diameter 60. Determine the volume of a cylindrical sunbeam that is 5 A in diameter. 61. Biology. An average of 4.55 x 10 11 bacteria live in each pound of U.S. mud.* There are 60.0 drops in one teaspoon and 6.0 teaspoons in an ounce. How many bacteria live in a drop of U.S. mud? *Harper's Magazine, April 1996, p. 13. 1.7 SCIENTIFIC NOTATION page 63 62. Astronomy If a star 5.9 x 10 14 mi from the earth were to explode today, its light would not reach us for 100 years. How far does light travel in 13 weeks? 63. Astronomy. The diameter of Jupiter is about 1.43 x 10 5 km. A day on Jupiter lasts about 10 hr. At what speed is Jupiter's equator spinning? 10+5 = 6 1.43 x 6 64. Finance. A mil is one thousandth of a dollar. The taxation rate in a certain school district is 5.0 mils for every dollar of assessed valuation. The assessed valuation for the district is 13.4 million dollars. How much tax revenue will be raised? 65. Write a problem for a classmate to solve. Design the problem so the solution is "The area of the pin- head is 3.14 x 10-4 cm2." 10 - 4=3 3.14 x 3 66. List two advantages of using scientific notation. Answers may vary. SKILL MAINTENANCE Evaluate. 67. 3x - 7y, for x = 5 and y = 1 68. 2a - 5b, for a = 1 and b = -6 SYNTHESIS 69. A criminal claims to be carrying $5 million in twenty-dollar bills in a briefcase. Is this possible? Why or why not? (Hint: See Exercise 53.) No. The briefcase is too small in size. 70. When a calculator indicates that 5 17 = 7.629394531 x 10 11, an approximation is being made. How can you tell? (Hint: What should the ones digit be?). 7.62 10 + 11 = 22 71. The Sartorius Microbalance Mode1 4108 can weigh objects to an accuracy of 3.5 x 10 -10 oz (Source: Guinness World Records 2000). A chemical com- pound weighing 1.2 x 10 -9 oz is split in half and weighed on the microbalance. Give a weight range for the actual weight of each half. 72. Write the reciprocal of 2.57 x 10 -17 in scientific notation. 73. Compare 8 * 10 -90 and 9 * 10 -91. Which is the larger value? How much larger is it? Write scientific notation for the difference. 74. Write the reciprocal of 8.00 x 10 -23 in scientific notation. 75. Evaluate: (4096)0.05(4096)0.02. 76. What is the ones digit in 513 128? 5.13 77. A grain of sand is placed on the first square of a chessboard, two grains on the second square, four grains on the third, eight on the fourth, and so on. Without a calculator, use scientific notation to approximate the number of grains of sand required for the 64th square. (Hint: Use the fact that 2 10 = 10 3.) 64 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Paired Problem Solving Focus: Problem solving, scientific notation, and unit conversion Time: 15-25 minutes Group size: 3 ACTIVITY Given that the earth's average distance from the sun is 1.5 x 10 11 meters, determine the earth's orbital speed around the sun in miles per hour. Assume a circular orbit and use the following guidelines. 1. Two students should attempt to solve this problem while the third group member silently observes and writes notes describing the efforts of the other two. 2. After 10-15 minutes, all observers should share their observations with the class as a whole, answering these three questions: a) What successful strategies were used? b) What unsuccessful strategies were used? c) What recommendations do the observers have to make for students working in pairs to solve a problem? Summary and Review 1 Key Terms Variable, p. 2 Rational numbers, p. 7 Constant, p. 2 Fraction notation, p. 8 Algebraic expression, p. 2 Decimal notation, p. 8 Equation, p. 2 Irrational numbers, p. 8 Solution, p. 3 . Real numbers, p. 8 Factors, p. 4 Subset, p. 9 Substituting, p. 4 Absolute value, p. 11 Evaluating, p. 4 Inequality, p. 12 Exponent, or power, p. 5 Opposite, p. 13 Base, p. 5 Additive inverse, p. 13 Natural numbers, p. 6 Reciprocal, p. 16 Whole numbers, p. 6 Multiplicative inverse, p. 16 Integers, p. 6 Indeterminate, p. 17 Roster notation, p. 7 Equivalent expressions, p. 18 Set-builder notation, p. 7 Factoring, p. 19 Element, p. 7 Equivalent equations, p. 22 Term, p. 24 Like, or similar, terms, p. 24 Combine, or collect, like terms, p. 24 Linear equation, p. 26 Identity, p. 26 Contradiction, p. 26 Conditional equation, p. 26 Solution set, p. 27 Empty set, p. 27 Perimeter, p. 34 Formula, p. 39 Area, p. 40 Mathematical model, p. 42 Scientific notation, p. 56 Significant digits, p. 58 SUMMARY AND REVIEW: CHAPTER 1 Page 65 Area of a rectangle: A = lw Area of a square: A = s2 Area of a parallelogram: A = bh Area of a trapezoid: A = h 2 (bl + b2) Area of a triangle: A = 1 2 bh Area of a circle: A = ttr2 Circumference of a circle: C = ttd Volume of a cube: V = s3 Volume of a right circular cylinder: V = ttr2h Perimeter of a square: P = 4s Distance traveled: d = rt Simple interest: I = Prt For any number a and any nonzero number b. -a = a = a b -b b Addition of Two Real Numbers 1. Positive numbers: Add the numbers. The result is positive. 2. Negative numbers: Add absolute values. Make the answer negative. 3. A negative and a positive number: If the numbers have the same absolute value, the answer is 0. Otherwise, subtract the smaller absolute value from the larger one: a) If the positive number is further from 0, make the answer positive. b) If the negative number is further from 0, make the answer negative. 4. One number is zero: The sum is the other number. Multiplication of Two Real Numbers 1. To multiply two numbers with unlike signs, multiply their absolute values. The answer is negative. 2. To multiply two numbers with the same sign, multiply their absolute values. The answer is positive. Division of Two Real Numbers 1. To divide two numbers with unlike signs, divide their absolute values. The answer is negative. 2. To divide two numbers with the same sign, divide their absolute values. The answer is positive. The law of opposites: a + (-a) = 0 The law of reciprocals: a * 1 = 1.a = 0 a Absolute value: |x| = x, if x >_ 0, -x, ifx<0 -a = a -a b -b b a(bc) _ (ab)c ab + ac signs, multiply their absolute values, Rules for Order of Operations 1. Simplify within any grouping symbols. 2. Simplify all exponential expressions. 3. Perform all multiplication and division, working from left to right. 4. Perform all addition and subtraction, working from left to right. Commutative laws: a + b = b + a, ab = ba Associative laws: a + (b + c) = (a+b)+c, a(bc) = (ab)c Distributive law: a(b + c) = ab + ac The Addition Principle for Equations a = b is equivalent to a + c = b + c. The Multiplication Principle for Equations For c = 0, a = b is equivalent to ac = bc. page 66 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Five Steps for Problem Solving Review Exercises The following review exercises are for practice. Answers Perform the indicated operation. are at the back of the book. If you need to, restudy the builder notation. Use a commutative law to write an equivalent Find the absolute value. 1. Familiarize yourself with the problem. For any integers m and n (assuming 0 is 2. Translate to mathematical language. not raised to a nonpositive power): 3. Carry out some mathematical Zero as an exponent: a° = 1 manipulation. 4. Check your possible answer in the orig- Negative exponents: inal problem. 5. State the answer clearly. Zero as a exponent a0 =1 Negative exponents a-n = 1 * a-n= bm; a -n= b n a n b-m an b a Multiplying with like bases am * an = am+n product rule. 9. (- 5~ + (7~ section indicated alongside the answer. Review Exercises 1. Translate to an algebraic expression: Three less than the quotient of two numbers. 2. Evaluate 7x2 - 5 divide zx for x = -2,y = 3 and z = -5. 3. Name the set consisting of the first six even natural numbers using both roster notation and set builder notation. 4. Find the area of a triangular sign that has a base of 50 cm and a height of 70 cm. Find the absolute value. 5. |-7.3| 6. |4.09| 7. |0| 8. -6.5 + (-3.7) 9. (-4) + (1) 5 7 10. ( 1) + 4 -3 5 11. -7.9 - 3.6 12. -2 - (-1) 3 2 13. 12.5- 17.9 14. (-4.2) (-3) 15. (-2) (5) 3 8 16. (1.2) (-4) 17. -18 -3 18. 72.8 -8 19. -7 divide 4 3 20. Find -a if a = -4.01. 21. 7 + a 22. 7y 23. 5x + y To solve a formula for a specified letter: Dividing with like bases: am n = am-n, a ~ 0 (Quotient Rule) a 1. Get all terms with the letter being solved for on one side of the equation and all other terms on the other side, using the addition principle. To do this may require removing parentheses. • To remove parentheses, divide both Raism g a pomn r to a power: sides by the multiplier in front of the (a ) - a (Power Rule) parentheses or use the distributive Raising a quotient to a power: law raln=an~b~0 2. When all terms with the specified letter `b 1 bn ~ are on the same side, factor (if neces- sary) to combine like terms. Scientific notation for a number is an 3. Solve for the specified letter by dividing expression of the form N x lOm, where both sides by the multiplier of that 1 < N < 10, N is in decimal notation, and letter. m is an integer. Raising a product to a power: (ab)n - anbn Use an associative law to write an equivalent expression, Definitions and Rules for Exponents 24. (4 + a) + b 25. (xy)7 REVIEW EXERCISES: CHAPTER 1 page 67 26. Obtain an expression that is equivalent to 7mn + 14m by factoring. Simplify and write scientific notation for each answer. Use the correct number of significant digits. 27. Combine like terms: 5x3 - 8x2 + x3 + 2. 28. Simplify: 7x - 4[2x + 3(5 - 4x)]. Solve. If the solution set is 0/ or R classify the equation as a contradiction or as a identity. 29. x - 3.9 = 2.7 30. 2 3a =9 31. -9x + 4(2x - 3) = 5(2x - 3) + 7 32. 3(x - 4) + 2 = x + 2(x - 5) 33. 5t - (7 - t) = 4t + 2(9 + t) 34. Translate to an equation but do not solve: 13 more than twice a number is 21. 35. A number is 17 1ess than another number. The sum of the numbers is 115. Find the smaller number. 36. One angle of a triangle measures three times the second angle. The third angle measures twice the second angle. Find the measures of the angles. 37. Solve for m: P = m/S. 38. Solve for x: c = mx - rx. 39. The volume of a film canister is 28.26 cm3. If the radius of the canister is 1.5 cm, determine the height. 40. Multiply and simplify: (5a2b7) (-2a3b). 41. Divide and simplify: 12x3y8 3x2y2 42. Evaluate a0, a2, and -a2 for a = -5.3. Simplify. Do not use negative numbers. 43. 3 -4 * 3 7 44. (5a2)3 45. (-2a-3b2)-3 46. x2y3 -2 z4 47. 2a-2b 4 4a3b -3 Simplify. 48. 4(9 - 2 * 3) - 3 2is upper right. 4 2 - 3 2is upper right 49. 1 - (2 - 5)2 + 5 divide 10 * 4 2 50. Convert 0.000000103 to scientific notation. 51. One parsec a unit that is used in astronomy is 30,860,000,000,000 km. Write a scientific notation for this number. 386. 52. (8.7 x 10 -9) x (4.3 x 10 15) 53. 1.2 x 10 -12 6.1 x 10 -7 54. A sheet of plastic has a thickness of 0.00015 mm. The sheet is 1.2 m by 79 m. Use scientific notation to find the volume of the sheet. SYNTHESIS 55. Describe a method that could be used to write equations that have no solution. 56. Explain how the distributive law can be used when combining like terms. 57. If the smell of gasoline is detectable at 3 parts per billion, what percent of the air is occupied by the gasoline? 58. Evaluate a + b(c - a2)° + (abc)-1 for a = 2, b = -3, and c = -4. 59. What's a better deal: a 13-in. diameter pizza for $8 or a 17-in. diameter pizza for $11? Explain. You get more pizza for a 17in. diameter pizza. 60. The surface area of a cube is 486 cm2. Find the vol- ume of the cube. 61. Solve for z: m = x y-z 62. Simplify: (3 -2)a * (3b)-2a (3-2)b * (9-b)-3a 63. Each of Ray's test scores counts three times as much as a quiz score. If after 4 quizzes Ray's aver- age is 82.5 what score does he need on the first test in order to raise his average to 85? 64. Fill in the following blank so as to ensure that the equation is an identity. 5x -7(x + 3)-4 = 2 (7 - x) + _____ 65. Replace the blank with one term to ensure that the equation is a contradiction. 20-7[3(2x + 4)-l0]=9 - 2(x-5)+ _____ 66. Use the commutative law for addition once and the distributive law twice to show that a2 + cb + cd + ad = a(d + 2) + c(b + d). 67. Find an irrational number between 1 and 3. 2 4 page 68 CHAPTER 1 ALGEBRA AND PROBLEM SOLVING Chapter Test 1 1. Translate to an algebraic expression: Three more than the product of two numbers. 2. Evaluate a3 - 5b + b divide ac for a = -2, b = 6, and c=3. 3. A triangular stamp's base measures 3 cm and its height 2.5 cm. Find its area. 4. -15 + (-16) 5. -7.5 + 3.8 6. 3.21 + (-8.32) 7. 29.5 - 43.7 8. -16.8 - 26.4 9. -6.4(5.3) 10. -7 - (-3) 3 4 11. -2 (-5) 7 14 12. -42.6 7.1 13. 2 divide (- 3) 5 10 14. Simplify: 7 + (1 - 3)2 - 9 divide 2 2 * 6. 15. Use a commutative law to write an expression equivalent to 7x + y. Combine like terms. 16. 5y - 14y + 19y 17. 6a2b - 5ab2 + ab2 - 5a2b + 2 18. Simplify: 9x - 3(2x - 5) - 7. Solve. If the solution set is R or Z, classify the equation as an identity or a contradiction. 19.1Ox - 7= 38x + 49 20. 13t - (5 - 2t) = 5(3t - 1) 21. Solve for Pz: P1V1=P2V2 T1 T2 22. Greg's scores on five tests are 84, 80, 76, 96, and 80. What must Greg score on the sixth test so that his average will be 85? 23. Find three consecutive odd integers such that the sum of four times the first, three times the second, and two times the third is 167. Simplify. Do not use negative exponents in the answer. 24. -5(x - 4) - 3(x + 7) 25. 6b - [7 - 2(9b - 1)] 26. (7x-4y -7) (-6x-6y) 27. -3 -2 28. (-6x 2y -4)-2 29. (2x3y-6)2 -4y -2 30. (7x3Y)° Simplify and write scientific notation for the answer. Use the correct number of significant digits. 31. (9.05 x 10 -3) (2.22 x 10 -5) 32. 5.6 x 10 7 2.8 x 10 -3 33. 1.8 x 10 -4 4.8 x 10 -7 Solve. 34. The average distance from the planet Venus to the sun is 6.7 x 10 7 mi. About how far does Venus travel in one orbit around the sun? (Assume a circular orbit.) 10 + 7 =8 SYNTHESIS Simplify. 35. (4x3ayb+1)zc 36. -27a x+1 3a x-2 37. (-16x x-1y y-2)(2x x+1y y+1) (-7x x+2y y+2)(8x x-2y y-1) ® technology ~~ connection B Often, graphs are drawn with the aid of a grapher. As tion that is entered. By adjusting the TABLE SETUP, we can shown above, determining just a few ordered pairs can be control the smallest x-value listed (using Table Min) and quite time-consuming. With a grapher, thousands of the difference between successive x-values (using ~TBL). ordered pairs can be found at the touch of a button. Most The up and down arrow keys allow us to scroll up and graphers can display a table of ordered pairs for any equa- down the table. For the table shown, we used y1=-4x+3. Graph each equation using a [-10,10, -10, 10] win-dow. Then create a table of ordered pairs in which the x-values start at -1 and are 0.1 unit apart. TABLE MIN = 1.4 OTBL = .1 Page 74 Example 5 Graph the equation y= -1 x 2 if we choose 4 for x we get y = (-1)(4) or -2 2 When x is -6 we get =(-1) (-6)or 3.several ordered pairs plot them and draw the line. x y (x,y) 4 -2 (4,-2) -6 3 (-6,3) 0 0 (0,0) 2 -1 (2, -1) Examp1e 7 Next, we plot the points. The more points plotted, the clearer the shape of the graph becomes. Since the value of x2 - 5 grows rapidly as x moves away from the origin, the graph rises steeply on either side of the y-axis. Graph: y = |x|. Solution We select numbers for x and find the corresponding values for y. For example, if we choose -1 for x, we get y = |-1| = 1. Several ordered pairs are listed in the table below x y -3 3 -2 2 -1 1 0 0 1 1 2 2 3 3 We plot these points, noting that the absolute value of a positive number is the same as the absolute value of its opposite. The x-values 3 and -3 both are paired with the y-value 3. The graph is V shaped, as shown above. Page 76 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Exercise Set 2.1 FOR EXTRA HELP Digital Video Tutor CD 1 InterACt Math Math Tutor Center MathXL MyMathlab.com Videotape 3 Throughout this text, selected exercises are marked with the icon Aha~' . These `Aha!" exercises can be answered quite easily if the stu-dent pauses to inspect the exercise rather than proceed mechanically. This is done to discourage rote memoriza-tion. Some `Aha!" exercises are left unmarked to encour-age students to always pause before working a problem. Give the coordinates of each point. Determine if each vrdered pair is a solution of the given equation. 1. y=5x-3 2. y=xz-4x+3 3. y=(x+4)2 4. y= x+2 5. y=~x+2~ 1. A, B, C, D, E and F 2. G, H, I, J, K, and L Plot the points. Label each point with the indicated letter. 3. a(3,0), B(4,2), C(5,4), d(6,6), E(3,-4), F(3,-3) g(3,-2), H(3,-1) 4. a(1,1), b(2,3), c(3,5) d(4,7) e(-2,1) f(-2,2), g(-2,3), h(-2,4) j(-2,5) k(-2,6) 5. Plot the points M(2,3), N(5,-3) and P(-2,-3). Draw MN, NP, and MP. MN means the line seg- ment from m to n. What kind of geometric figure is formed? What is its area? 6. Plot the points Q(-4,3),R(5,3), S(2,-1) and T(-7,-1). Draw QR, RS, St and TQ. What kind of figure is formed? What is its area? Name the quadrant in which each point is located. 7. (-4, -9) 8. (2,17) 9. (-6,1) 10. (4, -8) 11. (3, 1) 2 12. (-1, -7) 13. (6.9, -2) 14. (-4, 31) 15. (1,-1);y=3x-4 16. (2,5);y=4x-3 17. (2, 4); 5s - t = 8 18. (1, 3); 4p + q = 5 19. (3, 5); 4x - y = 7 20. (2, 7); 5x - y = 3 21. (0,3 5); 6a + 5b = 3 22. (O,3 2); 3f + 4g = 6 23. (2, -1); 4r + 3s = 5 24. (2, -4); 5w + 2z = 2 25. (5, 3); x - 3y = -4 26. (l, 2); 2x - 5y = -6 27. (-1, 3); y = 3x2 28. (2, 4); 2r2 - s = 5 29. (2, 3); 5s2 - t = 7 30. (2, 3); Y = x3 - 5 Graph. 31. y=x+1 32. y=x+3 33. y = -x 34. y = 3x 35. y = 3x - 2 36. y = -4x + 1 37. y = -2x + 3 38. y = -3x + 1 39. y+2x=3 40. y+3x=1 41. y = -3x + 5 2 42. y = 2 -3x - 2 43. y = 3 4x + 1 44. y = 3 4x + 2 45. y = x2 + 1 46. y = x2 + 2 47. y = x2 - 3 48. y = x2 - 1 49. y = 5 - x2 50. y = 4 - x2 51• y = |x| + 1 52. y = |x| + 2 53• Y = |x| - 3 54. y = |x| - 1 55. What can be said about the location of two points that have the same first coordinates and second coordinates that are opposites of each other? 56. Examine Example 7 and explain why it is unwise to draw a graph after plotting just two points. SKILL MAINTENANCE Evaluate. 57. 5s - 3t, for s = 2 and t =,4 2.1 GRAPHS page 77 58. 2a + 7b, for a = 3 and b = 1 59. (3x - y)2, for x = 4 and y = 2 60. (2m + n)2, for m = 3 and n = 1 61. (5 - x)4(x + 2)3, for x = -2 62. 2x2 + 4x - 9, for x = -3 SYNTHESIS 63. Without making a drawing, how can you tell that the graph of y = x - 30 passes through three quadrants? 64. At what point will the line passing through (a, -1) and (a, 5) intersect the line that passes through (-3, b) and (2, b)? Why? 1 65. Graph y = 6x, y = 3x, y = 2 x, y = -6x, y = -3x, 1 and y = - 2 x using the same set of axes and com- pare the slants of the lines. Describe the pattern that relates the slant of the line to the multiplier of x. 66. Using the same set of axes, graph y = 2x, y = 2x - 3, and y = 2x + 3. Describe the pattern relating each line to the number that is added to 2x. 67. Match each sentence with the most appropriate of the four graphs shown. a) Roberta worked part time until September, full time until December, and overtime until Christmas. b) Clyde worked full time until September, half time until December, and full time until Christmas. c) Clarissa worked overtime until September, full time until December, and overtime until Christmas. d) Doug worked part time until September, half time until December, and full time until Christmas. Hours of work per week 10 20 30 40 50 60 September - December 68. Match each sentence with the most appropriate of the four graphs shown below. a) Carpooling to work, Terry spent 10 min on local streets, then 20 min cruising on the freeway, and then 5 min on local streets to his office. b) For her commute to work, Sharon drove 10 min to the train station, rode the express for 20 min, and then walked for 5 min to her office. c) For his commute to school, Roger walked 10 min to the bus stop, rode the express for 20 min and then walked for 5 min to his class. d) Coming home from school, Krisiy waited 10 min for the school bus, rode the bus for 20 min, and then walked 5 min to her house. Time from the start (in minutes) Time from the start (in minutes) I II III 70T IV 70 T , I 60 3 50 a. 40 3 30 0 20 10 x II a 60 ~ 60 60 ~ 50 ~ 50 3 50~ 0. 40 ~1 a dn 40 ----- ---- ~~ 30 y 30 30 ~ 20 c 20 0 20 '~ 10 y to LLI o ~ 0 5 10 15 20 25 30 35 40 `n 0 5 10 15 20 25 30 35 40 Sept. Dec. x Sept. Dec. Time from the start (in minutes) Time from the start (in minutes) page 77 page 78 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS 69. Which of the following equations have (- 1. 1) as a solution? 3 4 a)-3x - 3y= 1 2 -4 b) 8y - 15x = 7 2 c) 0.16y = -0.09x + 0.1 d) 2(-y + 2) - 1(3x - 1) = 4 4 ® Graph each equation after plotting at least 10 points. 70. y = 1/x; use x-values from -4 to 4 71. y = 1/(x - 2); use values of x from -2 to 6 72. y = / use values of x from 0 to 10 73. y = 1/x2; use values of x from -4 to 4 74. If (2, -3) and (-5, 4) are the endpoints of a diagonal of a square, what are the coordinates of the other two vertices? What is the area of the square? 75. If (-10, -2), (-3, 4), and (6, 4) are the coordinates of three vertices of a parallelogram, determine the coordinates of three different points that could serve as the fourth vertex. Note: Throughout this text, the icon ~i' is used to indi-cate exercises designed for graphers (graphing calcula-tors or computers). ~~' In Exercises 76 and 77, use a grapher to draw the graph of each equation. For each equation, select a window that shows the curvature of the graph. Then, if possible, create a table of ordered pairs in which x-values extend, by tenths, from 0 to 0.6. 76. a) y = -12.4x + 7.8 b) y=-3.5x2+6x-8 c) y=(x-3.4)3+5.6 77. a) y=2.3x4 + 3.4x2+ 1.2x -4 b) y=12.3x-3.5 c) y=3(x + 2.3)2 + 2.3 Functions Functions and Graphs • Function Notation and Equations • Applications: Interpolation and Extrapolation We now develop the idea of a function-one of the most important concepts in mathematics. In much the same way that the ordered pairs of Section 2.1 formed correspondences between first coordinates and second coordinates, a function is a special kind of correspondence between two sets. For example, To each person in a class there corresponds his or her biological mother. To each item in a shop there corresponds its price. To each real number there corresponds the cube of that number. In each example, the first set is called the domain. The second set is called the range. For any member of the domain, there is just one member of the range to which it corresponds. This kind of correspondence is called a function. Conespondence page 79 2.2 FUNCTIONS H7 Example Find the domain and the range of the function f shown here. . -5 -4 -3 -2 -1 1 2 3 4 5 . Solution Here f can be written {(-3, 1),(1,-2),(3,0),(4,5). The domain is the set of all first coordinates {-3,1,3,4) and the range is the set of all second coordinates (1,-2,0,5}. We can also find the domain and the range directly without first writing f. page 87 FOR EXTRA HELP ®. Exercise Set 2,2 Digital Video Tutor CD 2 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 3 ; Exercise Set 2.2 Determine whether each correspondence is a function. 1. 3 > a 5 > b 7 c 9 d e 2. 1 > a 2 b 3 c 4 d 5 3. Girl's Age (in months) Average Daily Weight Gain (in grams) 2 > 21.8 9 > 11.7 16 > 8.5 23 > 7.0 Source: American Family Physician, December 1993, p. 1435. 4. Boy's Age Average Daily (in months) Weight Gain (in grams) 2 > 24.3 9 > 11.7 16 > 8.2 23 > 7.0 Source: American FamilyPhysician, December 5 x 1993, p. 1435. 5. Predator Prey cat dog fish worm dog > cat tiger fish bat > mosquito 6. Olympics Site Year Lake Placid 1980 Calgary 2002 Squaw Valley 1960 Salt Lake City 1988 1932 Determine whether each of the following is a function. Identify any relations that are not functions. Domain Correspondence Range 7. A yard full of Each tree's A set of Christmas trees price prices 8. The swordfish Each fish's A set of stored in a boat weight weights 9. The members An instrument A Set of of a rock band the person instruments can play 10. The students Each person's A set of in a math class seat number numbers 11. A set of Square each A set of numbers number and numbers then add 4. 12. A set of The area A set of shapes of each shape numbers For each graph of a function, determine (a) f(1); (b) the domain; (c) any x-values for which f(x) = 2; and (d) the range. 13 through 34 are all graphs. a graph is y 5 4 3 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x -1 -2 -3 -4 -5 page 88 Find the function values. 35. g(x) = x + 3 a) g(0) b) g(-4) c) g(-7) d) g(8) e) g(a + 2) 36. h(x)=x-2 a) h(4) b) h(8) c)h(-3) d) h(-4) e) h(a - 1) 37. f(n) =5n2 + 4n a) f(0) b) f(-1) c) f(3) d) f(t) e) f(2a) 38. g(n) = 3n2 - 2n a) g(0) b) g(-1) c) g(3) d) g(t) e) g(2a) 39. f(x) = x - 3 2x - 5 a) f(0) b)f(4) c) f(-1) d) f(3) e)f(x + 2) 40. s(x) = 3x - 4 2x + 5 a) s(10) b) s(2) c) s(1) 2 d) s(-1) e) s(x + 3) 41. Find the domain of f. a) f(x) = 5 b) f(x) = 7 x-3 6-x page 88 page 89 2.2 Functions c) f(x) = 2x + 1 d) f(x) = x2 + 3 e) f(x) = 3 f) f(x) = |3x - 4| 2x - 5 42. Find the domain of g. a) g(x) = 3 b) g(x) =|5 - x| x-1 c) g(x) = 9 d) g(x)= 4 x+3 3x + 4 e) g(x) = x3 - 1 f) g(x) = 7x - 8 The function A described by A(s)= s2 V3 gives the area of a equilateral triangle with side s. 4 43. Find the area when a side measures 4 cm. 44. Find the area when a side measures 6 in. 45. Find the area when the radius is 3 in. 46. Find the area when the radius is 5 cm. Chemistry. The function F described by F(C) = 9 C + 32 5 gives the Fahrenheit temperature corresponding to the Celsius temperature C. 47. Find the Fahrenheit temperature equivalent to -10°C. 48. Find the Fahrenheit temperature equivalent to 5°C. Archaeoloay The function H described by H(x) 2.75x + 71.48 can be used to predict the height in centimeters of a woman whose humerus the bone from the elbow to the shoulder is x cm long. Predict the height of a woman whose humerus is the length given. 49. 32 cm 50. 35 cm Heart attacks and cholesterol. For Exercises 51 and 52 use the following graph which shows the annual heart attack rate per 10,000 men as a function of blood choles- teral level. 200 150 100 50 0 100 150 200 250 300 Blood cholesterol in milligrams per deciliter. 51. Approximate the annual heart attack rate for those men whose blood cholesterol level is 225 mg/dl. That is, find H(225). 52. Approximate the annual heart attack rate for those men whose blood cholesterol level is 275 mg/dl. That is, find H(275). Voting attitudes For Exercises 53 and 54, use this graph, which shows the percentage of people responding yes to the question, "If your (political) party nominated a generally well-qualified person for president who hap- pened to be a woman, would you vote for that person?" (Source: The New York Times, August 13, 2000) 100% 80 60 40 20 1940 1960 1980 2000 Year 2020 53. Approximate the percentage of Americans willing to vote for a woman for president in 1960. That is, find P(1960). 54. Approximate the percentage of Americans willing to vote for a woman for president in 2000. That is, fmd P(2000). *Copyright 1989, CSPI. Adapted from Nutrition Action Healthletter (1875 Connecticut Avenue, N.W, Suite 300, Washington, DC 20009-5728. $24 for 10 issues). page 90 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Blood alcohol level The following table can be used to predict the number of drinks required for a person of a specified weight to be considered legally intoxicated (blood alcohol level of 0.08 or above). One 12-oz glass of beer, a 5-oz glass of wine, or a cocktail containing 1 oz of a distilled liquor all count as one drink. Assume that all drinks are consumed within one hour. 12 oz 5 oz 1 oz Input, Body Weight Output, (in pounds) Number of Drinks 100 2.5 160 4 180 4.5 200 5 55. Use the data in the table above to draw a graph and to estimate the number of drinks that a 140-1b per-son would have to drink to be considered intoxicated. 56. Use the graph from Exercise 55 to estimate the number of drinks a 120-1b person would have to drink to be considered intoxicated. Incidence of AIDS The following table indicates the number of cases of AIDS reported in each of several years (Source: U.S. Centers for Disease Control and Prevention). Input Year Output, Number of Cases Reported 1994 77,103 1996 66,497 1998 48,269 57. Use the data in the table above to draw a graph and to estimate the number of cases of AIDS reported in 2001. 58. Use the graph from Exercise 57 to estimate the number of cases of AIDS reported in 1997. Population growth. The town of Falconburg recorded the following dates and populations. Input, Output Population (in Year tens of thousands) 1995 5:8 1997 6 1999 7 2001 7.5 59. Use the data in the table above to draw a graph of the population as a function of time. Then estimate what the population was in 1998. 60. Use the graph in Exercise 59 to predict Falconburg's population in 2003. 61. Retailing Shoreside Gifts is experiencing con-stant growth. They recorded a total of $250,000 in sales in 1996 and $285,000 in 2001. Use a graph that displays the store's total sales as a function of time to predict total sales for 2005. 62. Use the graph in Exercise 61 to estimate what the total sales were in 1999. Researchers at Yale University have suggested that the following graphs* may represent three different aspects of love. Passion Intimacy Commitment Level Time 63. In what unit would you measure time if the hori- zontal length of each graph were ten units? Why? 64. Do you agree with the researchers that these graphs should be shaped as they are? Why or why not? *From "A Triangular Theory of Love," by R. J. Sternberg, 1986, Psy-chological Review, 93(2), 119-135. Copyright 1986 by the Ameri-can Psychological Association, Inc. Reprinted by permission. 2.2 FUNCTIONS page 91 65. 10 - 3 2 9 - 2 . 3 66. 2 4 - 10 6 - 4 . 3 67. The surface area of a rectangular solid of length l, width w, and height h is given by S = 2lh + 2lw + 2wh. Solve for l. 68. Solve the formula in Exercise 67 for w. Solve for y. 69.2x + 3y=6 70.5x - 4y = 8 SYNTHESIS 71. Which would you trust more and why: estimates made using interpolation or those made using extrapolation? 72. Explain in your own words why every function is a relation, but not every relation is a function. For Exercises 73 and 74, let f(x) = 3x2 - 1 and g(x) = 2x +5. Simplify. 73. Find f(g(-4)) and g( f(-4)). 74. Find f(g(-1)) and g( f(-1)). Programming For Exercises 75-78, use the following graph of a woman's "stress test." This graph shows the size of a pregnant woman's contractions as a function of time. 25 20 was up higher 3 and 5 1/2 minutes. 75. How large is the largest contraction that occurred during the test? 76. At what time during the test did the largest con- traction occur? 77. On the basis of the information provided, how large a contraction would you expect 60 seconds after the end of the test? Why ? 78. What is the frequency of the largest contraction? 79. The greatest integer function f(x) = [x] is defined as follows: [x] is the greatest integer that is less than or equal to x. For example, if x = 3.74, then [x] = 3; and if x = -0.98, then [x]= -1. Graph the greatest integer function for -5 <- x < 5. (The no- tation f(x) = INT[x] is used in many graphers and computer programs.) 80. Suppose that a function g is such that g(-1) = -7 and g(3) = 8. Find a formula for g if g(x) is of the form g(x) = mx + b, where m and b are constants. 81. Energy expenditure. On the basis of the informa- tion given below, what burns more energy: walking 4 1 2 mph for two hours or bicycling 14 mph for one hour? Approximate Energy Expenditure by a 150-Pound Person in Various Activities Calories Activity per Hour Walking, 2 1 2 mph 210 Bicycling, 5 1 2 mph 210 Walking, 3 3 4 mph 300 Bicycling, 13 mph 660 Source: Based on material prepared by Robert E. Johnson, M.D., Ph.D., and colleagues, University of Illinois. Time (in minutes) 92 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS CORNER Calculating License Fees . , Focus: Functions group member can find the fee for a few dif- W Time. 15-20 minutes ferent years. Group size: 3-4 3. Graph the results from part (2). On the x-axis, plot years beginning with the year of pur- chase, and on the y-axis, plot V(x), the VLF as a function of year. 4. What is the lowest VLF that the owner of this car will ever have to pay, according to this schedule? Compare your group's answer with other groups' answers. 5. Does your group feel that California's method for calculating registration fees is fair? Why or why not? How could it be improved? The California Department of Motor Vehicles calculates automobile registration fees (VLF) according to the schedule shown below. ACTIVITY 1. Determine the original sale price of the oldest vehicle owned by a member of your group. If necessary, use the price and age of a family member's vehicle. Be sure to note the year in 6. Try, as a group, to find an algebraic form for which the car was purchased. the function y = V(x). 2. Use the schedule below to calculate the vehi- 7. Optional out-of-class extension: Create a pro- cle license fee (VLF) for the vehicle in part (1) gram for a grapher that accepts two inputs above for each year from the year of purchase (initial value of the vehicle and year of pur- to the present. To speed your work, each chase) and produces V(x) as the output. DMAV VEHICLE LICENSE FEE INFORMATION A Public Service Agency The 2% Vehicle License Fee (VLF) is in lieu of a personal PERCENTAGE SCHEDULE property tax on vehicles. Most VLF revenue is returned to Rev. & Tax. Code Sec. 10753.2 City and County Local Governments (see reverse side). (Trailer coaches have a different schedule) The license fee charged is based upon the sale price or 1st Year 100% 7th Year 40% vehicle value when initially registered in California. The 2nd Year 90% 8th Year 30% vehicle value is adjusted for any subsequent sale or 3rd Year 80% 9th Year 25% transfer, that occurred 8/19/91 or later, excluding sales 4th Year 70% 10th year 20% or transfers between specified relatives. 5th Year 60% 11th Year 6th Year 50% onward 15% The VLF is calculated by rounding the sale price to the nearest odd hundred dollar. That amount is reduced by VLF CALCULATION EXAMPLE a percentage utilizing an eleven year schedule (shown to the right), and 2% of that amount is the fee charged. Purchase Price: $9,199 See the accompanying example for a vehicle purchased Rounded to: $9,100 last year for $9,199. This would be the second registration Times the Percentage: 90% year following that purchase. Equals Fee Basis of: $8,190 Times 2% Equals: $163.80 WHERE DO YOUR DMV FEES GO? SEE REVERSE SIDE. Rounded to: $164 DMV77 S(REV8/95) 9530123 2.3 LINEAR FUNCTIONS: SLOPE, GRAPHS, AND MODELS page 99 E x a m p 1 e 1 1 ` Cellular phone use. In 1999, there were approximately 86 million cellular . phone customers in the United States. By 2001, the figure had grown to ` 110 million (Source: based on data from the Cellular Telecommunications Industry Association). At what rate is the number of cellular phone customers changing? Solution The rate at which the number of customers is changing is given b~ Rate of change = change in number of customers change in time 110 million customers - 86 million customers 2001 - 1999 _ 24 million customers 2 years = 12 million customers per year. Between 1999 and 2001, the number of cellular phone customers grew at a rate of approximately 12 million customers per year. Example 12 Running speed. Stephanie runs 10 km during each workout. For the first 7 km, her pace is twice as fast as it is for the last 3 km. Which of the following graphs best describes Stephanie's workout? 6 V ° 4 :°. /~ ~0 2 ,/ j ;.d 2 0 4 'c 0 10 20 30 40 50 60 "c 0 10 20 30 40 50 60 ' a c Time (in minutes) ~ Time (in minutes)  2 ~d 2 c a 'c 0 10 20 30 40 50 60 's~ 0 10 20 30 40 50 60 a c~ 4  8    Time (in minutes) ~ Time (in minutes) Solution The slopes in graph A increase as we move to the right. This would indicate that Stephanie ran faster for the last part of her workout. Thus graph A is not the correct one. The slopes in graph B indicate that Stephanie slowed down in the middle of her run and then resumed her original speed. Thus graph B does not cor-rectly model the situation either.  100 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS  According to graph C, Stephanie slowed down not at the 7-km mark, but at the 6-km mark. Thus graph C is also incorrect. Graph D indicates that Stephanie ran the first 7 km in 35 min, a rate of 0.2 km/min. It also indicates that she ran the final 3 km in 30 min, a rate of 0.1 km/min. This means that Stephanie's rate was twice as fast for the first 7 km, so graph D provides a correct description of her workout. Linear functions arise continually in today's world. As with the rate problems appearing in Examples 1Q-12, it is critical to use proper units in all answers. Example 1 3 ; Salvage value. Tyline Electric uses the function S(t) _ -700t + 3500 to ; determine the salvage value S(t), in dollars, of a color photocopier t years after its purchase. a) What do the numbers -700 and 3500 signify? b) How long will it take the copier to depreciate completely? c) What is the domain of S?  Solution Drawing, or at least visualizing, a graph can be useful here. 5(t)   Number of years of use a) At time t = 0, we have S(0) _ -700 - 0 + 3500 = 3500. Thus the number 3500 signifies the original cost of the copier, in dollars. This function is written in slope-intercept form. Since the output is measured in dollars and the input in years, the number -700 signifies that the value of the copier is declining at a rate of $700 per year. b) The copier will have depreciated completely when its value drops to 0. To learn when this occurs, we determine when S(t) = 0: S(t) = 0 A graph is not always available. -700t + 3500 = 0 Substituting -700t + 3500 for S(t) -700t = -3500 Subtracting 3500 from both sides t = 5. Dividing both sides by -700 The copier will have depreciated completely in 5 yr. 2.3 LINEAR FUNCTIONS: SLOPE, GRAPHS, AND MODELS 1 O1 c) Neither the number of years of service nor the salvage value can be nega-tive. In part (b) we found that after 5 yr the salvage value will have dropped to 0. Thus the domain of S is {t~ 0 < t <- 5}. The graph above serves as a visual check of this result. Page 101 Exercise Set 2,3 Digital Video Tutor CD 2 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 3 Graph. 1. f(x)= 2x - 7 7 -2 =5 2. g(x)= 3x - 7 7 -3 =4 3. g(x)=-1 3x + 2 3+1=4 4 +2 =6 4. f(x)=-1 2x - 5 2-1=1 5 -1 =4 5. h(x)= 2 5x-4 5+2=7 7 -4 =3 6. h(x)= 4 5x + 2 Determine the y-intercept. 7. y=5x + 7 8. y=4x - 9 9. f(x) = -2x - 6 10. g(x) = -5x + 7 11. y=-3 5x - 4.5 12. y= 15 7 x + 2.2 13. g(x) = 2.9x - 9 14. f(x) = -3.1x + 5 15. y = 37x + 204 16. y = -52x + 700 For each pair of points, find the slope of the line containing them. 17. (6, 9) and (4, 5) 18. (8, 7) and (2, -1) 19. (3, 8) and (9, -4) 20. (17, -12) and (-9, -15) 21. (-16.3,12.4) and (-5.2, 8.7) 22. (12.4, -5.8) and (-14.5, -15.6) appropriate units. See Example 10. 23. (-9.7, 43.6) and (4.5, 43.6) 24. (-2.8, -3.1) and (-1.8, -2.6) 25. y= 5 2x+3 26. y= 2 5x + 4 Determine the slope and the y-intercept. Then draw a graph. Be sure to check as in Example 5 or Example 8. 27. f(x)=-5 2x+ l 28. f(x)=-2 5x + 3 29. 2x - y= 5 30. 2x + y= 4 Number of years of use Number of seconds 31. f (x) = 1 3 x + 2 32. f (x) = -3x + 6 spent running 33.7y + 2x=7 34.4y + 20=x Ah~ 35. f(x) = -0.25x 36. f(x) = 1.5x - 3 37. 4x - 5y = 10 38. 5x + 4y = 4 39. f(x) = 5 4x - 2 40. f(x) = 4 3x + 2 41. 12 - 4f(x) = 3x 42. 15 + 5f(x) = -2x Aha~~ 43. g(x) = 2.5 44. g(x) = 3 4x Find a linear function whose graph has the given slope and y-intercept. 2 45. Slope 3, y-intercept (0, -9) 3 46. Slope -4, y-intercept (0, 12) 47. Slope -5, y-intercept (0, 2) 48. Slope 2, y-intercept (0, -1) 7 49. Slope -9, y-intercept (0, 5) 4 50. Slope -11, y-intercept (0, 9) 1 51. Slope 5, y-intercept (0, 2) 2 52. Slope 6, y-intercept (0, 3) For each graph, ~nd the rate of change. Remember to use 53. 36 30 24 18 12 6 0 1 2 3 54. 120 100 80 60 40 20 0 2 4 6 8 10 12 page 1O2 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS 55. through 60. are graphs 55. Four dots left lower corner to upper right corner. 2,0 4,1 8,2 10, 3 56. 7 dots 1,0 1,1 2,2 2,3 3,4 3,5 4,6 dots 57. 5 dots 50,0 120, 1 200, 2 300, 3 350, 4 58. 4 dots 5,0 6,4 7,8 8,12 Number of minutes spent running. 59. 4 dots 470,15 490,35 510,55 530,75 60. 4 dots 500,35 510,45 520,55 530,65 540 530 520 510 500 490 480 15 25 35 45 55 65 75 85 61. Skiing rate. A cross-country skier reaches the 3-km mark of a race in 15 min and the 12-km mark 45 min later. Find the speed of the skier. 12km 45km 62. Running rate. An Olympic marathoner passes the 5-mi point of a race after 30 min and reaches the 25-mi point 2 hr later. Find the speed of the marathoner. 63. Rate of computer hits. At the beginning of 1999, SciMor.com had already received 80,000 hits at their website. At the beginning of 2001, that num- ber had climbed to 430,000. Calculate the rate at which the number of hits is increasing. 64. Work rate. As a painter begins work, one fourth of a house has already been painted. Eight hours later, the house is two-thirds done. Calculate the painter's work rate. 65. Rate of descent. A plane descends to sea level from 12,000 ft after being airborne for 1 1 hr. The 2 entire flight time is 2 hr and 10 min. Determine the average rate of descent of the plane. 66. Growth in overseas travel. In 1988, the number of U.S. visitors overseas was about 11.6 million. In 1995, the number grew to 18.7 million (Source: Statistical Abstract of the United States). Deter- mine the rate at which the number of U.S. visitors overseas was growing. 67. Nursing. Match each sentence with the most appropriate of the four graphs shown. a) The rate at which fluids were given intra- venously was doubled after 3 hr. b) The rate at which fluids were given intra- venously was gradually reduced to 0. c) The rate at which fluids were given intra- venously remained constant for 5 hr. d) The rate at which fluids were given intra- venously was gradually increased. 500 y y 400 300 II i, .a 900 , ^, 800 °-" .. 700 b ~ 600 ^ 500 y y 400 ,q 300 o " 200 loo 1 5 9 1 5 9 -._. ._ ~ ~ ~- P.M. P.M. P.M. ~ P.M. P.M. P.M. *Based on data from the College Board Online. Time of day Time of day g ~ zoo 3 0 1 2 3 4 5 6 Number of bags of feed used ~. 540 sso 520 510 500 ~ 490 ~ 4so I ,D 900 Q, 800 700 600 page 103 2.3 LINEAR FUNCTIONS: SLOPE, GRAPHS, AND MODELS 1O3 III 68. Market Research Match each sentence with the most appropriate graph below. a) After January 1, daily sales continued to rise, but at a slower rate. Graph I b) After January 1, sales decreased faster than they ever grew. Graph IV. c) The rate of growth in daily sales doubled after January 1. Graph III d) After January 1, daily sales decreased at half the rate that they grew in December. Graph II In Exercises 69-78, each model is of the form f(x) = mx + b. In each case, determine what m and b signify. 69. Catering When catering a party for x people, Jennette's Catering uses the formula C(x) = 25x + 75, where C(x) is the cost of the party, in dollars. 70. Weekly Pay Each salesperson at Knobby's Furni- ture is paid P(x) dollars, where P(x) = 0.05x + 200 and x is the value of the salesperson's sales for the week. 71. Hair growth. After Tina gets a "buzz cut," the length L(t) of her hair, in inches, is given by L(t) = 1 2 t + 1, where t is the number of months after she gets the haircut. 72. Landscaping After being cut, the length G(t) of the lawn, in inches, at Great Harrington Commu- nity College is given by G(t) = 1 8 t + 2, where t is the number of days since the lawn was cut. 73. Life expectancy of American women. The life ex- pectancy of American women t years after 1950 is given by A(t) = 3 20 t + 72. 74. Natural gas demand. The demand, in quadrillions of joules, for natural gas is approxi- mated by D(t) = 1 5 t + 20, where t is the number of years after 1960. 75. Sales of cotton goods. The function given by f(t) = 2.6t + 17.8 can be used to estimate the yearly sales of cotton goods, in billions of dollars, t years after 1975. 76. Cost of a movie ticket. The average price P(t), in dollars, of a movie ticket is given by P(t) = 0.1522t + 4.29, where t is the number of years since 1990. 77. Cost of a taxi ride. The cost, in dollars, of a taxi ride in Pelham is given by C(d) = 0.75d + 2, where d is the number of miles traveled. 0.75 + 2 =77 78. Cost of renting a truck. The cost, in dollars, of a one-day truck rental is given by C(d) = 0.3d + 20, where d is the number of miles driven. $9 s 6 G s 4 °' 3 ;, 2 i Dec. Jan. Feb. ` De c. Jan. Feb. \ i f > > i i Dec. Jan. Feb. \ i i i i I i Dec. Jan. Feb. Iv 900 soo P a .. 700 boo :- 500 ~ 400 ~ soo -- is given by 200 E v ioo 1 5 9 \ 1 5 9 P.M. P.M. P.M. P.M. P.M. P.M. Time of day Time of day page 1O4 CHAPTER 2 GRAPHS, FUNCTIQNS, AND LINEAR EQUATIONS 79. Salvage value. Green Glass Recycling uses the function given by F(t) = -5000t + 90,000 to de- termine the salvage value F(t), in dollars, of a waste removal truck t years after it has been put into use. a) What do the numbers -5000 and 90,000 signify? b) How long will it take the truck to depreciate completely? -5000 can represent number of days or years. c) What is the domain of F? 80. Salvage value Consolidated Shirt Works uses the function given by V(t) = -2000t + 15,000 to determine the salvage value V(t), in dollars, of a color separator t years after it has been put into use. a) What do the numbers -2000 and 15,000 signify? -2000 is how long in days or in years. b) How long will it take the machine to depreci-ate completely? -2000 c) What is the domain of V? 81. Trade-in value. The trade-in value of a Homelite snowblower can be determined using the func- tion given by v(n) = -150n + 900. Here v(n) is the trade-in value, in dollars, after n winters of use. a) What do the numbers -150 and 900 signify? b) When will the trade-in value of the snow- blower be $300? c) What is the domain of v? 82. Trade in value The trade-in value of a John Deere riding lawnmower can be determined using the function given by T(x) = -300x + 2400. Here T(x) is the trade-in value, in dollars, after x summers of use. a) What do the numbers -300 and 2400 signify? b) When will the value of the mower be $1200? c) What is the domain of T? 83. Economics. In 2000, the federal debt could be modeled using D(t) = mt + 6000, where D(t) is in billions of dollars tyears after 2000, and m is a constant. If you were president of the United States, would you want m to be positive or negative? Why? 84. Economics Examine the function given in Exer- cise 83. What units of measure must be used for m? Why? SKILL MAINTENANCE Solve. 85.2x - 5= 7x + 3 86.4t + 9=t - 6 87.1x + 7 =2 5 2 88. -3t + 4=t - 1 89. 3 * 0 - 2y= 9 90. 4x - 7 * 0= 3 SYNTHESIS 91. Hope Diswerkz claims that her firm's profits con- tinue to go up, but the rate of increase is going down. a) Sketch a graph that might represent her firm's profits as a function of time. b) Explain why the graph can go up while the rate of increase goes down. 92. Belly Up, Inc., is losing $1.5 million per year while Spinning Wheels, Inc., is losing $170 an hour. Which firm would you rather own and why? In Exercises 93 and 94, assume that r, p, and s are con- stants and that x and y are variables. Determine the slope and the y-intercept. 93. rx + py = s 94. rx + py = s - ry 95. Let (xl, yl) and (x2, y2) be two distinct points on the graph of y = mx + b. Use the fact that both pairs are solutions of the equation to prove that m is the slope of the line given by y = mx + b. (Hint: Use the slope formula.) Given that f(x) = mx + b, classify each of the following as true or false. 96. f(c + d) = f(c) + f(d) 97. f(cd) = f(c)f(d) 98. f(kx) = kf(x) 99. f(c - d) = f(c) - f(d) 100. Find k such that the line containing (-3, k) and (4, 8) is parallel to the line containing (5, 3) and (1 -6). page 105 z•3 LINEAR FUNCTIONS: SLOPE, GRAPHS, AND MODELS 101. Match each sentence with the most appropriate graph below. a) Annie drove 2 mi to a lake, swam 1 mi, and then drove 3 mi to a store. is graph III b) During a preseason workout, Rico biked 2 mi, ran for 1 mi, and then walked 3 mi. is graph I c) James bicycled 2 mi to a park, hiked 1 mi over the notch, and then took a 3-mi bus ride back to the park. is graph IV d) After hiking 2 mi, Marcy ran for 1 mi before catching a bus for the 3-mi ride into town. is graph II 4 III i 8 .~ ~ I c 61 5 '~ 2 u 1 c y3 Y - b 0 10 20 30 40 50 60 Time from the start F°-~ (in minutes) 102. Find the slope of the line that contains the given pair of points. a) (5b, -6c), (b, -c) b) (b, d), (b, d + e) c) (c+f, a + d),(c-f,-a - d) 103. Cost of a speeding ticket. The penalty schedule shown at the top of the next column is used to determine the cost of a speeding ticket in certain states. Use this schedule to graph the cost of a speeding ticket as a function of the number of miles per hour over the limit that a driver is going. 104. Graph the equations y1 = 1.4x + 2. y2 = 0.6x + 2 y3 = 1.4x + 5, and y4 = 0.6x + 5. using a grapher. If possible, use the SIMULTANEOUS mode so that you cannot tell which equation is being graphed first. Then decide which line corresponds to each equation. Time from the start Time from the start ~in minutes) a ~in minutes) ~ 105. A student makes a mistake when using a grapher to draw 4x + 5y = 12 and the following screen appears. 10 10 10 10 Use algebra to show that a mistake has been made. What do you think the mistake was? 106. A student makes a mistake when using a grapher to draw 5x - 2y = 3 and the following screen appears. io io 10 10 Use algebra to show that a mistake has been made. What do you think the mistake was? Time from the start F (in minutes) STANDING VIOLATION gpEE FINES 1 _ 10 mPh overll m!R: $6.OOImPh Pilus $17.50 surchar9e 1 ~ _ 20 mPh over limit: $7.OOImP p 21 -30 mp h lus $17.50 surchar9e 31+ mph over limit $9.OOImPh Ptus $17.50 surcharge Officer wilt enter mph over limtt ln tine 5a page 106 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Another Look at 2.4 Linear Graphs Zero Slope and Lines with Undefined Slope • Graphing Using Intercepts • Solving Equations Graphically • Recognizing Linear Equations In Section 2.3, we graphed linear equations using slopes and y-intercepts. We now graph lines that have slope 0 or that have an undefined slope. We also graph lines using both x- and y-intercepts and learn how to use graphs to solve certain equations. Zero Slope and Lines with Undefined Slope If two different points have the same second coordinate, what is the slope of the line joining them? In this case, we have y2 = yl, so m = y2 - y1 = 0 = 0 x2 - x1 = x2 - x1 Horizontal line: slope=0 y x1 - y1 x2 - y1 Slope of a Horizontal Line Every horizontal line has a slope of 0. Example 1 Graph: f(x) = 3. Solution Recall from Example 5(c) in Section 2.2 that a function of this type is called a constant function. Writing slope-intercept form, f(x)=0 * x + 3, we see that the y- intercept is (0, 3) and the slope is 0. we can graph f by plotting the point (0,3) and, from there, determining a slope of 0. Because 0 = 0/2 (any nonzero number could be used in place of 2), we can draw the graph by going up 0 units and to the right 2 units. As a check, we also find some ordered pairs. Note that for any choice of x-value, f (x) must be 3. page 113 Example 3x + 6 = 2x - 1 -4 3x + 7 = 2x Adding 1 to both sides. -4 7 = 11 x 3 4 Adding 4 x to both sides. 28 = x. Multiplying both sides by 4 11 11 The solution is 28 or about 2.55. 11 page 115 Digital Video Tutor CD 2 InterAd Math Math Tutor Center MathXL MyMathLab.com Videotape 4 For each equation, find the slope. If the slope is unde- 51. x - 7 = 3x - 5 . 52. x + 3 = 5 - x fined, state this. 53. 3-x=2 x-3 54. 5-Zx=x-4 page 115 1. y-7=5 2. x + 1 = 7 3. 3x = 6 4. y - 3=5 5. 4y= 20 6. 19= -6y 7. 9 + x=12 would be 9 + 3= 12 since x becomes a 3. 8. 2x = 18 9. 2x - 4=3 10. 5y - 1 =16 11. 5y - 4 =35 12. 2x - 17 =3 13. 3y + x=3y + 2 14. x-4y=12 - 4y 15. 5x - 2 =2x - 7 16. 5y + 3=y + 9 AW~ 17. y=- 2 3x + 5 18. y=-3 2x + 4 Graph. 19. y = 4 20. x = -1 21. x = 2 22. y = 5 23. 4 * f(x) = 20 24. 6 * g(x) = 12 25. 3x = -15 26. 2x = 10 27. 4 - g(x) + 3x= 12 + 3x 28. 3-f(x)=2 29. x + y=5 30. x + y=4 31. y = 2x + 8 32. y = 3x + 9 33. 3x + 5y = 15 34. 5x - 4y = 20 35. 2x - 3y = 18 36. 3x + 2y = 12 37. 7x = 3y - 21 38. 5y = -15 + 3x 39. f(x) = 3x - 8 40. g(x) = 2x - 9 41. 1.4y - 3.5x = -9.8 42. 3.6x - 2.1y = 22.68 43. 5x + 2g(x)=7 44. 3x - 4f(x) = 11 45. x - 3 = 4 46. x + 4 = 6 47. 3x - 4 = 1 48. 2x + 1 = 7 49. 1x + 3 = -1 2 50. 1 3 x - 2 = 1 51. x - 7 = 3x - 5 52. x + 3 = 5 - x 53. 3 - x = 1x - 3 2 54. 5 - 1x = x - 4 2 55. 2x + 1= -x + 7 56. -3x + 4= 3x - 4 Use a graph to estimate the solution in each of the following. Be sure to use graph paper. 2.4 ANOTHER LOOK AT LINEAR GRAPHS 11 5 57. Telephone charges Skytone Calling charges $50 for a telephone and $25 per month under its economy plan. Estimate the time required for the total cost to reach $150. 58. Cellular phone charges. The Cellular Connection charges $60 for a cellular phone and $40 per month under its economy plan. Estimate the time required for the total cost to reach $260. ` 40 divide into 260 6 months 59. Parking fees, Karla's Parking charges $3.00 to park plus 50 cents for each 15-min unit of time. Esti- mate how long someone can park for $7.50.* 60. Cost of a road call. Dave's Foreign Auto Village charges $35 for a road call plus $10 for each 15-min unit of time. Estimate the time required for a road call that cost $75. 61. Cost of a FedEx delivery In 2001, for Standard de- ivery, within 150 mi, of packages weighing from 6 to 16 1b, FedEx charged $18.75 for the first 6 1b plus $0.75 for each additional pound. Estimate the weight of a package that cost $24 to ship. page 116 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS 62. Copying costs A local Mailboxes Etc.® store charges $2.25 for binding plus 5cents per page for each spiralbound copy of a town report. Estimate the length of a spiralbound report that cost $3.50. Determine whether each equation is linear. Find the slope of any nonvertical lines. 63. 5x - 3y = 15 64. 3x + 5y + 15 = 0 65. 16 + 4y = 10 66. 3x - 12 = 0 67. 3g(x) = 6 x 2 68. 2x + 4f(x) = 8 69. 3y = 7(2x - 4) 70. 2(5 - 3x) = 5y 71. g(x) - 1 = 0 8 72. f(x) + 1 = 0 x 73. f(x) = x2 5 74. g(x) = 3 + x 2 75. Meteorology. Wind chill is a measure of how cold the wind makes you feel. Below are some measurements of wind chill for a 15-mph breeze. How can you tell from the data that a linear function will give an approximate fit? Temperature 15-mph wind chill 30° 9° 25° 2° 20° -5° 15° -11° 10° -18° 5° -25° 0° -31° 76. Engineering Wind friction, or air resistance, in-creases with speed. At the top of the next column are some measurements made in a wind tunnel. Plot the data and explain why a linear function does or does not give an approximate fit. Force of Velocity Resistance (in kilometers per hour) (in newtons) 10 3 21 4.2 34 6.2 40 7.1 45 15.1 52 29.0 Simplify. SKILL MAINTENANCE 77. - 3 * 7 7 3 78. 5 (-4) 4 5 79. -5[x - (-3)] 80. -2[x - (-4)] 81. 2[x-(-1)]-1 3 2 82. - 3(x - 2)- 3 2 5 SYNTHESIS 83. Jim tries to avoid fractions as often as possible. Under what conditions will graphing using inter-cepts allow him to avoid fractions? Why? 84. Under what condition(s) will the x- and y-intercepts of a line coincide? What would the equation for such a line look like? 85. Give an equation, in standard form, for the line whose x-intercept is 5 and whose y-intercept is -4. 86. Find the x-intercept of y = mx + b, assuming that m =/ 0. In Exercises 87-90, assume that r, p, and s are nonzero constants and that x and y are variables. Determine whether each equation is linear. 87. rx + 3y=p - s 88. py= sx - ry + 2 89. r2x = py + 5 90. x - py = 17 r 91. Suppose that two linear equations have the same y-intercept but that equation A has an x-intercept that is half the x-intercept of equation B. How do the slopes compare? Consider the linear equation ax + 3y = 5x - by + 8. 92. Find a and b if the graph is a horizontal line passing through (0, 2). page 117 93. Find a and b if the graph is a vertical line passing through (4, 0). 94. (Refer to Exercise 59.) It costs as much to park at Karla's for 16 min as it does for 29 min. The linear graph drawn in the solution of Exercise 59 is not a precise representation of the situation. Draw a graph with a series of "steps" that more accurately reflects the situation. 95. (Refer to Exercise 60.) A 32-min road call with Dave's costs the same as a 44-min road call. The linear graph drawn in the solution of Exercise 60 is not a precise representation of the situation. Draw a graph with a series of "steps" that more accurately reflects the situation. 96. 5x + 3 = 7 - 2x 97. 4x - 1 = 3 - 2x 98. 3x - 2 = 5x - 9 99. 8 - 7x = -2x - 5 Solve using a grapher. 100. Weekly pay at Bikes for Hikes is $219 plus a 3.5% sales commission. If a salespersoris pay was $370.03, what did that salesperson's sales total? 2.5 OTHER EQUATIONS OF LINES 1.I7 101. It costs Bert's Shirts $38 plus $2.35 a shirt to print tee shirts for a day camp. Camp Weehawken paid Bert's $623.15 for shirts. How many shirts were printed? Other Equations 2.5 Of Lines Point-Slope Equations • Parallel and Perpendicular Lines Specifying a line's slope and one point through which the line passes enables us to draw the line. In this section, we study how this same information can be used to produce a equation of the line. The ability to do this is important in more advanced courses. Point-Slope Equations Suppose that a line of slope m passes through the point (xl, yl). For any other point (x, y) to lie on this line, we must have Y - Y1 = (x - xl) m It is tempting to use this last equation as an equation of the line of slope m that passes through (x1,yl). The only problem with this form is that when x and y are replaced with xl and yl, we have 0 = m, a false 0 equation. To avoid this difficulty we multiply both sides by x - xl and simplify: (x - xl) y - yl = m x - xl. Multiplying both sides by x - xl x-xl x - x1 y - y1 m(x - x1). Removing a factor equal to 1: x - x1 = x - x1 1 This is the point-slope form of a linear equation. page 118 11H CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Point-Slope equation The point-slope equation of a line with slope m, passing through (x1 y1), is y - y1=m(x - xl) Example 1 Find and graph an equation of the line passing through (3, 4) with slope - 1. 2 If you are finding it difficult to master a particular topic or concept, talk about it with a classmate. Verbalizing your questions about the material might help clarify it for you. If your classmate is also having difficulry, it is possible that a majority of your classmates are confused and you can ask your instructor to explain the concept again. Solution We substitute in the point-slope equation: y-y1 = m(x - x1) 1 y - 4 = -2 (x - 3). Substituting 1 To graph this point-slope equation, we count off a slope of -2, starting at (3, 4). Then we draw the line. y y - 4 = 1(x-3) 2 -5 -4 -3 -2 -1 1 2 3 4 5 Example 2 Find a linear function that has a graph passing through the points (-1, -5) and (3, -2). Solutiorr We first determine the slope of the line and then use the point-slope equation. Note that m = -5 - (-2) = -3 = 3 -1 -3 -4 4 Since the line passes through (3, -2), we have y - (-2) = 3 4 (x - 3) Substituting into the point-slope equation y + 2 = 3 9 4 x - 4. Using the distributive law Before using function notation, we isolate y: y = 3 9 4 x - 4 - 2 Subtracting 2 from both sides y = 3 17 4 x 4 - 9 - 8 =17 4 4 4 f (x) = 3 17 4 x - 4 . Using function notation You can check that substituting (-1, -5) instead of (3, -2) in the point-slope equation will yield the same expression for f (x). page 119 Tattoo removal. In 1996, an estimated 275,000 Americans visited a doctor for tattoo removal. That figure was expected to grow to 410,000 in 2000 (Source: Mike Meyers, staff writer, Star-Tribune Newspaper of the Twin Cities Minneapolis-St. Paul, copyright 2000). Assuming constant growth since 1995, how many people will visit a doctor for tattoo removal in 2005? 1. Familiarize. Constant growth indicates a constant rate of change, so a linear relationship can be assumed. If we let n represent the number of people, in thousands, who visit a doctor for tattoo removal and t the number of years since 1995, we can form the pairs (1, 275) and (5, 410). After choosing suitable scales on the two axes, we draw the graph. Tattoo Removai 300 275 250 number of people 1 2 3 4 5 Number of years since 1995. 2.5 OTHER EQUATIONS OF LINES page 119 2. Translate. To find an equation relating n and t, we first find the slope of the line. This corresponds to the growth rate: 410 thousand people - 275 thousand people m= 5 years - 1 year =135 thousand people 4 years = 33.75 thousand people per year. Next, we use the point-slope equation and solve for n: n - 275 = 33.75(t - 1) Writing point-slope form n - 275 = 33.75t - 33.75 Using the distributive law n = 33.75t + 241.25. Adding 275 to both sides 3. Carry out. Using function notation, we have n(t) = 33.75t + 241.25. page 12O CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS To predict the number of people who will visit a doctor for tattoo removal in 2005, we find n(10) = 33.75 * 10 + 241.25 2005 is 10 years from 1995. = 578.75. This represents 578,750 people. 4. Check. To check, we can repeat our calculations. We could also extend the graph to see if (10, 578.75) appears to be on the line. 5. State. Assuming constant growth, there will be about 578,750 people visiting a doctor for tattoo removal in 2005. Parallel and Perpendicular Lines If two lines are vertical, they are parallel. How can we tell whether nonvertical lines are parallel? The answer is simple: We look at their slopes (see Examples I and 2 in Section 2.3). Slope and Parallel Lines Two lines are parallel if they have the same slope. Example 4 ® technology ~~ connection To check that y 7 x -3 and y=-8x + 6 8 7 7 are perpendicular, we can use a grapher. To do so, we use the ZSQUARE option of the ZOOM menu to create a "squared" window. This corrects distortion that would otherwise result from differing scales on the axes. 1. Use a grapher to check that y= 3 4 4 x + 2 and y= -3x - 1 are perpendicular. 2. Use a grapher to check that y= 2 5 -5x - 4 and y=2x + 3 are perpendicular. 3. To see that this type of check is not foolproof, graph Y= 31 40 40x + 2 and y= -30x - 1. Are the lines perpendicular? Why or why not? Determine whether the line passing through the points (1, 7) and (4, -2) is par-allel to the line given by f (x) = -3x + 4.2. Solution The slope of the line passing through (1, 7) and (4, -2) is given by m= 7 - (-2)= 9 1 - 4 -3 -3 Since the graph of f(x) = -3x + 4.2 also has a slope of -3, the lines are parallel. If one line is vertical and another is horizontal, they are perpendicular. There are other instances in which two lines are perpendicular. Y S' b -a Slope - b a b R 2.5 OTHER EQUATIONS OF LINES page 121 Consider a line RS, as shown on p. 120, with slope a/b. Then think of rotating the figure 90° to get a line R'S' perpendicular to RS. For the new line, the rise and the run are interchanged, but the run is now negative. The slope of the new line is -b/a. Let's multiply the slopes: a(-b) =-1 b a This can help us determine which lines are perpendicular. Slope and Perpendicular Lines Two lines are perpendicular if the product of their slopes is -l. (If one line has slope m, the slope of a line perpendicular to it is -1/m. That is, we take the reciprocal and change the sign.) Lines are also perpendicular if one is vertical and the other is horizontal. Example 5 Consider the line given by the equation 8y = 7x - 24. a) Find an equation for a parallel line passing through (-1, 2). b) Find an equation for a perpendicular line passing through (-1, 2). Sol ution To find the slope of the line given by 8y = 7x - 24, we solve for y to fmd slope-intercept form: 8y = 7x - 24 y = -7 x - 3. Multiplying both sides by 1 8 8 7 8 The slope is 7. 8 7 a) The slope of any parallel line will be 8. The point-slope equation yields 7 y - 2 = 8 [x - (-1)] Substituting 8 for the slope and (-1, 2) for the point y-2 = 7[x + 1] 8 y = 7x + 7 + 2 8 8 y = 7x + 23 8 8 b) The slope of a perpendicular line is given by the opposite of the reciprocal of 7, or - 8. The point-slope equation yields 8 7 y - 2 = - 8 [x-(-1)] 7 Using the distributive law and adding 2 to both sides y=8x+ 8. y - 2 = -8[x + 1] Substituting -~ for the slope and (-1, 2) for the point 7 y = - 8 x - 8 + 2 Using the distributive law and adding 2 to both sides 7 7 y = -8 x + 6 7 7 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS page 122 CONNECTing THE CONCEPTS We have now studied the slope-intercept, on what information we are given and what in- point-slope, and standard forms of a linear formation we are seeking, one form may be equation. These are the most common ways in more useful than the others. A referenced sum- which linear equations are written. Depending mary is given below. Slope-intercept form, = Useful when an equation is needed and the slope and y-intercept are y = mx + b or given. See Example 7 on p. 96. f(x) = mx + b Useful when a line's slope andy-intercept are needed. See Example 6 on p. 96. - Useful when solving equations graphically. See F.~cample 6 on p. 110. • Commonly used for linear functions. Standard form, Allows for easy calculation of intercepts. See Example 4 on p. 109. Ax + By = C • Will prove useful in future work. See Sections 3.1-3.3. Point-slope form, = Useful when an equation is needed and the slope and a point on the y - yl = m(x - xl) line are given. See Example 1 on p. 118. 7 Useful when a linear function is needed and two points on its graph are given. See Example 2 on p. 118. • Will prove useful in future work with curves and tangents in calculus. Exercise Set 2.5 FOR EXTRA HELP Digital Video Tutor CD 2 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 4 Find an equation in point-slope form of the line having 1. m = -2, (2, 3) 2. m = 5, (7, 4) 3. m= 3,(4,7) 4. m= 2,(7,3) Find an eqvcation of the line having the specified slope 5. m = 1, (-2, -4) 2 6. m = 1, (-5, -7) and containing the indicated point. Write your final an- swer as a linear function in 7. m = -1, (8, 0) 8. m = -3, (-2, 0) slope-intercept form. 2 9. m = 5, (-3, 8) 3 10. m = 4, (1, -5) 2 11. y - 4=7(x - 1) 12. y - 3=9(x - 2) 13. y + 2 = -5(x - 7) 2 14. y - 1 = - 9 (x + 5) the specified slope and containing the point indicated. 5 15. y - 1 = - 3 (x + 2) 16. y + 7 = -4(x - 9) Then graph the line. 17. y = 4x 7 18. y = 3x 19. m = 4, (2, -3) 20. m = -4, (-1, 5) For each point-slope equation listed, state the slope and 3 21. m = - 5, (4, -7) 1 22. m = - 5, (-2,1) a point on the graph. 23. m = -0.6, (-3, -4) 24. m = 2.3, (4, -5) Aha~ 2 25. m=7,(0,-5) 1 26. m=4,(0,3) page 123 Find an equation of the line containing each pair of points. Write your final answer as a linear function in slope-intercept form. 27. (1, 4) and (5, 6) 28. (2, 6) and (4,1) 29. (2.5, -3) and (6.5, 3) 30. (2, -1.3) and (7,1.7) Aha~ 31. (l, 3) and (0, -2) 32. (-3, 0) and (0, -4) 33. (-2, -3) and (-4, -6) 34. (-4, -7) and (-2, -1) In Exercises 35-44, assume that a constant rate of change exists for each model formed. 35. Records in the 400- meter run. 1930, the record for the 400-m run was 46.8 sec. In 1970, it was 43.8 sec. Let R( t) represent the record in the 400-m run and t the number of years since 1930. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the record in 2003; in 2006. c) When will the record be 40 sec? 36. Records in the 1500- meter run. In 1930, the record for the 1500-m run was 3.85 min. In 1950, it was 3.70 min. Let R( t) represent the record in the 1500-m run and t the number of years since 1930. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the record in 2002; in 2006. c) When will the record be 3.1 min? 37. PAC cnntriblrtinns. In 1992, Political Action Com- mittees (PACs) contributed $178.6 million to con- gressional candidates. In 2000, the figure rose to $243.1 million (Source: Congressional Research Service and Federal Election Commission). Let A(t) represent the amount of PAC contributions, in mil- lions, and t the number of years since 1992. $178.6 million 1992 $243.1 million 2000 a) Find a linear function that fits the data. b) Use the function of part (a) to predict the ammount of PAC contributions in 2006. 2.5 OTHER EQUATIONS OF LINES 123 38. Consumer Demand Suppose that 6.5 million lb of coffee are sold when the price is $8 per pound, and 4.0 million lb are sold when it is $9 per pound. a) Find a linear function that expresses the amount of coffee sold as a function of the price per pound. b) Use the function of part (a) to predict how much consumers would be willing to buy at a price of $6 per pound. 39. Recycling. In 1993, Americans recycled 43.8 mil- lion tons of solid waste. In 1997, the figure grew to 60.8 million tons. (Source: Statistical Abstract of the United States, 1999) Let N(t) represent the number of tons recycled, in millions, and t the number of years since 1993. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the amount recycled in 2005. 40. Seller's supply. Suppose that suppliers are willing to sell 5.0 million lb of coffee at a price of $8 per pound and 7.0 million lb at $9 per pound. a) Find a linear function that expresses the amount suppliers are willing to sell as a func- tion of the price per pound. b) Use the function of part (a) to predict how much suppliers would be willing to sell at a price of $6 per pound. 41. Life expectanacy of females iu the United States. In 1990, the life expectancy of females was 78.8 yr. In 1997, it was 79.2 yr. (Source: Statistical Abstract of the United States, 1999) Let E(t) represent life expectancy and t the number of years since 1990. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the life expectancy of females in 2008. 42. Life expectancy of males in the United States. In 1990, the life expectancy of males was 71.8 yr. In 1997, it was 73.6 yr. Let E(t) represent life expectancy and t the number of years since 1990. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the life expectancy of males in 2007. page 124 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS 43. National Park Land. In 1994, the National Park system consisted of about 74.9 million acres. By 1997, the figure had grown to 77.5 million acres. (Source: Statistical Abstract of the United States, 1999) Let A(t) represent the amount of land in the National Park system, in millions of acres, t years after 1994. a) Find a linear function that fits the data. b) Use the function of part (a) to predict the amount of land in the National Park system in 2006. 44. Pressure at sea depth The pressure 100 ft beneath the ocean's surface is approximately 4 atm (atmos- pheres), whereas at a depth of 200 ft, the pressure is about 7 atm. a) Find a linear function that expresses pressure as a function of depth. b) Use the function of part (a) to determine the pressure at a depth of 690 ft. 45. x + 8=y, Y - x =-5 46. 2x - 3=y, y - 2x=9 47. y + 9 = 3x, 3x - y=-2 48. y + 8 = -6x, -2x + y = 5 49. f(x) = 3x + 9, 2y=8x - 2 50. f(x) = -7x - 9, -3y=21x + 7 51. (3,7), x + 2y = 6 52. (0,3), 3x - y = 7 53. (2, -1), 5x - 7y = 8 54. (-4, -5), 2x + y = -3 55. (-6,2), 3x - 9y = 2 56. (-7,0), 5x + 2y = 6 57. (-3, -2), 3x + 2y = -7 58. (-4,3), 6x - 5y = 4 Aha 59. (0, -7), Y = 2x + 3 60. (0, 4), y = -x + 2 61. f(x) = 4x - 3, 4y= 7 - x 62. 2x - 5y = -3, 2x + 5y = 4 63. x + 2y = 7, 2x + 4y = 4 64. y = -x + 7, f(x)=x + 3 Write an equation of the line containing the specified point and perpendicular to the indicated line. 65. (2, 5), 2x + y = -7 66. (4, 0), x - 3y = 0 67. (3, -2), 3x + 4y = 5 68. (-3, -5), 5x - 2y = 4 69. (0.9), 2x + 5y = 7 70. (-3, -4), -3x + 6y = 2 71. (-4, -7), 3x - 5y = 6 72. (-4,5), 7x - 2y = 1 of equations are parallel. Without graphing, tell whether the graphs of each pair of equations are perpendicular. 73. (0, 6), 2x - 5 = y 74. (0, -4), -x + 3 = y Without graphing, tell whether the graphs of each pair 75. Rosewood Graphics recently promised its employ- ees 6% raises each year for the next five years. Amy currently earns $30,000 a year. Can she use a linear function to predict her salary for the next five years? Why or why not? 76. If two lines are perpendicular, does it follow that the lines have slopes that are negative reciprocals of each other? Why or why not? SKILL MAINTENANCE 77. (3x2 + 5x) + (2x - 4) 78. (5t2 - 2t) - (4t + 3) 2.5 OTHER EQUATIONS OF LINES page 125 Evaluate. 79. 2t - 6 for t = 3 4t + 1 80. (3x - 1) (4t + 20), for t = -5 81.2x - 5y,for x = 3 and y = -1 82. 3x - 7y, for x = -2 and y = 4 SYNTHESIS 83. In 1986, Political Action Committees contributed $132.7 million to congressional candidates. Does this information make your answer to Exercise 37(b) seem too low or too high? Why? 84. On the basis of your answers to Exercises 41 and 44, would you predict that at some point in the fu- ture the life expectancy of males will e~eed that of females? Why or why not? For Exercises 85-89, assume that a linear equation models each situation. 85. Depreciation of a computer. After 6 mos of use, the value of Pearl's computer had dropped to $900. After 8 mos, the value had gone down to $750. How much did the computer cost originally? 86. Temperature conversion. Water freezes at 32° Fahrenheit and at 0° Celsius. Water boils at 212°F and at 100°C. What Celsius temperature corre- sponds to a room temperature of 70° F? 87. Cellular phone charges. The total cost of Mel's cel- lular phone was $230 after 5 mos of service and $390 after 9 mos. What costs had Mel already in- curred when his service just began? 88. Operating expenses The total cost for operating Ming's Wings was $7500 after 4 mos and $9250 after 7 mos. Predict the total cost after 10 mos. 89. Medical insurance. In 1993, health insurance companies collected $124.7 billion in premiums and paid out $103.6 billion in benefits. In 1996, they collected $137.1 billion in premiums and paid out $113.8 billlon ln benefits. What percentage Of premiums will be paid out in benefits in 2005? 90. Based on the information given in Exercises 38 and 40, at what price will the supply equal the demand? 91. Specify the domain of your answer to Exercise 38(a). 92. Specify the domain of your answer to Exercise 40(a). 93. For a linear function g, g(3) = -5 and g(7) = -1. a) Find an equation for g. b) Find g(-2). c) Find a such that g(a) = 75. 94. Find the value of k such that the graph of 5y - kx = 7 and the line containing the points (7, -3) and (-2, 5) are parallel. 95. Find the value of k such that the graph of 7y - kx = 9 and the line containing the points (2, -1) and (-4, 5) are perpendicular. 96. Use a grapher with a squared window to check your answers to Exercises 61-74. 97. When several data points are available and they ap- pear to be nearly collinear, a procedure known as linear regression can be used to find an equation for the line that most closely fits the data. a) Use a grapher with a LINEAR REGRESSION option and the table that follows to find a linear func- tion that predicts a woman's life expectancy as a function of the year in which she was born. Compare this with the answer to Exercise 41. b) Predict the life expectancy in 2008 and compare your answer with the answer to Exercise 41. Which answer seems more reliable? Why? Life Expectancy of Women Year. x Life Expectancy, y (in years) 1920 54.6 1930 61.6 1940 65.2 1950 71.1 1960 73.1 1970 74.7 1980 77.5 1990 78.8 1999 79.2 Write an equation of the line containing the specified point and parallel to the indicated line. Simplify. page 126 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS The Algebra of Functions The Sum, Difference, Product, or Quotient of Two Functions • Domains and Graphs We now return to the idea of a function as a machine and examine four ways in which functions can be combined. The Sum, Difference, Product, or Quotient of Two Functions Suppose that a is in the domain of two functions, f and g. The input a is paired with f (a) by f and with g(a) by g. The outputs can then be added to get f(a) + g(a). Example 1 Let f(x) = x + 4 and g(x) = x2 + 1. Find f(2) + g(2). Solution We visualize two function machines. Because 2 is in the domain of each function, we can compute f(2) and g(2). f(x)= x + 4= 6 Since f(2)=2 + 4 = 6 and g(2)=2 2 + 1 = 5, we have f(2) + g(2) = 6 + 5 = 11. In Example l, suppose that we were to write f(x) + g(x) as (x + 4) + (x2 + 1), or f(x) + g(x) = x2 + x + 5. This could then be regarded as a "new" function. The notation (f + g) (x) is generally used to denote a func tion formed in this manner. Similar notations exist for subtraction, multiplication, and division of functions. page 127 The Algebra of Functions If f and g are functions and x is in the domain of both functions, then: 1. ( f + g) (x) = f(x) + g(x); 2. ( f - g) (x) = f(x) - g(x) 3. ( f . g) (x) = f(x) . g(x); 4. ( f/g) (x) = f(x)/g(x), provided g(x) = 0. Example 2 For f (x) = x2 - x and g(x) = x + 2, find the following. a) (f + g)(3) b) (f - g)(x)and(f - g)(-1) c) (f/g) (x) and (f/g) (-4) d) (f * g) (3) Solution a) Since f(3) = 3 2 - 3 = 6 and g(3) = 3 + 2 = 5, we have (f + g)(3) = f(3) + g(3) = 6 + 5 Substituting = 11. Alternatively, we could first find ( f + g) (x): (f + g)(x) = f(x) + g(x) =x2 - x + x + 2 = x2 + 2. Combining like terms ( f + g) (3 ) = 3 2 + 2 = 11. Our results match. b) We have (f - g) (x) = f(x) - g(x) = x2 - x - (x + 2) Substituting = x2 - 2x - 2. Removing parentheses and combining like terms ( f - g) (-1) = (-1)2 - 2(-1) - 2 Using ( f - g) (x) is faster than using f (x) - g(x). = 1. Simplifying c) We have (f/g) (x) = f(x)/g(x) = x2 - x x + 2 We assume that x =/ -2. (f/g)(-4) = (-4)2 - (-4) -4 +2 Substituting =20 = -10. -2 THE ALGEBRA OF FUNCTIONS page 127 page 128 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS d) Using our work in part (a), we have (f * g) (3) = f(3) * g(3) =6 * 5 = 30. It is also possible to compute ( f * g) (3) by first multiplying x2 - x and x + 2 using methods we will discuss in Chapter 5. Domains and Graphs Although applications involving products and quotients of functions rarely appear in newspapers, situations involving sums or differences of functions often do appear in print. For example, the following graphs are similar to those published by the California Department of Education to promote breakfast programs in which students eat a balanced meal of fruit or juice, toast or cereal, and 2% or whole milk. The combination of carbohydrate, protein, and fat gives a sustained release of energy, delaying the onset of hunger for several hours. Graphs shown below. 150 120 90 60 30 0 Carbohydrate C at 150 line. 0 to 120.number of minutes after breakfast. 60 to 180 number of minutes graph 2. protein 120 to 240 number of minutes in graph 3.Fat 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 r Number of minutes after breakfast c p '~ ~ 150 ~ 120 o ~ 90 ~ 60 ~ 30 a G zw 0 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 t Number of minutes after breakfast b v a~ ° 'o A 150 ~ l20 o p 90 ~ 60 ~ 30 z ~ 0 0 _ _ ,-" \ Fat 15 30 45 60 75 90 105 120 135 I50 165 180 195 Z10 225 240 t Number of minutes after breakfast When the three graphs are superimposed, and the calorie expenditures added, it becomes clear that a balanced meal results in a steady, sustained supply of energy. 2.6 THE ALGEBRA OF FUNCTIONS page 131 A partial check of Example 4 can be performed by setting up a table so the TABLE MINIMUM is 0 and the increment of change (~Tbl) is 0.7. (Other choices, like 0.1, will also work.) Next, we let yl = 1/x and y2 = 2x - 7. Using the Y-VARS key to write y3 = yl + y2 and y4 = y1/y2, we can create the table of values shown here. Note that when x is 3.5, a value for y3 can be found, but y4 is undefined. When set-ting up these functions, you may wish to enter yl and y2 without "selecting" either. Otherwise, the table's columns must be scrolled to display y3 and y4. Division by 0 is not the only condition that can force restrictions on the domain of a function. In Chapter 7, we will examine functions similar to that given by f(x) = v/x, for which the concern is taking the square root of a negative number. FOR EXTRA HELP Use a similar approach to partially check Example 3. Exercise Set 2,6 Let f(x) = -3x + 1 and g(x) = x2 + 2. Find the following. 1. f(2) + g(2) 2. f(-1) + g(-1) 3. f(5) - g(5) 4. f(4) - g(4) 5. f(-1) * g(-1) 6. f(-2) * g(-2) 7. f(-4)/g(-4) 8. f(3)/g(3) 9. g(1) - f(1) 10. g(2)/ f(2) 11. (f + g)(x) 12. (g - f)(x) Let F(x) = x2 - 2 and G(x) = 5 - x. Find the following. 13. (F + G) (x) 14. (F + G) (a) 15. (F + G) (-4) 16. (F + G) (-5) 17. (F - G)(3) 18. (F - G)(2) 19. (F * G) (-3) 20. (F * G) (-4) 21. (F/G) (x) 22. (G - F) (x) 23. (F/G) (-2) 24. (F/G) (-1) Digital Video Tutor CD 2 InterAct Math Math Tutor Center MathXL MyMathLab.com 1980 1990 1998 Year 25. Use estimates of R(1980) and W(1980) to estimate N(1980) . 3 top line 4 1990 4 1998 N 2 line 2.5 2 1990 1998 W 1 line 1.5 1.5 1990 1998 R 26. Use estimates of R(1990) and W(1990) to estimate N(1990). page 132 CHAPTER 2 GRAPHS, FUNCTIONS, AND UNEAR EQUATIONS 27. Which group of women was responsible for the drop in the number of births from 1990 to 1998? 28. Which group of women was responsible for the rise in the number of births from 1980 to 1990? Often function addition is represented by stacking the individual functions directly on top of each other. The graph below indicates how the three major airports serv- icing New York City have been utilized. The braces indi- cate the values of the individual functions. Number of passengers (in millions) F(t) = Total in year 60 Kennedy Airport f(t) k(I) 45 LaGuardia Airport l(t) 10 Newark Airport n (t) 1986 - 1999 0 10 20 90 29. Estimate (n + I) ('98). What does it represent? 30. Estimate (k + I) ('98). What does it represent? 31. Estimate (k - l) ('94). What does it represent? 32. Estimate (k - n) ('94). What does it represent? 33. Estimate (n + l + k) ('99). What does it represent? 34. Estimate (n + l + k) ('98). What does it represent? For each pair of functions f and g, determine the domain of the sum, difference, and product of the two functions. 35. f(x) = x2, g(x) = 7x - 4 36. f(x) = 5x - 1, g(x) = 2x2 37. f(x) = 1 x - 3 g(x) = 4x3 38. f(x) = 3x2, g(x) 1 = x -9 39. f(x) = 2, x g(x) = x2 - 4 40. f(x) = x3 + 1, g(x) = 5 x 41. f(x) = x + 2 x - 1 42. f(x) = 9 - x2 g(x) = 3 + 2x x - 6 43. f(x) = 3 x - 2 g(x) = 5 4 - x 44. f(x) = 5 x - 3. g(x) = 1 x - 2 For each pair of functions f and g, determine the domain of f/g. 45. f(x) = x4, g(x)= x - 3 46. f(x) = 2 x 3, g(x) = 5 - x 47. f(x) = 3x - 2, g(x) = 2x - 8 48. f(x) = 5 + x, g(x)= 6 - 2x 49. f(x) = 3 x - 4. g(x) = 5 - x 50. f(x) = 1 2 - x. g(x) = 7 - x 51. f (x) = 2x x + 1 g(x)= 2x + 5 52. f (x) = 7x x - 2 g(x)= 3x + 7 For Exercises 53-60, consider the functions F and G as shown. 53. Determine (F + G) (5) and (F + G) (7). 54. Determine (F * G) (6) and (F * G) (9). 55. Determine (G - F) (7) and (G - F) (3). 56. Determine (F/G) (3) and (F/G) (7). 57. Find the domains of F G, F + G, and F/G. 58. Find the domains of F - G, F * G, and G/F. 59. Graph F + G. 60. Graph G - F page 133 In the following graph, W(t) represents the number of gallons of whole milk, L(t) the number of gallons of low- fat milk, and S(t) the number of galloras of skim milk consumed by the average American in year t.* 1970 1975 1980 1985 1990 1995 2000 61. Explain in words what ( W - S) (t) represents and what it would mean to have ( W - S) (t) < 0. 62. Consider (W + L + S) (t) and explain why you feel that total milk consumption per person did or did not change over the years 1970-1998. SKILL MAlNTENANCE Solve. 63. 4x - 7y = 8,for x 64. 3x - 8y = 5, for y 65. 5x + 2y = -3, for y 66. 6x + 5y = -2, for x Translate each of the following. Do not solve. 67. Five more than twice a number is 49. 68. Three less than half of some number is 57. 69. The sum of two consecutive integers is 145. 70. The difference between a number and its opposite is 20. SYNTHESIS 71. If f(x) = c, where c is some positive constant, describe how the graphs of y = g(x) and y = ( f + g) (x) will differ. 72. Examine the graphs following example 2 and ex- plain how they might be modified to represent the absorption of 200 mg of Advil taken four times a day. 73. Find the domain of f/g, if f(x) = 3x and g (x) = x4 - 1 2x + 5 3x + 9 74. Find the domain of F/G, if F(x) = 1 and G(x) = x2 - 4 x - 4 x - 3 75. Sketch the graph of two functions f and g such that the domain of f/g is {x|-2 < x < 3 and x =/1} 76. Find the domain of m/n, if m(x) = 3x for -1 < x < 5 and n(x) = 2x - 3. 77. Find the domains of f + g, f - g, f * g, and f/g, if f = {(-2,1), (-1, 2), (0, 3), (1, 4), (2, 5)} and g = {(-4, 4), (-3, 3), (-2, 4), (-1, o), (o, 5), (1, 6)}. 78. For f and g as defined in Exercise 77, find (f + g)(-2),(f*g)(0) and (f/g)(1). 79. Write equations for two functions f and g such that the domain of f + g is {x|x is a real number and x =/ -2 and x =/ 5). 80. Let y1 = 2.5x + 1.5 y2 = x - 3 and y3 = y1/y2. Depending on whether the CONNECTED or DOT mode is used the graph of y3 appears ass follows 10 10 10 10 2.6 ' THE AIGEBRA OF FUNCTIONS 133 Use algebra to determine which graph more accu- rately represents y3. page 134 CHAPTER 2 , GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS 81. Using the window[ -5, 5, -1,9], graph yl = 5, y2 = x + 2, and y3 = V-x. Then predict what shape the graphs of y1 + y2, yl + y3, and y2 + y3 will take. Use a grapher to check each prediction. 82. Use the TABLE feature on a grapher to check your answers to Exercises 37, 43, 45, and 51. (See the Technology Connection on p. 131.) CORNER Time On Your Hands Focus: The algebra of functions Time: 10-15 minutes Group size: 2-3 The graph and data at right chart the average Average Retirement age: 67.3 64.9 63.2 62.8 62.7 61.5 retirement age R(x) and life expectancy E(x) Of Average Life Expectancy: 73.9 75.5 77.2 78.5 79.1 80.1 ... If, You're Age 50, Consider This ACTIVITY 1. Working as a team, perform the appropriate calculations and then graph E - R. 2. What does (E - R) (x) represent? In what fields of study or business might the function E - R prove useful? 3. Should E and R really be calculated separately for men and women? Why or why not? 85 4. What advice would you give to someone con-sidering early retirement? Year; 1955 1965 1975 1985 1995 2005e Average Life Expectancy US citizens in year x e=esGmated Average Retirement Age_.,_,.__. „ 11955 1960 1965 1:9701975 1980 1985 1990 1995 2000e 2005e . . Years, Source. Bureau of Labor Statistics, courtesy of the Insurance Advisory Board page 135 Summary and Review 2 SUMMARY AND REVIEW: CHAPTER 2 'I3S partant Praperties and Formulas Key Terms Axes, axis, p. 70 Range, p. 78 Linear function, p. 93 x, y-coordinate system, p. 70 Function, p. 78 y-intercept, p. 94 Cartesian coordinate system, Projection, p. 81 Slope, p. 95 p~ 70 Relation, p. 83 Rise, p. 95 Ordered pair, p. 71 Input, p. 83 Run, p. 95 Origin, p. 71 Output, p. 83 Rate of change, p. 98 Coordinates, p. 71 Dependent variable, p. 84 Salvage value, p. 100 Quadrant, p. 71 Independent variable, p. 84 Zero slope, p. 106 Graph, p. 72 Constant function, p. 84 Undefined slope, p. 107 Linear equation, p. 74 Dummy variable, p. 84 x-intercept, p. 108 Nonlinear equation, p. 74 Interpolation, p. 85 Growth rate, p. 119 Domain, p. 78 Extrapolation, p. 85 The Verticat-Line Test A graph represents a function if it is not possible to draw a vertical line that inter-sects the graph more than once. Slope = m = rise = change in y = y2 - yl run = change in x = x2 - xl Every horizontal line has a slope of 0. The slope of a vertical line is undefined. The x-intercept is (a, 0). To find a, let y = 0 and solve the original equation for x. The y-intercept is (0, b). To find b, let x = 0 and solve the original equation for y. The slope-intercept equation of a line is y=mx + b. The point-slope equation of a line is y - y1 = mx + x 1. The standard form of a linear equation is Ax + By = C. Parallel lines: The slopes are equal. Perpendicular lines: The product of the slopes is -1. The Algebrcr of Functions 1. ( f + g) (x) = f(x) + g(x) 2. ( f - g) (x) = f(x) - g(x) 3. ( f * g) (x) = f(x) * g(x) 4. ( f/g) (x) = f(x)/g(x), provided g(x) =/ 0 page 136 G GHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Review Exercises Determine whether the ordered pair is a solution of the given equation. 1. (3, 7), 4p - q = 5 2. (-2, 4), x = 2y + 12 1 3. (0, -2), 3a - 4b = 2 4. (8, -2), 3c + 2d = 28 Graph 5. y = -3x + 2 6. y = -x2 + 1 7. 8x + 32 = 0 8. y - 2 = 4 9. For the following graph of f, determine (a) f(2); (b) the domain of f: (c) any x-values for which f(x) = 2 and (d) the range of f. y 5 4 3 2 1 -5 -4 - 3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 f line start from 1 -2 end at 5 4 10. The function A(t) = 0.233t + 5.87 can be used to estimate the median age of cars in the United States t years after 1990 (Source: The Polk Co.). (In this context, a median age of 3 yr means that half the cars are more than 3 yr old and half are less.) Pre- dict the median age of cars in 2010; that is, find A(20). Find the slope and the y-intercept. 11. g(x) = -4x - 9 12. -6y + 2x = 7 The following table shows the annual soft-drink production in the United States, in number of 12-oz cans per person (Sources: National Soft Drink Association; Beverage World). Output, Number of 12-oz Input, Year Cans per Person 1977 360 1987 480 1997 580 13. Use the data in the table at the bottom of the first column to draw a graph and to estimate the annual soft-drink production in 1990. 14. Use the graph from Exercise 13 to estimate the annual soft-drink production in 2005. 1977 360 output number of 12 oz cans 1987 480 1997 580 15. Containing the points (4,5) and (-3,1) 16. Containing (-16.4, 2.8) and (-16.4, 3.5) 17. Find the rate of change for the graph below Use appropriate units. start 15 25, 4 35, 8 45, 12 4 8 12 represent number of years since high school. personal income are 60 50 40 30 20 10 in thousands of dollars. 0 2 4 6 8 10 12 18. By March 1, 2000, U.S. builders had begun con- struction of 265,000 homes. By July 1, 2000, there were 795,000 begun (Source: U.S. Department of Commerce). Calculate the rate at which new homes were being started. 19. The average cost of tuition at a state university t years after 1997 can be estimated by C(t) = 645t + 9800. What do the numbers 645 and 9800 signify? 2 20. Find a linear function whose graph has slope 7 and y-intercept (0, -6). 21. Graph using intercepts: -2x + 4y = 8. 22. Solve 2 - x = 4 + x graphically. Then check your answer by solving the equation algebraically. 23. To join the Family Fitness Center, it costs $75 plus $15 a month. Use a graph to estimate the time re-quired for the total cost to reach $180. Determine whether each of these is a linear equation. 24.2x - 7=0 25.3x - 8f(x)=7 page 137 26. 2a + 7b2 = 3 27. 2p - 7 = 1 a 28. Find an equation in point-slope form of the line with slope -2 and containing (-3, 4). 29. Using function notation, write a slope-intercept equation for the line containing (2, 5) and (-4, -3). Determine whether each pair of lines is parallel, perpen-dicular, or neither. 30. y + 5= -x, x - y= 2 31. 3x - 5= 7y, 7y - 3x = 7 32. In 1955, the U.S. minimum wage was $0.75, and in 1997, it was $5.15. Let W represent the minimum wage, in dollars, t years after 1955. a) Find a linear function W(t) that fits the data. b) Use the function of part (a) to predict the mini-mum wage in 2005. Find an equation of the line. 33. Containing the point (2, -5) and parallel to the line 3x - 5y =9 34. Containing the point (2, -5) and perpendicular to the line 3x - 5y = 9 Let g(x) = 3x - 6 and h(x) = x2 + 1. Find the following. 35. g(0) 36. h(-5) 37. (g * h) (4) 38. (g - h) (-2) 39. (g/h) (-1) 40. g(a + b) 41. The domains of g + h and g * h 42. The domain of h/g SYNTHESIS 43. Explain why every function is a relation, but not every relation is a function. 44. Explain why the slope of a vertical line is unde-fined whereas the slope of a horizontal line is 0. 45. Find the y-intercept of the function given by f(x) + 3 = 0.17 x 2 + (5 - 2x)x - 7. 46. Determine the value of a such that the lines 3x - 4y = 12 and ax + 6y = -9 are parallel. 47. Homespun Jellies charges $2.49 for each jar of pre-serves. Shipping charges are $3.75 for handling, plus $0.60 per jar. Find a linear function for determining the cost of shipping x jars of preserves. 48. Match each sentence with the most appropriate graph below a) Joni walks for 10 min to the train station, rides the train for 15 min, and then walks 5 min to the office. graph II is 12, 14 b) During a workout, Phil bikes for 10 min, runs for 15 min, and then walks for 5 min. graph IV is 7, 12 c) Sam pilots his motorboat for 10 min to the mid-dle of the lake, fishes for 15 min, and then mo-tors for another 5 min to another spot. graph III is 2, 14, 15 d) Patti waits 10 min for her train, rides the train for 15 min, and then runs for 5 min to her job. graph I 6, 9 I 16 14 i: 12 „ ~ lo 0 8 ~ 6 ~ 4 ro:. 0 2i ~ 0 2 0~ 5 10 15 20 25 30 ~ F 0 5 10 15 20 25 30 ' Time (in minutes) Time (in minutes) 0 5 10 IS 20 25 30 ~ 0 5 10 15 20 25 30 Time (in minutes) Time (in minutes) REVIEW EXERCISES: CHAPTER 2 page 137 page 138 CHAPTER 2 GRAPHS, FUNCTIONS, AND LINEAR EQUATIONS Chapter Test 2 Determine whether the ordered pair is a solution of the given equation. 1. (0, -5), x + 4y = -20 2. (1, -4), -2p + 5q = 18 Graph. 3. y = -5x + 4 4. y = -2x2 + 3 5. f(x)=5 6. 3 - x = 9 7. For the following graph of f determine (a) f(-2); (b) the domain of f; (c) any x-value for which f(x) = 1; and (d) the range of f. 2 Y 2,-3 4,-1 8. The function S(t) = 1.2t + 21.4 can be used to esti- mate the total U.S. sales of books, in billions of dol- lars, t years after 1992. a) Predict the total U.S. sales of books in 2008. b) What do the numbers 1.2 and 21.4 signify? 9. There were 43.3 million international visitors to the United States in 1995, and 48.5 million in 1999 (Source: Tourism Industries, International Trade Administration, Department of Commerce). Draw a graph and estimate the number of international visitors in 1997. y 5 4 3 2 . 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 Find the slope and the y-intercept. 3 10. f(x) = -5x + 12 11. -5y - 2x = 7 Find the slope of the line containing the following points. If the slope is undefined state this. 12. (-2, -2) and (6, 3) 13. (-3.1, 5.2) and (-4.4, 5.2) 14. Find the rate of change for the graph below Use appropriate units. calories burned 400 300 . 200 . 100 0 30 60 90 120 150 Minutes in exercise. dot at 150 60 dot at 300 120 15. Find a linear function whose graph has slope -5 and y-intercept (0, -1). 16. Graph using intercepts: -2x + 5y = 12. 17. Solve x + 3 = 2x graphically. Then check your an-swer by solving the equation algebraically. 18. Which of these are linear equations? a) 8x - 7 = 0 b) 4b - 9a2 = 2 c) 2x - 5y = 3 19. Find an equation in point-slope form of the line with slope 4 and containing (-2, -4). 20. Using function notation, write a slope-intercept equation for the line containing (3, -1) and (4, -2). Determine without graphing whether each pair of lines is parallel, perpendicular, or neither. 21. 4y + 2 = 3x, -3x + 4y = -12 22. y = -2x + 5, 2y - x = 6 Find ara equation of the line. 23. Containing (-3, 2) and parallel to the line 2x - 5y = 8 24. Containing (-3, 2) and perpendicular to the line 2x - 5y = 8 Find the slope of the line containing the following 25. Find the following, given that g(x) = -3x - 4 and h(x) = x2 + 1 a) h(-2) b) (g * h) (3) c) The domain of h/g page 139 26. If you rent a van for one day and drive it 250 mi, the cost is $100. If you drive it 300 mi, the cost is $115. Let C(m) represent the cost, in dollars, of driving m miles. a) Find a linear function that fits the data. b) Use the function to find how much it will cost to rent the van for one day and drive it 500 miles. 27. The function f(t) = 5 + 15t can be used to deter-mine a bicycle racer's location, in miles from the starting line, measured t hours after passing the 5-mi mark. a) How far from the start will the racer be 1 hr and 40 min after passing the 5-mi mark? b) Assuming a constant rate, how fast is the racer traveling? 28. The graph of the function f(x) = mx + b contains the points (r, 3) and (7, s). Express s in terms of r if the graph is parallel to the line 3x - 2y = 7. 29. Given that f(x) = 5x2 + 1 and g(x) = 4x - 3, find an expression for h(x) so that the domain of f/g/h is {x|x is a real number and x =/ 3 and x =/ 2. 4 7 Answers may vary. page 142 Own 183 shopping centers in 32 states. Twice as many before merger. How many properties did the 2 each company own before the merger? page 143 Translating x + y = 183 x is number of properties y is number of properties total = 183. x + y = 183 x = 2y. 120 stamps for $30.30 If stamps were 20 cent postcard stamps and 34 cent stamps how many of each type were bought? 60 * $0.20 + 60 * $0.34 = $12.00 + $20.40 or $32.40. p + f = 120 0.20p + 0.34f = $30.30 Example 3 -4, 7 x + y = 3. 5x - y = -27. x + y = 3 -4 + 7 ? 3 3 3 True page 147 page 147 3.45x + 4.21y = 8.39 7.12x - 5.43y = 6.18. page 148 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING Exercise Set 3,1 Digital Video Tutor CD 2 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 5 Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables. 1. (1,2); 4x- y = 2, lOx - 3y = 4 2. (-1, -2); 2x + y = -4, x - y = 1 3. (2, 5); y = 3x - 1, 2x + y = 4 4. (-1, -2); x + 3y = -7, 3x - 2y = 12 5. (1,5);x + y = 6, y = 2x + 3 6. (5,2); a + b = 7, 2a - 8 = b Aha' 7. (3,1); 3x + 4y = 13, 6x + 8y = 26 8. (4, -2); - 3x - 2y = -8, 8 = 3x + 2y Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. 9. x - y= 3, x + y= 5 10. x + y = 4, x - y = 2 11. 3x + y = 5, x - 2y = 4 12. 2x - y = 4, 5x - y = 13 13. 4y= x + 8, 3x - 2y = 6 14. 4x - y = 9, x - 3y = 16 15. x = y - 1, 2x = 3y 16. a = 1 + b, b = 5 - 2a 17. x = -3, y = 2 18. x = 4, Y = -5 19. t + 2s =-1, s = t + 10 20. b + 2a = 2, a = -3 -b 21. 2b + a = 11, a - b = 5 1 22. y= -3x -1, 4x - 3y = 18 1 23. y= -4x + 1, 2y = x - 4 24. 6x - 2y = 2, 9x - 3y = 1 25. y - x = 5, 2x - 2y = 10 26. y = -x - 1, 4x - 3y = 24 27. Y = 3 - x 2x + 2y = 6 28. 2x - 3y = 6, 3y - 2x = -6 29. For the systems in the odd-numbered exercises. 9-27, which are consistent? 30. For the systems in the even-numbered exercises. 10-28 which are consistent? 31. For the systems in the odd-numbered exercises 9 - 27 which contain dependent equations? 32. For the systems in the even-numbered exercises, 10 - 28 which contain dependent equations? Translate each problem situation to a system of equa-tions. Do not attempt to solve, but save for later use. 33. The difference between two numbers is 11. Twice the smaller plus three times the larger is 123. What are the numbers? 34. The sum of two numbers is -42. The first number minus the second number is 52. What are the numbers? 35. Retail sales. Paint Town sold 45 paintbrushes, one kind at $8.50 each and another at $9.75 each. In all, $398.75 was taken in for the brushes. How many of each kind were sold? 36. Retail sales. Mountainside Fleece sold 40 neck-warmers. Polarfleece neckwarmers sold for $9.90 each and wool ones sold for $12.75 each. In all, $421.65 was taken in for the neckwarmers. How many of each type were sold? 37. Geometry Two angles are supplementary.* One angle is 3° less than twice the other. Find the measures of the angles. x y *The sum of the measures of two supplementary angles is 180°. page 149 3.1 SYSTEMS OF EQUATIONS IN TWO VARIABLES page 149 38. Geometry Two angles are complementary.* The sum of the measures of the first angle and half the second angle is 64°. Find the measures of the angles. 39. Basketball scoring. Wilt Chamberlain once scored 100 points, setting a record for points scored in an NBA game. Chamberlain took only two-point shots and (one-point) foul shots and made a total of 64 shots. How many shots of each type did he make? 40. Fundraising. The St. Mark's Communiry Barbecue served 250 dinners. A child's plate cost $3.50 and an adult's plate cost $7.00. A total of $1347.50 was col lected How many of each type of plate was served? 41. Sales of pharmaceuticals In 2001, the Diabetic Express charged $21.95 for a vial of Humulin in- sulin and $20.95 for a vial of Novolin insulin. If a total of $1077.50 was collected for 50 vials of in- sulin, how many vials of each type were sold? 42. Court dimensions The perimeter of a standard basketball court is 288 ft. The length is 44 ft longer than the width. Find the dimensions. P=288ft 43. Court dimensions The perimeter of a standard tennis court used for doubles is 228 ft. The width is 42 ft less than the length. Find the dimensions. 44. Basketball Scoring The Fenton College Cougars made 40 field goals in a recent basketball game, some 2-pointers and the rest 3-pointers. Altogether the 40 baskets counted for 89 points. How many of each type of field goal was made? 45. Hockey Rankings Hockey teams receive 2 points for a win and 1 point for a tie. The Wildcats once won a championship with 60 points. They won 9 more games than they tied. How many wins and how many ties did the Wildcats have? 46. Radio airplay Roscoe must play 12 commercials during his 1-hr radio show. Each commercial is either 30 sec or 60 sec long. If the total commercial time during that hour is 10 min, how many com-mercials of each type does Roscoe play? 47. Nontoxic floor wax. A nontoxic floor wax can be made from lemon juice and food-grade linseed oil. The amount of oil should be twice the amount of lemon juice. How much of each ingredient is needed to make 32 oz of floor wax? (The mix should be spread with a rag and buffed when dry.) 48. Lumber production. Denison Lumber can convert logs into either lumber or plywood. In a given day, the mill turns out 42 pallets of plywood and lumber. It makes a profit of $25 on a pallet of lumber and $40 on a pallet of plywood. How many pallets of each type must be produced and sold in order to make a profit of $1245? 49. Video rentals. J. P's Video rents general-interest films for $3.00 each and children's films for $1.50 each. In one day, a total of $213 was taken in from the rental of 77 videos. How many of each type of video was rented? 50. airplane seating. An airplane has a total of 152 seats. The number of coach-class seats is 5 more than six times the number of first-class seats. How many of each type of seat are there on the plane? 51. Write a problem for a classmate to solve that requires writing a system of two equations. Devise the problem so that the solution is "The Lakers made 6 three-point baskets and 31 two-point baskets." 52. Write a problem for a classmate to solve that can be translated into a system of two equations. De-vise the problem so that the solution is "Shelly gave 9 haircuts and 5 shampoos." SKILL MAINTENANCE Solve. 53. 2(4x - 3) - 7x = 9 54. 6y - 3(5 - 2y) = 4 55. 4x - 5x = 8x - 9 + llx page 150 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING *The sum of the measures of two complementary angles is 90°. 56. 8x - 2(5 - x) = 7x + 3 Solve. 57. 3x + 4y = 7, for y 58. 2x - 5y = 9, for y SYNTHESIS Technology in U.S. schools For Exercises 59-62, con-sider the following graph. Number of schools in thousands 70 cd-roms, Modems 60 50 networks 40 Interactive videodisk 30 20 10 1992 1994 1996 1998 Year Technology in U.S. Schools 1992 1994 1996 1998 Year Source: Quality Education Data, Denver, CO, National Education Database 59. Is it accurate to state that there have always been more schools with CD-ROMs than with networks? Why or why not? 60. Is it accurate to state that there have always been more schools with networks than with interactive videodisks? Why or why not? 61. During which year did the number of schools with CD-ROMs first exceed the number of schools with modems? 62. During which year did the number of schools own-ing an interactive videodisk increase the most? At what rate did it increase? 63. For each of the following conditions, write a system of equations. a) (5,1) is a solution. b) There is no solution. c) There is an infinite number of solutions. 64. A system of linear equations has (1, -1) and (-2, 3) as solutions. Determine. a) a third point that is a solution and b) how many solutions there are. 65. The solution of the following system is (4, -5). Find A and B. Ax - 6y = 13 x - By = -8. Translate to a system of equations. Do not solve. 66. Ages. Burl is twice as old as his son. Ten years ago, Burl was three times as old as his son. How old are they now? 67. Work experience. Lou and Juanita are mathemat-ics professors at a state university. Together, they have 46 years of service. Two years ago, Lou had taught 2.5 times as many years as Juanita. How long has each taught at the university? 68. Design. A piece of posterboard has a perimeter of 156 in. If you cut 6 in. off the width, the length be-comes four times the width. What are the dimensions of the original piece of posterboard? 69. Nontoxic scouring powder A nontoxic scouring powder is made up of 4 parts baking soda and 1 part vinegar. How much of each ingredient is needed for a 16-oz mixture? Solve graphically. 70. y = |x|, x + 4y = 15 71. x - y = 0, y = x2 In Exercises 72-75, use a grapher to solve each system of linear equations for x and y. Round all coordinates to the nearest hundredth. 72. y = 8.23x + 2.11, y = -9.l lx - 4.66 73. y = -3.44x - 7.72, y = 4.19x - 8.22 74. 14.12x + 7.32y = 2.98, 21.88x - 6.45y = -7.22 75. 5.22x - 8.21y = -10.21, -12.67x + 10.34y = 12.84 page 151 Solve the system x + y = 4 is equation 1 x = y + 1 is equation 2 x + y = 4 (y + 1) + y = 4 (y + 1) + y = 4 2y + 1 = 4 2y = 3 subtract 1 from both sides y = 3 2 divide by 2. Example 5 5x + 7y = 29 Multiply both sides by -2 > -10 -14y = -58 0 + y = 27 page 155 Example 6 Solve the system 3y - 2x = 6 equation 1 -12y + 8x = -24 equation 2. 12y - 8x = 24 Multiply both sides of equation 1 by 4. -12 + 8x = -24 0 =0 We obtain a identity 0 = 0 is always true. {(x,y)|3y - 2x = 6}. 12 x 4 = 48 3.2 SOLVING BY SUBSTITUTION OR ELIMINATION page 157 Before selecting a method to use, try to remember the strengths and weaknesses of each method. If possible, begin solving the system mentally before settling on the method that seems best suited for that particular system. Selecting the "best" method for a problem is a bit like selecting one of three different saws with which to cut a piece of wood. The "best" choice depends on what kind of wood is being cut and what type of cut is being made, as well as your skill level with each saw. Note that each of the three methods was introduced using a rather simple example. As the ex-amples became more complicated, additional steps were required in order to "turn" the new problem into a more familiar format. This is a common approach in mathematics: We perform one or more steps to make a "new" problem resemble a problem we already know how to solve. Exercise Set 3,2 FOR EXTRA HELP Digital Video Tutor CD 2 InterAd Math Math Tutor Center MathXL MyMathLab.com Videotape 5 For Exercises 1-48, if a system has an infinite number of 1. y= 5 - 4x, 2x - 3y = 13 2. x = 8 - 4y, 3x + 5y = 3 3. 2y + x = 9, x = 3y - 3 4. 9x - 2y = 3, 3x - 6 = y 5. 3s - 4t = 14, 5s + t = 8 6. m - 2n = 16, 4m + n = 1 7. 4x - 2y = 6, 2x - 3 = y 8. t = 4 - 2s, t + 2s = 6 9. -5s + t = 11, 4s + 12t = 4 10. 5x + 6y = 14, -3y + x = 7 11. 2x + 2y = 2, 3x - y = 1 12. 4p - 2q = 16, 5p + 7q = 1 13. 3a - b = 7, 2a + 2b = 5 14. 5x + 3y = 4, x - 4y = 3 15. 2x - 3 = y, y - 2x = 1 16. a - 2b = 3, 3a = 6b + 9 Solve using the elimination method. 17. x + 3y = 7, -x + 4y = 7 18. x + y = 9, 2x - y = -3 19. 2x + y = 6, x - y = 3 20. x - 2y = 6, -x + 3y = -4 21. 9x + 3y = -3, 2x - 3y = -8 22. 6x - 3y = 18, 6x + 3y = -12 23. 5x + 3y = 19, 2x - 5y = 11 24. 3x + 2y = 3, 9x - 8y = -2 25. 5r - 3s = 24, 3r + 5s = 28 26. 5x - 7y = -16, 2x + 8y = 26 27. 6s + 9t = 12, 4s + 6t = 5 28. 10a + 6b = 8, 5a + 3b = 2 1 1 29. 2x - 6y = 3, 2 1 5x + 2y = 2 30. 1x + 1y = 7, 3 5 1x - 2y = -4 6 5 31. x y 7 2 + 3 = 6. 2x 3y 5 3 + 4 =4, 32. 2x + 3y = 11 3 4 12 x + 7y = 1 3 18 = 2 Aha' 33. 12x - 6y = -15, -4x + 2y = 5 34. 8s + 12t = 16, 6s + 9t = 12 35. 0.2a + 0.3b = 1, 0.3a - 0.2b = 4 36. -0.4x + 0.7y = 1.3, 0.7x - 0.3y = 0.5 Solve using any appropriate method. 37. a - 2b = 16, b + 3 = 3a 38. 5x - 9y = 7, 7y - 3x = -5 39. lOx + y = 306, l0y + x = 90 40. 3(a - b) = 15, 4a = b + 1 41. 3y = x - 2, x = 2 + 3y 42. x + 2y = 8, x = 4 - 2y page 158 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 43. 3s - 7t = 5, 7t - 3s = 8 44. 2s - 13t = 120, -14s + 91t = -840 45. 0.05x + 0.25y = 22, 0.15x + 0.05y = 24 46. 2.1x - 0.9y = 15, -1.4x + 0.6y = 10 47. 13a - 7b = 9, 2a - 8b = 6 48. 3a - 12b = 9, 14a - llb = 5 49. Describe a procedure that can be used to write a inconsistent system of equations. 50. Describe a procedure that can be used to write a system that has a infinite number of solutions. Skill Maintenance 51. The fare for a taxi ride from Johnson Street to Elm Street is $5.20. If the rate of the taxi is $1.00 for the first 1 mi and 30cents for each additional 1 mi. how far 2 4 is it from Johnson Street to Elm Street? 52. A student's average after 4 tests is 78.5. What score is needed on the fifth test in order to raise the aver age to 80? 53. Home remodeling. In a recent year, Americans spent $35 billion to remodel bathrooms and kitchens. Twice as much was spent on kitchens as on bathrooms. (Source: Indianapolis Star) How much was spent on each? 54. A 480-m wire is cut into three pieces. The second piece is three times as long as the first. The third is four times as long as the second. How long is each piece? 55. Car Rentals. Badger Rent-A-Car rents a compact car at a daily rate of $34.95 plus 10cents per mile. A businessperson is allotted $80 for car rental. How many miles can she travel on the $80 budget? 56. Car rentals. Badger rents midsized cars at a rate of $43.95 plus 10 cents per mile. A tourist has a car-rental budget of $90. How many miles can he travel on the $90? SYNTHESIS 57. Some systems are more easily solved by substitu tion and some are more easily solved by elimina- tion. Write guidelines that could be used to help someone determine which method to use. 58. Explain how it is possible to solve Exercise 33 mentally. 59. If (1, 2) and (-3, 4) are two solutions of f(x) = mx + b, find m and b. 3 60. If (0, -3) and (-2, 6) are two solutions of px - qy = -1, find p and q. 61. Determine a and b for which (-4, -3) is a solution of the system. ax + by = -26 bx - ay = 7. 62. Solve for x and y in terms of a and b: 5x + 2y = a, x - y = b. Solve. 63. x + y - x - y = 1. 2 5 x - y + x + y = -2. 2 6 64. 3.5x - 2.1y = 106.2 4.1x + 16.7y = -106.28 Each of the following is a system of nonlinear equations. However each is reducible to linear since a appropri- ate substitution say u for 1/x and u for 1/y yields a linear system. Make such a substitution solve for the new variables and then solve for the original variables. 65. 2 + 1 = 0. x y 5 + 2 = -5. x y 66. 1 - 3 = 2. x y 6 + 5 = -34. x y 67. A student solving the system. 17x + 19y = 102. 136x + 152y = 826. graphs both equations on a grapher and gets the following screen. 10 10 10 10 The student then incorrectly concludes that the equations are dependent and the solution set is infinite. How can algebra be used to convince the student that a mistake has been made? page 160 x + y = 183 x = 2y Equations solve. x + y = 183 2y + y = 183 substituting 2y for x 3y = 183 combining like terms y = 61 x = 2y = 2 * 61 122. page 161 Number of stamps f + p = 45 + 75 = 120 Cost of first class stamps $0.34f = 0.34 x 45 = $15.30. Cost of postcard stamps $0.20p = 0.20 x 75 = $15.00 Total $30.30 Example 3 Blending teas Tara's Tea Terrace sells loose Black tea for 95 cents a ounceand Lapsang Souchong for $1.43 a ounce. Tara wants to make 1 lb mixture of the two types called Imperial Blend sells for $1.10 a ounce. How much tea of each type should Tara use? $1.43 $1.10 Multiply 16 ounces 1 lb times $1.10 per ounce for the total value. 95b + 143l = 1760. 95(16 - 1) + 143l = 1760 1520 - 95l + 143l = 1760 48l = 240 t=5 equation solving g + s = 100 5g + 15s = 1200 Total amount of mixture g + s = 30 + 70 = 100 Total amount of nitrogen 5% of 30 + 15% of 70 = 1.5 + 10.5 = 12 3.3 SOLVING APPLICATIONS: SYSTEMS OF TWO EQUATIONS page 169 Exercise Set 3.3 FOR EXTRA HELP 1.-18. For Exercises 1-18, solve Exercises 33-50 from pp. 148-149. 19. Sales Staples® recently sold a box of Flair felt-tip pens for $12 and a four-pack of Sanford® Uni-ball© pens for $8. At the start of a recent fall semester, a combination of 40 boxes and four-packs of these pens was sold for a total of $372. How many of each type were purchased? 20. Sales. Staples recently sold a wirebound graph- paper notebook for $2.50 and a college-ruled note- book made of recycled paper for $2.30. At the start of a recent spring semester, a combination of 50 of these notebooks was sold for a total of $118.60. How many of each type were sold? 21. Blending coffees. The Coffee Counter charges $9.00 per pound for Kenyan French Roast coffee and $8.00 per pound for Sumatran coffee. How much of each type should be used to make a 20-1b blend that sells for $8.40 per pound? 22. Mixed nuts. The Nutry Professor sells cashews for $6.75 per pound and Brazil nuts for $5.00 per pound. How much of each type should be used to make a 50-1b mixture that sells for $5.70 per pound? 23. Catering Casella's Catering is planning a wedding reception. The bride and groom would like to serve a nut mixture containing 25% peanuts. Casella has available mixtures that are either 40% or 10% peanuts. How much of each type should be mixed to get a 10-1b mixture that is 25% peanuts? 24. Livestoek feed. Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed to get a 350-1b mixture that is 12% protein? 25. Ink remover. Etch Clean Graphics uses one cleanser that is 25% acid and a second that is 50% acid. How many liters of each should be mixed to get 10 L of a solution that is 40% acid? Digital Video Tutor Cp 2 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 5 26. Blending granola Deep Thought Granola is 25% nuts and dried fruit. Oat Dream Granola is 10% nuts and dried fruit. How much of Deep Thought and how much of Oat Dream should be mixed to form a 20-1b batch of granola that is 19% nuts and dried fruit? 27. Student loans Lomasi's two student loans totaled $12,000. One of her loans was at 6% simple interest and the other at 9%. After one year, Lomasi owed $855 in interest. What was the amount of each loan? 28. Investments. An executive nearing retirement made two investments totaling $15,000. In one year, these investments yielded $1432 in simple interest. Part of the money was invested at 9% and the rest at 10%. How much was invested at each rate? 29. Automotive maintenance. "Aretic Antifreeze" is 18% alcohol and "Frost No-More" is 10% alcohol. How many liters of each should be mixed to get 20 L of a mixture that is 15% alcohol? page 17O CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 30. Food Service. The following bar graph shows the milk fat percentages in three dairy products. many pounds each of whole milk and cream should be mixed to form 2001b of milk for cream cheese? 31. Real estate The perimeter of an oceanfront lot is 190 m. The width is one fourth of the length. Find the dimensions. 32. Architecture. The rectanular round floor of the John Hancock building has a perimeter of 860 ft. The length is 100 ft more than the width. Find the length and the width. 33. Making change Cecilia makes a $9.25 purchase at the bookstore with a $20 bill. The store has no bills and gives her the change in quarters and fifty-cent pieces. There are 30 coins in all. How many of each kind are there? 34. Teller work. Ashford goes to a bank and gets change for a $50 bill consisting of all $5 bills and $1 bills. There are 22 bills in all. How many of each kind are there? 35. Train travel. A train leaves Danville Junction and travels north at a speed of 75 km/h. Two hours later, an express train leaves on a parallel track and travels north at 125 km/h. How far from the station will they meet? 36. Car travel. Two cars leave Salt Lake City, traveling in opposite directions. One car travels at a speed of 80 km/h and the other at 96 km/h. In how many hours will they be 528 km apart? 37. Canoeing Alvin paddled for 4 hr with a 6-km/h current to reach a campsite. The return trip against the same current took 10 hr. Find the speed of Alvin's canoe in still water. 38. Boating. Mia's motorboat took 3 hr to make a trip downstream with a 6-mph current. The return trip against the same current took 5 hr. Find the speed of the boat in still water. 39. Point of no return A plane flying the 3458-mi trip from New York City to London has a 50-mph tail- wind. The flight's point of no return is the point at which the flight time required to return to New York is the same as the time required to continue to London. If the speed of the plane in still air is 360 mph, how far is New York from the point of no return? 40. Point of no return. A plane is flying the 2553-mi trip from Los Angeles to Honolulu into a 60-mph headwind. If the speed of the plane in still air is 310 mph, how far from Los Angeles is the plane's point of no return? (See Exercise 39.) 41. Write at least three study tips of your own for someone beginning this exercise set. 42. In what ways are Examples 3 and 4 similar? In what sense are their systems of equations similar? SKILL MAINTENANCE Evaluate. 43. 2x - 3y + 12,for x = 5 and y = 2 44. 7x - 4y + 9,for x = 2 and y = 3 45. 5a - 7b + 3c, for a = -2, b = 3, and c = 1 46. 3a - 8b - 2c, for a = -4, b = -l, and c = 3 1 1 47. 4 - 2y + 3z, for y = 3 and z = 4 1 1 48. 3 - 5y + 4z, for y = 2 and z = 5 page 171 49. Suppose that in Example 3 you are asked only for the amount of Black tea needed for the Imperial Blend. Would the method of solving the problem change? Why or why not? 50. Write a problem similar to Example 2 for a class- mate to solve. Design the problem so that the solu- tion is "The florist sold 14 hanging plants and 9 flats of petunias." 51 - 54 For Exercises 51-54 solve Exercises 66-69 from Exercise Set 3.1. 66 67. 68. 69. 55. Retail. Some of the world's best and most expen- sive coffee is Hawaii's Kona coffee. In order for cof- fee to be labeled "Kona Blend," it must contain at least 30% Kona beans. Bean Town Roasters has 401b of Mexican coffee. How much Kona coffee must they add if they wish to market it as Kona Blend? 56. Automotive maintenance The radiator in Michelle's car contains 6.3 L of antifreeze and water. This mixture is 30% antifreeze. How much of this mixture should she drain and replace with pure antifreeze so that there will be a mixture of 50% antifreeze? 57. Exercise. Natalie jogs and walks to school each day. She averages 4 km/h walking and 8 km/h jog- ging. From home to school is 6 km and Natalie How many makes the trip in 1 hr. How far does she jog in a trip? 58. Book sales. A limited edition of a book published by a historical society was offered for sale to mem-bers. The cost was one book for $12 or two books for $20 (maximum of two per member). The society sold 880 books, for a total of $9840. How many members ordered two books? 59. The tens digit of a two-digit positive integer is 2 more than three times the units digit. If the digits are interchanged, the new number is 13 1ess than half the given number. Find the given integer. (Hint: Let x = the tens-place digit and y = the units-place digit; then lOx + y is the number.) 3.3 SOLVING APPLICATIONS: SYSTEMS OF TWO EQUATIONS page 171 60. Train travel. A train leaves Union Station for Cen tral Station, 216 km away, at 9 A.M. One hour later, a train leaves Central Station for Union Station. They meet at noon. If the second train had started at 9 A.M. and the first train at 10:30 a.m., they would still have met at noon. Find the speed of each train. 61. Wood stains. Williams' Custom Flooring has 0.5 gal of stain that is 20% brown and 80% neutral. A customer orders 1.5 gal of a stain that is 60% brown and 40% neutral. How much pure brown stain and how much neutral stain should be added to the original 0.5 gal in order to make up the order? 62. Fuel economy Grady's station wagon gets 18 miles per gallon (mpg) in ciry driving and 24 mpg in highway driving. The car is driven 465 mi on 23 gal of gasoline. How many miles were driven in the city and how many were driven on the highway? 63. Biochemistry. Industrial biochemists routinely use a machine to mix a buffer of 10% acetone by adding 100% acetone to water. One day, instead of adding 5 L of acetone to create a vat of buffer, a machine added 10 L. How much additional water was needed to bring the concentration down to 10%? 64. Gender: Phil and Phyllis are siblings. Phyllis has twice as many brothers as she has sisters. Phil has the same number of brothers as sisters. How many girls and how many boys are in the family? 65. See Exercise 61 above. Let x = the amount of pure brown stain added to the original 0.5 gal. Find a function P(x) that can be used to determine the percentage of brown stain in the 1.5-gal mixture. On a grapher, draw the graph of P and use INTERSECT to confirm the answer to Exercise 61. *This problem was suggested by Professor Ghris Burditt of page 172 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING Systems of Equations 3•4 in Three Variables Identifying Solutions • Solving Systems in Three Variables • Dependency, Inconsistency, and Geometric Considerations CONNECTING THE CONCEPTS As often happens in mathematics, once an idea be further extended to systems with four equa- is thoroughly understood, it can be extended to tions in four unknowns, five equations in five increasingly more complicated problems. This unknowns, and so on. is precisely the situation for the material in Sec- Another common occurrence in mathemat- tions 3.4-3.7: We will extend the elimination ics is the streamlining of a sequence of steps method of Section 3.2 to systems of three equa- that are used repeatedly. In Sections 3.6 and 3.7, tions in three unknowns. Although we will not we develop different notations that streainline do so in this text, the approach that we use can the calculations of Sections 3.2 and 3.4. Some problems translate directly to two equations. Others more naturally call for a translation to three or more equations. In this section, we learn how to solve systems of three linear equations. Later, we will use such systems in prob-lem-solving situations. Identifying Solutions A linear equation in three variables is an equation equivalent to one in the form Ax + By + Cz = D, where A, B, C and D are real numbers. We refer to the form Ax + By + Cz = D as standard form for a linear equation in three variables. A solution of a system of three equations in three variables is an ordered triple (x, y, z) that makes all three equations true. Example 1 3 Determine whether 2, -4, 3 is a solution of the system 4x - 2y - 3z = 5, -8x - y + z = -5, 2x + y + 2z = 5. 3.4 SYSTEMS OF EQUATIONS IN THREE VARIABLES page 179 Determine whether 2, -1,-2 is a solution of the system. 1. x + y - 2z = 5 2x - y - z = 7 -x - 2y +3z = 6 Determine whether 1,-2,3 is a solution of the system. 2. x + y + z = 2 x - 2y - z = 2 3x + 2y + z = 2 Solve each system. If a system's equations are dependent or if there is no solution state this. 3. x + y + z = 6, 2x - y + 3z = 9 -x + 2y + 2z = 9. 4. 2x - y + z = 10, 4x + 2y - 3z = 10, x - 3y + 2z = 8 5. 2x - y - 3z = -1 2x - y + z = -9 x + 2y - 4z = 17 6. x - y + z = 6 2x + 3y + 2z = 2 3x + 5y + 4z = 4 7. 2x - 3y + z = 5, x + 3y + 8z = 22, 3x - y + 2z = 12 8. 6x - 4y + 5z = 31, 5x + 2y + 2z = 13 x + y + z = 2 9. 3a - 2b + 7c = 13, a + 8b - 6c = - 47 7a - 9b - 9c = -3 10. x + y + z = 0, 2x + 3y + 2z = -3, -x + 2y - 3z = -1 11. 2x + 3y + z = 17, x - 3y + 2z = -8, 5x - 2y + 3z = 5 12. 2x + y - 3z = -4 4x - 2y + z = 9 3x + 5y - 2z = 5 13. 2x + y + z = -2, 2x - y + 3z = 6, 3x - 5y + 4z = 7 14. 2x + y + 2z = 11 3x + 2y + 2z = 8 x + 4y + 3z = 0 15. x - y + z = 4, 5x + 2y -3z = 2, 4x + 3y - 4z = -2 16. -2x + 8y + 2z = 4, x + 6y + 3z = 4, 3x - 2y + z = 0 17. a + 2b + c = 1, 7a + 3b - c = -2 a + 5b + 3c = 2 18. 4x - y - z = 4, 2x + y + z = -1 6x - 3y - 2z =3 1 7 19. 5x + 3y + 2z = 2, 0.5x - 0.9y - 0.2z = 0.3, 3x- 2.4y + 0.4z = -1 3 20. r + 2S + 6t = 2, 2r - 3s + 3t = 0.5, r + s + t = 1 21. 3p + 2r = 11, q - 7r = 4, p - 6q = 1 22. 4a + 9b = 8, 8a + 6c = -1, 6b + 6c = -1 23. x + y + z = 105, l0y - z = 11, 2x - 3y = 7 24. x + y + z = 57, -2x + y = 3, x - z = 6 25. 2a - 3b = 2, ~ 3 7a + 4c = 4, 2c - 3b = 1 26. a - 3c = 6, b + 2c = 2, 7a - 3b - 5c = 14 Aha~ 27. x + y + z = 182, y = 2 + 3x z = 80 + x 28. l + m = 7, 3m + 2n = 9 4l + n = 5 29. x + y = 0, x + z = 1 2x + y + z = 2 30. x + + z = 0, x + y + 2z = 3 y + z = 2 31. y + z = 1, x + y + z = 1 x + 2y + 2z = 2, 32. x + y + z = 1, -x + 2y + z = 2 2x - y = -1 33. Abbie recommends that a frustrated classmate double- and triple-check each step of work when attempting to solve a system of three equations. Is this good advice? Why or why not? 34. Describe a method for writing a inconsistent sys tem of three equations in three variables. SKILL MAINTENANCE Translate each sentence to mathematics. 35. One number is twice another. 36. The sum of two numbers is three times the first number. 37. The sum of three consecutive numbers is 45. 38. One number plus twice another number is 17. 39. The sum of two numbers is five times a third number. page 180 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 40. The product of two numbers is twice their sum. SYNTHESIS 41. Is it possible for a system of three equations to have exactly two ordered triples in its solution set? Why or why not? 42. Describe a procedure that could be used to solve system of four equations in four variables. Solve. 43. x + 2 - y + 4 + z + 1 = 0 3 2 6 x - 4 + y + 1 - z - 2 = -1 3 4 2 x + l + y + z - 1 = 3 2 + 2 + 4 4 44. w + x + y + z = 2. w + 2x + 2y + 4z = 1. w - x + y + z = 6. w - 3x - y + z = 2. 45. w + x - y + z = 0 w - 2x - 2y - z = -5 w - 3x - y + z = 4 2w - x - y + 3z = 7 For Exercises 46 and 47 let u represent 1/x.u represent 1/y and w represent 1/z. Solve for u v and w and then solve for x y and z. 46. 2 - 1 - 3 = -1. x y z 2 - 1 + 1 = -9. x y z 1 + 2 - 4 = 17 x y z 47. 2 + 2 - 3 = 3. x y z 1 - 2 - 3 = 9. x y z 7 - 2 + 9 = -39 x y z Determine k so that each system is dependent. 48. x - 3y + 2z = 1, 2x + y - z = 3, 9x - 6y + 3z = k 49. 5x - 6y + kz = -5, x + 3y - 2z = 2, 2x - y + 4z = -1 In each case, three solutions of an equation in x, y, and z are given. Find the equation. 50. Ax + By + Cz = 12; 3 4 (1,4, 3), (3,1,2), and (2, 1, 1) 51. z = b - mx - ny; 3 (1, 1, 2), (3, 2, -6), and (2, 1, 1) 52. Write an inconsistent system of equations that con tains dependent equations. CORNER Fi Focus: Systems of three linear equations Time. 10-15 minutes Group size: 3 should begin by eliminating x, one should first eliminate y, and one should first elimi-nate z. Write neatly so that others can follow your steps. Consider the six steps outlined on p. 174 along 2• Once all group members have solved the sys- with the following system: tem, compare your answers. If the answers do not check, exchange notebooks and check 2x + 4y = 3 - 5z, each other's work. If a mistake is detected, 0.3x = 0.2y + 0.7z + 1.4, allow the person who made the mistake to make the repair. 0.04x + 0.03y = 0.07 + 0.04z. 3. Decide as a group which of the three ap- ACTIVITY proaches above (if any) ranks as easiest and which (if any) ranks as most difficult. Then 1. Working independently, each group member compare your rankings with the other groups should solve the system above. One person in the class. page 184 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING Exercise Set 3,5 Digital Video Tutor CD 3 InterAct Math Math Tutor Center Videotape 6 MathXL MyMathLah.com Solve. 1. The sum of three numbers is 57. The second is 3 more than the first. The third is 6 more than the first. Find the numbers. 2. The sum of three numbers is 5. The first number minus the second plus the third is 1. The first minus the third is 3 more than the second. Find the numbers. 3. The sum of three numbers is 26. Twice the first minus the second is 2 1ess than the third. The third is the second minus three times the first. Find the numbers. 4. The sum of three numbers is 105. The third is 11 less than ten times the second. Twice the first is 7 more than three times the second. Find the numbers. 5. Geometry In triangle ABC, the measure of angle B is three times that of angle A. The measure of angle C is 20° more than that of angle A. Find the angle measures. 6. Geometry In triangle ABC, the measure of angle B is twice the measure of angle A. The measure of angle C is 80° more than that of angle A. Find the angle measures. 7. Automobile pricing. A recent basic model of a particular automobile had a price of $12,685. The basic model with the added features of automatic transmission and power door locks was $14,070. The basic model with air conditioning (AC) and power door locks was $13,580. The basic model with AC and automatic transmission was $13,925. What was the individual cost of each of the three options? 8. Lens production When Sight-Rite's three polish- ing machines, A, B, and C, are all working, 5700 lenses can be polished in one week. When only A and B are working, 3400 lenses can be pol- ished in one week. When only B and C are working, 4200 lenses can be polished in one week. How many lenses can be polished in a week by each machine? Aha' v 9. Welding rates Elrod, Dot, and Wendy can weld 74 linear feet per hour when working together. Elrod and Dot together can weld 44 linear feet per hour, while Elrod and Wendy can weld 50 linear feet per hour. How many linear feet per hour can each weld alone? 10. Telemarketing Sven, Tillie, and Isaiah can process 740 telephone orders per day. Sven and Tillie to- gether can process 470 orders, while Tillie and Isaiah together can process 520 orders per day. How many orders can each person process alone? Sven, 470 Tillie, 520 and Isaiah 740 11. Restaurant management. Kyle works at Dunkin Donuts, where a 10-oz cup of coffee costs $1.05, a 14-oz cup costs $1.35, and a 20-oz cup costs $1.65. During one busy period, Kyle served 34 cups of cof- fee, emptying five 96-oz pots while collecting a total of $45. How many cups of each size did Kyle fill? 10 oz 14 oz 20 oz $1.05 $1.35 $1.65 12. Advertising In a recent year, companies spent a total of $84.8 billion on newspaper, television, and radio ads. The total amount spent on television and radio ads was only $2.6 billion more than the amount spent on newspaper ads alone. The amount spent on newspaper ads was $5.1 billion more than what was spent on television ads. How How much was spent on each form of advertising? Example 200 + 50 + 52 = 302 2 * 50 + 3 * 52 = 256. page 185 3.5 SOLVING APPLICATIONS: SYSTEMS OF THREE EQUATIONS of pizza, he or she takes in 65 mg of cholesterol. By eating 2 eggs and 1 cupcake, a child consumes 567 mg of cholesterol. How much cholesterol is in each item? Solution 1. Familiarize. After we have read the problem a few times, it becomes clear that an egg contains considerably more cholesterol than the other foods. Let's guess that one egg contains 200 mg of cholesterol and one cup cake contains 50 mg. Because of the third sentence in the problem, it would follow that a slice of pizza contains 52 mg of cholesterol since 200 + 50 + 52 = 302. To see if our guess satisfies the other statements in the problem, we find the amount of cholesterol that 2 cupcakes and 3 slices of pizza would contain: 2 ~ 50 + 3 ~ 52 = 256. Since this does not match the 65 mg listed in the fourth sentence of the problem, our guess was incorrect. Rather than guess again, we examine how we checked our guess and let e, c, and s = the number of milligrams of cholesterol in an egg, a cupcake, and a slice of pizza, respectively. 2. Translate. By rewording some of the sentences in the problem, we can translate it into three equations. The amount of the amount of the amount of cholesterol in cholesterol in 1 cholesterol in 1 1 egg plus cupcake plus slice of pizza is 302 mg.   The amount of cholesterol the amount of cholesterol in 2 cupcakes plus in 3 slices of pizza is 65 mg. 2c 3s = 65 The amount of cholesterol the amount of cholesterol in 2 eggs plus in 1 cupcake is 56, 7 mg.    2e + c = 567 We now have a system of three equations: e+c+ s=302, 2c + 3s = 65, 2e + c = 567. 3. Carry out. We solve and get e = 274, c = 19, s = 9, or (274, 19, 9?.  4. Check. The sum of 274, 19, and 9 is 302 so the total cholesterol in 1 egg, 1 cupcake, and 1 slice of pizza checks. Two cupcakes and three slices of pizza would contain 2 ~ 19 + 3 ~ 9 = 65 mg, while two eggs and one cup cake would contain 2 ~ 274 + 19 = 567 mg of cholesterol. The answer checks. 5. State. An egg contains 274 mg of cholesterol, a cupcake contains 19 mg of cholesterol, and a slice of pizza contains 9 mg of cholesterol. 3.5 SOLVING APPLICATIONS: SYSTEMS OF THREE EQUATIONS page 185 13. Investments A business class divided an imagi- nary investment of $80,000 among three mutual funds. The first fund grew by 10%, the second by 6%, and the third by 15%. Total earnings were $8850. The earnings from the first fund were $750 more than the earnings from the third. How much was invested in each fund? 14. Restaurant management McDonald'sg recently sold small soft drinks for 87 cents, medium soft drinks for $1.08, and large soft drinks for $1.54. During a lunch-time rush, Chris sold 40 soft drinks for a total of $43.40. The number of small and large drinks, combined, was 10 fewer than the number of medium drinks. How many drinks of each size were sold? small medium large $0.87 $1.08 $1.54 15. Nutrition A dietician in a hospital prepares meals under the guidance of a physician. Suppose that for a particular patient a physician prescribes a meal to have 800 calories, 55 g of protein, and 220 mg of vitamin C. The dietician prepares a meal of roast beef, baked potatoes, and broccoli according to the data in the following table. Protein Vitamin C Calories (in grams) (in milligrams) Roast Beef 3 oz. 300 20 0 Baked Potato 100 5 20 Brocoli 156 g 50 5 100 How many servings of each food are needed in order to satisfy the doctor's orders? 16. Nutrition Repeat Exercise 15 but replace the broccoli with asparagus, for which a 180-g serving contains 50 calories, 5 g of protein, and 44 mg of vitamin C. Which meal would you prefer eating? 17. Crying rare. The sum of the average number of times a man, a woman, and a one-year-old child cry each month is 71.7. A one-year-old cries 46.4 more times than a man. The average number of times a one-year-old cries per month is 28.3 more than the average number of times combined that a man and a woman cry. What is the average number of times per month that each cries? 18. Obstetrics. In the United States, the highest inci- dence of fraternal twin births occurs among Asian- Americans, then African-Americans, and then Cau casians. Out of every 15,400 births, the total number of fraternal twin births for all three is 739, where there are 185 more for Asian-Americans than African-Americans and 231 more for Asian- Americans than Caucasians. How many births of fraternal twins are there for each group out of every 15,400 births? 19. Basketball scoring. The New York Knicks recently scored a total of 92 points on a combination of 2-point field goals, 3-point field goals, and 1-point foul shots. Altogether, the Knicks made 50 baskets and 19 more 2-pointers than foul shots. How many shots of each kind were made? 20. History Find the year in which the first U.S. transcontinental railroad was completed. The fol- lowing are some facts about the number. The sum of the digits in the year is 24. The ones digit is 1 more than the hundreds digit. Both the tens and the ones digits are multiples of 3. 21. Problems like Exercises 11 and 12 could be classi- fied as total-value problems. How do these prob lems differ from the total-value problems of Section 3.3? page 186 CHAPTER 3 - SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 22. Write a problem for a classmate to solve. Design the problem so that it translates to a system of three equations in three variables. SKILL MAINTENANCE 23. 5(-3) + 7 24. -4(-6) + 9 25. -6(8) + (-7) 26. 7(-9} + (-8) 27. -7(2x - 3y + 5z) 28. -6(4a + 7b - 9c) 29. -4(2a + 5b) + 3a + 20b 30. 3(2x - 7y) + 5x + 21y Simplify. SYNTHESIS 31. Consider Exercise 19. Suppose there were no foul shots made. Would there still be a solution? Why or why not? 32. Consider Exercise 11. Suppose Kyle collected $46. Could the problem still be solved? Why or why not? 33. Find a three-digit positive integer such that the sum of all three digits is 14, the tens digit is 2 more than the ones digit, and if the digits are reversed, the number is unchanged. 34. Ages Tammy's age is the sum of the ages of Car- men and Dennis. Carmeri s age is 2 more than the sum of the ages of Dennis and Mark. Dennis's age is four times Mark's age. The sum of all four ages is 42. How old is Tammy? 35. Ticket revenue. The Pops concert audience of 100 people consists of adults, students, and chil- dren. The ticket prices are $10 for adults, $3 for stu- dents, and 50 cents for children. The total amount of money taken in is $100. How many adults, stu- dents, and children are in attendance? Does there seem to be some information missing? Do some more careful reasoning. 36. Sharing raffle tickets. Hal gives T as many raffle tickets as T first had and Gary as many as Gary first had. In like manner, T then gives Hal and Gary as many tickets as each then has. Similarly, Gary gives Hal and T as many tickets as each then has. If each finally has 40 tickets with how many tickets does T begin? 37. Find the sum of the angle measures at the tips of the star in this figure. A E B D C Elimination Using 3 • Matriees Matrices and Systems • Row-Equivalent Operations In solving systems of equations, we perform computations with the constants. The variables play no important role until the end. we can simplify writ-ing a system by omitting the variables. For example, the system 3x + 4y = 5, 3 4 5 simplifies to x - 2y = 1 1 -2 1 if we do not write the variables, the operation of addition, and ttae equals signs. page 189 3.6 ELIMINATION USING MATRICES 189 Next, we multiply the first row by -2, add it to the second row, and replace Row 2 with the result: 1 0 -4 5 -2(1 0 -4 |5) = (-2 0 8 | -10 and 0 -1 12 -13 (-2 0 8 | -10 + (2 -1 4| -3 = 6 -1 2 10 (0 -1 12| -13) Now we mu]tiply the first row by -6, add it to the third row, and replace Row 3 with the result: 1 0 -4 5 -6(1 0 -4 ; 5) = (-6 0 24 ; -30) 0 -1 12 ; -13 . (-6 0 24 ; -30) + (6 -1 2 ; 10) = 0 -1 26 | -20 (0 -1 26 | -20) Next, we multiply Row 2 by -1, add it to the third row, and replace Row 3 with the result: x - 4z = 5, -y + 12 | = -13, 14z=- 1 0 -4 | 5 -1(0 -1 12 | -13) = (0 1 -12 | 13) 0 -1 12 | -13 and (0 1 -12 | l3) + (0 -1 26 | -20) = 0 0 14 | -7 (0 0 14 | -7) Reinserting the variables gives us x - 4z = 5, -y + 12z = -13, 14z = -7. 1 We now solve this last equation for z and get z = - 2. Next, we 1 substitute - 2 for 1 z in the preceding equation and solve for y: -y + 12(-2) = -13, so y = 7. Since there is no y-term in the first equation of this last system, we need only substitute 1 1 -2 for z to solve for x: x - 4(-2) = 5, so x = 3. The solution is 1 (3, 7, -2). The check is left to the student. Technology connection Row-equivalent operations can be performed on a grapher. rowSwap([A],1,2)-~[B] For example, to interchange the first and second rows of [[1 0 -4 5] the matrix, as in step (1) of Example 2 above, we enter the [2 -1 4 -3] matrix as matrix A and select "rowSwap" from the MATRIX ~ [6 -1 2 10]] MATH menu. Some graphers will not automatically store the matrix produced using a row-equivalent operation, so when several operations are to be performed in succes-sion, it is helpful to store the result of each operation as it is produced. In the window at right, we see both the matrix produced by the rowSwap operation and the indication 1. Use a grapher to proceed through all the steps in that this matrix is stored as matrix B. Example 2. page 19O CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING The operations used in the preceding example correspond to those used to produce equivalent systems of equations. We call the matrices row-equivalent and the operations that produce them row-equivalent operations. Row-Equivalent Operations Row-Equivalent Operations Each of the following row-equivalent operations produces a row-equivalent matrix: a) Interchanging any two rows. b) Multiplying all elements of a row by a nonzero constant. c) Replacing a row with the sum of that row and a multiple of another row The best overall method for solving systems of equations is by row-equivalent matrices; even computers are programmed to use them. Matrices are part of a branch of mathematics known as linear algebra. They are also studied in many courses in fmite mathematics. Exercise Set 3.6 FOR EXTRA HELP - Digital Video Tutor CD 3 InterAct Math Math Tutor Center MaYhXL MyMathLab rom Videotape 6 Solve using matrices. 1. 9x - 2y = 5, 3x - 3y = 11 2. 4x + y = 7, 5x - 3y = 13 3. x + 4y = 8, 3x + 5y = 3 4. x + 4y = 5, -3x + 2y = 13 5. 6x - 2y = 4, 7x + y = 13 6. 3x + 4y = 7, -5x + 2y = 10 7. 3x + 2y + 2z = 3, x + 2y - z = 5, 2x - 4y + z = 0 8. 4x - y -3z = 19, 8x + y - z = 11, 2x + y + 2z = -7 Solve using matrices. 9. p - 2q -3r = 3, 2p - q -2r = 4, 4p + 5q + 6r = 4 10. x + 2y - 3z = 9, 2x - y + 2z = -8, 3x - y - 4z = 3 11. 3p + 2r = 11, q - 7r = 4, P - 6q = 1 12. 4a + 9b = 8, 8a + 6c = -1, 6b + 6c = -1 13. 2x + 2y -2z -2w = -10, w + y + z + x = -5, x - y + 4z + 3w = -2, w - 2y + 2z + 3x = -6 14. -w - 3y + z + 2x = -8, x + y - z - w = -4, w + y + z + x = 22, x - y - z - w = -14 Solve using matrices 15. Coin value. A collection of 34 coins consists of dimes and nickels. The total value is $1.90. How many dimes and how many nickels are there? page 191 chapter 3 16. Coin value A collection of 43 coins consists of dimes and quarters. The total value is $7.60. How many dimes and how many quarters are there? 17. Mixed granola Grace sells two kinds of granola. One is worth $4.05 per pound and the other is worth $2.70 per pound. She wants to blend the two granolas to get a 15-1b mixture worth $3.15 per pound. How much of each kind of granola should be used? 18. Trail mix Phil mixes nuts worth $1.60 per pound with oats worth $1.40 per pound to get 20 1b of trail. mix worth $1.54 per pound. How many pounds of nuts and how many pounds of oats should be used? 19. Investments. Elena receives $212 per year in simple interest from three investments totaling $2500. Part is invested at 7%, part at 8%, and part at 9%. There is $1100 more invested at 9% than at 8%. Find the amount invested at each rate. 20. Investments Miguel receives $306 per year in simple interest from three investments totaling $3200. Part is invested at 8%, part at 9%, and part at 10%. There is $1900 more invested at 10% than at 9%. Find the amount invested at each rate. 21. Explain how you can recognize dependent equations when solving with matrices. 22. Explain how you can recognize an inconsistent sys- tem when solving with matrices. SKILL Maintenance Simplify. 23. 5(-3) - (-7)4 24. 8(-5) - (-2)9 25. -2(5 * 3 - 4 * 6) - 3(2 * 7 - 15) + 4(3 * 8 - 5 * 4) 26. 6(2* 7 -3(-4)) -4(3(-8)-10)+ 5(4 * 3- (-2)7) SYNTHESIS 27. If the matrices a1 b1 c1 and a2 b2 c2 d1 e1 f1 d2 e2 f2 share the same solution, does it follow that the cor- responding entries are all equal to each other (al = a2, bl = b2, etc.)? Why or why not? 28. Explain how the row-equivalent operations make use of the addition, multiplication, and distributive properties. 29. The sum of the digits in a four-digit number is 10. Twice the sum of the thousands digit and the tens digit is 1 less than the sum of the other two digits. The tens digit is twice the thousands digit. The ones digit equals the sum of the thousands digit and the hundreds digit. Find the four-digit number. 30. Solve for x and y: ax + by = c, dx + ey = f. Determinants of 2 x 2 Matrices When a matrix has m rows and n columns, it is called an "m by n" matrix its dimensions are denoted by m x n. If a matrix has the same number of rows and columns, it is called a square matrix. Associated with every square matrix is a number called its determinant, defined as follows for 2 x 2 matrices. 3.7 DETERMINANTS AND CRAMER'S RULE page 192 Determinants and Cramer's Rule Determinants of 2 x 2 Matrices • Cramer's Rule: 2 x 2 Systems • - Cramer's Rule: 3 x 3 Systems 192 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 2 x 2 Determinants The determinant of a two-by-two matrix a c is denoted a c and is defined as follows: b d b d and is defined as follows a c = ad - bc b d E x a m p 1 e 1 Evaluate: I6 7I. Solution We multiply and subtract as follows: 2 5 6 7 =2*7-6*(-5)=14 + 30 = 44. Cramer's Rule: 2 x 2 Systems One of the many uses for determinants is in solving systems of linear equations in which the number of variables is the same as the number of equations and the constants are not all 0. Let's consider a system of two equations: a1x + bly = cl, a2x + b2y = c2. If we use the elimination method, a series of steps can show that c1b2 - c2b1 and a1c2 - a2c1 x = alb2 - a2bl and y = a1b2 - a2b1. Determinants can be used in these expressions for x and y. Cramer's Rule: 2 x 2 Systems The solution of the system alx + bly = c1, a2x + b2y = c2, if it is unique, is given by x = c1 b1 a1 c1 c2 b2 a2 c2 x = ai b1 y= a1 b1 a2 b2 a2 b2. 3.7 DEfERMINANTS AND CRAMER'S RULE page 195 E x a m p 1 e 4 Solve using Cramer's rule: Solution We compute D, Dx, Dy, and Dz: Determinants can be evaluated on most graphers using the MATRIX package. After entering a matrix on the grapher, we select the determinant operation from the MATRIX MATH menu and enter the name of the matrix. The grapher will return the value of the determinant of the matrix. For example, for 1 6 -1 A = -3 -5 3 , 0 4 2 we have det [A] 26 Use a grapher to confirm The calculations in Example 4. Cramer's Rule: 3 x 3 Systems The solution of the system replace the a s. alx + bly + clz = d1, a2x + b2y + c2Z = d2, a3x + b3y + c3z = d3 is found by considering the following determinants. a1 b1 c1 d1 b1 c1 D contains only D = D = a2 b2 c2 , Dx = d2 b2 c2 , coefficients. a3 b3 c3 d3 b3 c3 In Dx, the d's a1 d1 c1 a1 b1 d1 Dy = a2 d2 c2 , Dz = a2 b2 d2 a3 d3 c3 a3 b3 d3 If a unique solution exists, it is given by Dx, y = Dy, z = Dz x = D D D Solve using Cramer's rule x - 3y + 7z = 13. x + y + z = 1. x + 2y + 3z = 4. Solution We compute D D D and D. 1 -3 7 13 -3 7 D = 1 1 1 = -10; Dx = 1 1 1 = 20; 1 -2 3 4 -2 3 1 13 7 1 -3 13 Dy = 1 1 1 = -6 1 1 1 = -24 1 4 3 1 -2 4. Dx = 20 x = D -10 = -2 Dy = -6 = 3 y = D -10 5 Dz -24 12 z = D -10 5 3 12 The solution is -2 5 5 page 196 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING In Example 4, we need not have evaluated Dz. Once x and y were found, we could have substituted them into one of the equations to find z. To use Cramer's rule, we divide by D, provided D =A 0. If D = 0 and at least one of the other determinants is not 0, then the system is inconsistent. If all the determinants are 0, then the equations in the system are dependent. Exercise Set 3,7 FOR EXTRA HELP Digital Video Tutor CD 3 InterAct Math Math Tutor Center MathXL MyMathLab.com 3 Videotape 6 2 21. What is it about Cramer's rule that makes it useful? Evaluate. 1. |5 1 |2 4| 2. 3 2 2 -3 3. 6 -9 2 3 4. 3 2 -7 5 5. 1 4 0 0 -1 2 3 -2 1 6. 3 0 -2 5 1 2 2 0 -1 7. -1 -2 -3 3 4 2 0 1 2 8. 1 2 2 2 1 0 3 3 1 9. -4 -2 3 -3 1 2 3 4 -2 10. 2 -1 1 1 2 -1 3 4 -3 Solve using Cramer's rule. 11. 5x + 8y = 1, 3x + 7y = 5 12. 3x - 4y = 6 5x + 9y = 10 13. 5x - 4y = -3, 7x + 2y = 6 14. -2x + 4y = 3, 3x - 7y = 1 15. 3x - y + 2z = 1, x - y + 2z = 3, -2x + 3y + z = 1 16. 3x + 2y - z = 4, 3x - 2y + z = 5, 4x - 5y - z = -1 17. 2x - 3y + 5z = 27, x + 2y - z = -4 5x - y + 4z = 27 18. x - y + 2z = -3, x + 2y + 3z = 4 2x + y + z = -3 19. r - 2s + 3t = 6, 2r - s - t = -3, r + s + t = 6 20. a - 3c= 6, b + 2c = 2, 7a - 3b - 5c = 14 21. Which is it about Cramer's rule that makes it useful? 22. Which version of Cramer's rule do you find more useful: the version for 2 x 2 systems or the version for 3 x 3 systems? Why? SKILL MAINTENANCE Solve. 23. 0.5x - 2.34 + 2.4x = 7.8x - 9 24. 5x + 7x = -144 25. A piece of wire 32.8 ft long is to be cut into two pieces, and those pieces are each to be bent to make a square. The length of a side of one square is to be 2.2 ft greater than the length of a side of the other. How should the wire be cut? 26. Inventory. The Freeport College store paid $1728 for a order of 45 calculators. The store paid $9 for each scientific calculator. The others all graphing calculators cost the store $58 each. How many of each type of calculator was ordered? 27. Insulation The Mazzas' attic required three and a half times as much insulation as did the Kranepools'. Together, the two attics required 36 rolls of insulation. How much insulation did each attic require? 28. Sales of food. High Flyin' Wings charges $12 for a bucket of chicken wings and $7 for a chicken din- ner. After filling 28 orders for buckets and dinners, High Flyin' Wings had collected $281. How many buckets and how many dinners did they sell? page 198 y + 3z = 1 (4) y + 3z = 4 Multiplying both sides of equation 1 by -1 -y -3z = -4 y + 3z = 1 y + 3z = 1 This is a contradiction > 0 = -3 The sum of the three numbers is 4. x + y + z =4 x - 2y - z = 1. 2x - y - 2z = -1. x + y + z = 180 x + 70 = z 2y = z page 187 5x - 4y = -1 -2x + 3y = 2. or a b c 0 d e Multiply row 2 by 5. 5 -4 -1 -10 15 10 The profits be P(3000) = 11 * 3000 - 90,000 = - $57,000. page 168 Distance Rate Time Freight Train d 60 t > d = 60t d= 60t. page 200 Supply and Demand As the price of coffee varies the amount sold varies. The table and graph show consumers will demand less as the price goes up. Demand Function D Price, p, Quantity D(p) in per Kilogram millions of kilograms $8.00 25 $9.00 20 $10.00 15 $11.00 10 $12.00 5 As the price of coffee varies the amount available varies. The table below show sellers will supply less as the price goes down. Supply function S Price, p, Quantity S(p) in per Kilogram millions of kilograms $9.00 5 $9.50 10 $10.00 15 $10.50 20 $11.00 25 R(x) = 26x. C(x) = 90,000 + 15x. d = 26x. (1) d = 90,000 + 15x (2) solve using substitution 26x = 90,000 + 15x 11x = 90,000 x = 8181.8. page 201 3.8, ,BUSINESSANDECONOMICAPPLICATIONS ZO'I E x a m p 1 e 2 Find the equilibrium point for the demand and supply functions given: D( p) = 1000 - 60p, (1) S( p) = 200 + 4p. (2) Solution Since both demand and supply are quantities and they are equal at the equilibrium point, we rewrite the system as q = 1000 - 60p, (1) q = 200 + 4p. (2) We substitute 200 + 4p for q in equation (1) and solve: 200 + 4p = 1000 - 60p Substituting 200 + 4p for q in equation (1) 200 + 64p = 1000 Adding 60p to both sides 64p = 800 Adding -200 to both sides aoo = 12.5. P = 500 = 12.5 64 The equilibrium price is $12.50 per unit. To find the equilibrium quantity, we substitute $12.50 into either D(p) or S(p). We use S(p): S(12.5) = 200 + 4(12.5) = 200 + 50 = 250. Exercise Set 3,$ The equilibrium quantity is 250 units, and the equilibrium point is ($12.50, 250). FOR EXTRA HELP Digital Video Tutor CO 3 InterAct Math Math Tutor Center MathXl MvMathlab.com Videotape 6 exercise set 3.8 For each of the following pairs of total-cost and total- revenue fmactions, find (a) the total-profit function and R(x) = 70x (b) the break-even poii2t. 1. C(x) = 45x + 300,000; R(x) = 65x 2. C(x) = 25x + 270,000; R(x) = 70x 3. C(x) = lOx + 120,000; R(x) = 60x page 202 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING 4. C(x) = 30x + 49,500; R(x) = 85x 5. C(x) = 40x + 22,500; R(x) = 85x 6. C(x) = 20x + 10,000; R(x) = 100x 7. C(x) = 22x + 16,000; R(x) = 40x 8. C(x) = 15x + 75,000; R(x) = 55x 9. C(x) = 75x + 100,000; R(x) = 125x 10. C(x) = 20x + 120,000; R(x) = 50x Find the equilibrium point for each of the following pairs of demand and supply functions. 11. D(p) = 1000 - lOp, S(p) = 230 + p 12. D(p) = 2000 - 60p, S(p) = 460 + 94p 13. D(p) = 760 - 13p, S(p) = 430 + 2p 14. D(p) = 800 - 43p, S(p) = 210 + 16p 15. D(p) = 7500 - 25p, S(p) = 6000 + 5p 16. D(p) = 8800 - 30p, S(p) = 7000 + 15p 17. D(p) = 1600 - 53p, S(p) = 320 + 75p 18. D(p) = 5500 - 40p S(p) = 1000 + 85p 19. Computer manufacturing. Biz.com Electronics is planning to introduce a new line of computers. The fixed costs for production are $125,300. The vari- able costs for producing each computer are $450. The revenue from each computer is $800. Find the following a) The total cost C(x) of producing x computers b) The total revenue R(x) from the sale of x computers c) The total profit P(x) from the production and sale of x computers d) The profit or loss from the production and sale of 100 computers; of 400 computers e) The break-even point 20. Manufacturing lamps. City Lights, Inc., is plan- ning to manufacture a new type of lamp. The fixed costs for production are $22,500. The variable costs for producing each lamp are estimated to be $40. The revenue from each lamp is to be $85. Find the following a) The total cost C(x) of producing x lamps b) The total revenue R(x) from the sale of x lamps c) The total profit P(x) from the production and sale of x lamps d) The profit or loss from the production and sale of 3000 lamps; of 400 lamps e) The break-even point 21. Manufacturing caps. Martina's Custom Printing is planning on adding painter's caps to its product line. For the first year, the fixed costs for setting up production are $16,404. The variable costs for pro- ducing a dozen caps are $6.00. The revenue on each dozen caps will be $18.00. Find the following. a) The total cost C(x) of producing x dozen caps b) The total revenue R(x) from the sale of x dozen caps c) The total profit P(x) from the production and sale of x dozen caps d) The profit or loss from the production and sale of 3000 dozen caps; of 1000 dozen caps e) The break-even point 22. Sport coat production. Sarducci's is planning a new line of sport coats. For the first year, the fixed costs for setting up production are $10,000. The variable costs for producing each coat are $30. The revenue from each coat is to be $80. Find the following. a) The total cost C(x) of producing x coats b) The total revenue R(x) from the sale of x coats c) The total profit P(x) from the production and sale of x coats d) The profit or loss from the production and sale of 2000 coats; of 50 coats following. e) The break-even point 23. In Example 1, the slope of the line representing Revenue is the sum of the slopes of the other two lines. This is not a coincidence. Explain why. 24. Variable costs and fixed costs are often compared to the slope and the y-intercept, respectively, of an equation for a line. Explain why you feel this analogy is or is not valid. SKILL MAIN Solve. solve. 25. 3x - 9 = 27 26. 4x - 7 = 53 SUMMARY AND REVIEW: CHAPTER 3 page 203 27. 4x - 5 = 7x - 13 28. 2x + 9 = 8x - 15 29. 7 - 2(x - 8) = 14 30. 6 - 4(3x - 2) = 10 SYNTHESIS 31. Ian claims that since his fixed costs are $1000, he need sell only 20 birdbaths at $50 each in order to break even. Does this sound plausible? Why or why not? 32. In this section, we examined supply and demand functions for coffee. Does it seem realistic to you for the graph of D to have a constant slope? Why or why not? 33. Yo-yo production. Bing Boing Hobbies is willing to produce 100 yo-yo's at $2.00 each and 500 yo-yo's at $8.00 each. Research indicates that the public will buy 500 yo-yo's at $1.00 each and 100 yo-yo's at $9.00 each. Find the equilibrium point. 34. Loudspeaker production. Fidelity Speakers, Inc., has fixed costs of $15,400 and variable costs of $100 for each pair of speakers produced. If the speakers sell for $250 a pair, how many pairs of speakers must be produced (and sold) in order to have enough profit to cover the fixed costs of two addi- tional facilities? Assume that all fixed costs are identical. Use a grapher to solve. 35. Dog food production. Puppy Love, Inc., will soon begin producing a new line of puppy food. The marketing department predicts that the demand function will be D(p) = -14.97p + 987.35 and the supply function will be S( p) = 98.55p - 5.13. a) To the nearest cent, what price per unit should be charged in order to have equilibrium between supply and demand? b) The production of the puppy food involves $87,985 in fixed costs and $5.15 per unit in vari- able costs. If the price per unit is the value you found in part (a), how many units must be sold in order to break even? 36. Computer production. Number Cruncher Com- puters, Inc., is planning a new line of computers, each of which will sell for $970. The fixed costs in setting up production are $1,235,580 and the vari- able costs for each computer are $697. a) What is the break-even point? (Round to the nearest whole number.) ( b) The marketing department at Number Cruncher is not sure that $970 is the best price. Their demand function for the new computers is given by D(p) = -304.5p + 374,580 and their supply function is given by S( p) = 788.7p - 576,504. To the nearest dollar, what price p would result in equilibrium between supply and demand? Summary and Review 3 Key Terms System of equations, p. 143 Matrix (plural, matrices), Cramer's rule, p. 192 Solution of a system, p. 144 p. 187 Total cost, p. 197 Consistent, p. 147 Elements, p. 187 Total revenue, p. 197 Inconsistent, p. 147 Entries, p. 187 Total profit, p. 197 Dependent, p. 147 Rows, p. 187 Fixed costs, p. 197 Independent, p. 147 Columns, p. 187 Variable costs, p. 197 Substitution method, p. 151 Row-equivalent, p. 190 Break-even point, p. 199 Elimination method, p. 153 Dimensions,.p. 191 Demand function, p. 200 Total-value problem, p. 160 Square matrix, p. 191 Supply function, p. 200 Mixture problem, p. 164 Determinant, p. 191 Equilibrium point, p. 200 Motion problem, p. 166 page 204 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING Important Properties and Formulas When solving a system of two linear equa- Determinant of a 3 x 3 Matrix tions in two variables: a1 b1 c1 b2 c2 - a2 b1 c1 a2 b2 c2 =a1 b3 c3 b3 c3 a3 b3 C3 b1 c1 +a3 b2 c2 1. If an identity is obtained, such as 0 = 0, then the system has an infinite number of solutions. The equations are depend-ent and, since a solution exists, the sys-tem is consistent. 2. If a contradiction is obtained, such as 0 = 7, then the system has no solution. The system is inconsistent. To use the elimination method to solve systems of three linear equations: _ bZ cz bl CI - al ~ b3 C3I - a21b3 C3 bl cl + a3lbz c2 l Cramer's Rule: 2 x 2 Systems The solution of the system alx + bly = cl, a2x + b2y = c2 c1 b1 a1 c1 c2 b2 y=a2 c2 x= a1 b1 a1 b1 a2 b2 a2 b2 1. Write all equations in the standard form if it is unique, is given by Ax + By + C1z = D1. 2. Clear any decimals or fractions. 3. Choose a variable to eliminate. Then se- lect two of the three equations and work get one equation in two variables. 4. Next, use a different pair of equations and eliminate the same variable that Cramer's Rule: 3 x 3 Systems . you did in step (3). The solution of the system 5. Solve the system of equations that re- sulted from steps (3) and (4). alx + bly + clz = dl, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3 6. Substitute the solution from step (5) into one of the original three equations and for the third variable. Then check. is found by considering the following determinants: Row-Equivalent Operations Each of the following row-equivalent operations produces a row-equivalent matrix: a1 b1 c1 dl bl cl D = a2 b2 c2, Dx = d2 b2 c2, a3 b3 C3 d3 b3 C3 Determinant of a 2 x 2 Matrix a1 d1 c1 a1 b1 d1 Dy = a2 d2 c2, Dz = a2 b2 d2. a3 d3 C3 a3 b3 d3 a) Interchanging any two rows. b) Multiplying all elements of a row by a nonzero constant. c) Replacing a row by the sum of that row and a multiple of another row. If a unique solution exists, it is given by Dx Dy Dz x= D Y= z D a c b d = ad - bc REVIEW EXERCISES: CHAPTER 3 page 205 Review Exercises For Exercises 1-9, if a system has an infinite number of Solve graphically. 1. 3x + 2y = -4, y = 3x + 7 2. 2x + 3y = 12, 4x - y = 10 Solve using the substitution method. 3. 9x - 6y = 2, x= 4y + 5 4. y = x + 2, y - x = 8 5. x - 3y = -2, 7y - 4x = 6 Solve using the elimination method. 6. 8x - 2y = 10, -4y - 3x = -17 7. 4x - 7y = 18, 9x + 14y = 40 8. 3x - 5y = -4, 5x - 3y = 4 9. 1.5x - 3 = - 2y, 3x + 4y = 6 Solve. 10. Glynn bought two DVD's and one videocassette for $72. If he had purchased one DVD and two video cassettes, he would have spent $15 less. What is the price of a DVD? What is the price of a videocassette? 11. A freight train leaves Chicago at midnight traveling south at a speed of 44 mph. One hour later, a pas- senger train, going 55 mph, travels south from Chicago on a parallel track. How many hours will the passenger train travel before it overtakes the freight train? 12. Yolanda wants 14 L of fruit punch that is 10% juice. At the store, she finds punch that is 15% juice and punch that is 8% juice. How much of each should she purchase? Solve. If a system's equations are dependent or if there is no solution, state this. 13. x + 4y + 3z = 2, 2x + y + z = 10, -x + y + 2z = 8 14. 4x + 2y -6z = 34, 2x + y + 3z = 3, 6x + 3y - 3z = 37 15. 2x - 5y - 2z = -4, 7x + 2y - 5z = -6, -2x + 3y + 2z = 4 16. -5x + 5y = -6, 2x - 2y = 4 17. 3x + y = 2, x + 3y + z = 0, x + z = 2 18. In triangle ABC, the measure of angle A is four times the measure of angle C, and the measure of angle B is 45° more than the measure of angle C. What are the measures of the angles of the triangle? 19. Find the three-digit number in which the sum of the digits is 11, the tens digit is 3 less than the sum of the hundreds and ones digits, and the ones digit is 5 less than the hundreds digit. 20. Lynn has $159 in her purse, consisting of $20, $5, and $1 bills. The number of $20 bills is the same as the total number of $1 and $5 bills. If she has 14 bills in her purse, how many of each denomination does she have? Solve using matrices. Show your work. 21. 3x + 4y = -13, 5x + 6y = 8 22. 3x - y + z = -1, 2x + 3y + z = 4, 5x + 4y + 2z = 5 Evaluate 23. -2 4 -3 5 24. 2 3 0 1 4 -2 2 -1 5 Solve using Cramer's rule. Show your work. 25. 2x + 3y = 6, x - 4y = 14 26. 2x + y + z = -2, 2x - y + 3z = 6, 3x - 5y + 4z = 7 27. Find the equilibrium point for the demand and supply functions S( p) = 60 + 7p and D( p) = 120 - 13p. 28. Robbyn is beginning to produce organic honey. For the first year, the fixed costs for setting up produc- tion are $18,000. The variable costs for producing each pint of honey are $1.50. The revenue from each pint of honey is $6. Find the following. a) The total cost C(x) of producing x pints of honey. page 206 CHAPTER 3 SYSTEMS OF LINEAR EQUATIONS AND PROBLEM SOLVING b) The total revenue R(x) from the sale of x pints of honey c) The total profit P(x) from the production and sale of x pints of honey d) The profit or loss from the production and sale of 1500 pints of honey; of 5000 pints of honey e) The break-even point SYNTHESIS 29. How would you go about solving a problem that in- volves four variables? 30. Explain how a system of equations can be both de- pendent and inconsistent. 31. Robbyn is quitting a job that pays $27,000 a year to make honey (see Exercise 28). How many pints of honey must she produce and sell in order to make the same amount that she made in the job she left? 32. Solve graphically: y = x + 2 y = x2 + 2, 33. The graph of f(x) = ax2 + bx + c contains the points (-2, 3), (1,1), and (0, 3). Find a, b, and c and give a formula for the function. Chapter Test 3 1. Solve graphically: 2x + y = 8, y - x - 2, Solve, if possible, using the substitution method. 2. x + 3y = -8, 4x - 3y = 23 3. 2x + 4y = -6, y = 3x - 9 Solve, if possible, using the elimination method. 4. 4x - 6y = 3, 6x - 4y = -3 5. 4y + 2x = 18, 3x + 6y = 26 6. The perimeter of a rectangle is 96. The length of the rectangle is 6 less than twice the width. Find the di- mensions of the rectangle. 7. Between her home mortgage (loan), car loan, and credit card bill (loan), Rema is $75,300 in debt. Rema's credit card bill accumulates 1.5% interest, her car loan 1% interest, and her mortgage 0.6% in- terest each month. After one month, her total accu- mulated interest is $460.50. The interest on Rema's credit card bill was $4.50 more than the interest on her car loan. Find the amount of each loan. 8. -3x + y - 2z = 8 -x + 2y - z = 5 2x + y + z = -3 9. 6x + 2y - 4z =15 -3x - 4y + 2z = -6 4x - 6y + 3z = 8 10. 2x + 2y = 0, 4x + 4z = 4, 2x + y + z = 2 11. 3x + 3z = 0, 2x + 2y = 2, 3y + 3z = 3 Solve using matrices. 12. 7x - 8y = 10, 9x + 5y = -2 13. x + 3y - 3z = 12, 3x - y + 4z = 0, -x + 2y - z = 1 Evaluate. 14. 4 -2 3 7 15. 3 4 2 2 -5 4 4 5 -3 16. Solve using Cramer's rule: 8x - 3y = 5, 2x + 6y = 3. 17. An electrician, a carpenter, and a plumber are hired to work on a house. The electrician earns $21 per hour, the carpenter $19.50 per hour, and the plumber $24 per hour. The first day on the job, they worked a total of 21.5 hr and earned a total of $469.50. If the plumber worked 2 more hours than the carpenter did, how many hours did the electri- cian work? 18. Find the equilibrium point for the demand and supply functions D( p) = 79 - 8p and S( p) = 37 + 6p. page 207 19. Complete Communications, Inc., is producing a new family radio service model. For the first year, the fixed costs for setting up production are $40,000. The variable costs for producing each radio are $25. The revenue from each radio is $70. Find the following. a) The total cost C(x) of producing x radios b) The total revenue R(x) from the sale of x radios c) The total profit P(x) from the production and sale of x radios d) The profit or loss from the production and sale of 300 radios; of 900 radios e) The break-even point 20. The graph of the function f(x) = mx + b contains the points (-1, 3) and (-2, -4). Find m and b. 21. At a county fair, an adult's ticket sold for $5.50, a senior citizeris ticket for $4.00, and a child's ticket for $1.50. On opening day, the number of adults' and senior citizens' tickets sold was 30 more than the number of children's tickets sold. The number of adults' tickets sold was 6 more than four times the number of senior citizens' tickets sold. Total re- ceipts from the ticket sales were $11,219.50. How many of each type of ticket were sold? Cumulative Review 1-3 Solve. 1. -14.3 + 29.17 = x 2. x + 9.4= -12.6 3. 3.9(-11) = x 4. -2.4x = -48 5. 4x + 7= -14 6. -3 + 5x = 2x + 15 7. 3n - (4n - 2) = 7 8. 6y - 5(3y - 4) = 10 9. 14 + 2c = -3(c + 4) - 6 10. 5x - [4 - 2(6x - 1)] = 12 Simplfy. Do not leave negative exponents in your answers. 11. x4 * x-6 * x13 12. (4x-3y2) (-lOx4 y -7) 13. (6x2y3)2(-2x0y4)3uppercorner 14. y4 Y-6 15. -10a7b-11 25a-4b22 16. (3x4y-2 )4 4x -5 17. (1.95 x 10-3) (5.73 x 10 8) 18. 2.42 x 10 5 6.05 x 10-2 1 19. Solve A = 2 h(b + t) for b. 20. Determine whether (-3, 4) is a solution of 5a - 2b = -23. Graph. 21. y = -2x + 3 22. y = x2 - 1 23. 4x + 16 = 0 24. -3x + 2y = 6 25. Find the slope and the y-intercept of the line with equation - 4y + 9x = 12. 26. Find the slope, if it exists, of the line containing the points (2, 7) and (-l, 3). 27. Find an equation of the line with slope -3 and containing the point (2, -11). 28. Find an equation of the line containing the points (-6, 3) and (4,2) 29. Determine whether the lines are parallel or perpendicular: 2x = 4y + 7, x - 2y = 5. 30. Find an equation of the line containing the point (2,1) and perpendicular to the line x - 2y = 5. CUMULATIVE REVIEW: CHAPTERS 1-3 page 208 CUMULATIVE REVIEW: CHAPTERS 1-3 31. For the graph of f shown, determine the domain, the range f(-3), and any value of x for which f(x) = 5. 4,1 5,3 1, -1 -3, -2 -5, -3 32. Determine the domain of the function given by 7 f(x) = 2x - 1. Given g(x) = 4x - 3 and h(x) = -2x2 + 1, find the following function values. 33. h(4) 34. -g(0) 35. (g * h)(-1) 36. g(a) - h(2a) Solve 37. 3x + y = 4, 6x - y = 5 38. 4x + 4y = 4, 5x - 3y = -19 39. 6x - l0y = -22, -llx - 15y = 27 40. x + y + z= -5, 2x + 3y - 2z = 8, x - y + 4z = -21 41. 2x + 5y - 3z = -11, -5x + 3y - 2z = -7, 3x - 2y + 5z = 12 Evaluate 42. 2 -3 4 1 43. 1 0 1 -1 2 1 2 1 3 44. The sum of two numbers is 26. Three times the smaller plus twice the larger is 60. Find the numbers. 13 + 13 = 26 45. In 1997, there were 652 endangered or threatened species of U.S. animals and plants that had recov- ery plans. By August 2000, there were 923 species with recovery plans. (Source: U.S. Fish and Wildlife Service. Department of the Interior. Find the rate at which recovery plans were being formed. 46. The number of U.S. pleasure trips, in millions, t years after 1994 can be estimated by P( t) = 9t + 616 (Source: Travel Industry Association of America). What do the numbers 9 and 616 signify? 47. In 1989, there were 6.6 million U.S. aircraft depar- tures, and in 1999, there were 8.6 million departures (Source: Air Transport Association of America). Let A(t) represent the number of departures, in mil- lions, t years after 1989. a) Find an equation for a linear function that fits the data. b) Use the function of part (a) to predict the number of departures in 2010. 48. "Soakem" is 34% salt and the rest water. "Rin- sem" is 61% salt and the rest water. How many ounces of each would be needed to obtain 120 oz of a mixture that is 50% salt? 49. Find three consecutive odd numbers such that the sum of four times the first number and five times the third number is 47. 50. Belinda's scores on four tests are 83, 92, 100, and 85. What must the score be on the fifth test so that the average will be 90? 51. The perimeter of a rectangle is 32 cm. If five times the width equals three times the length, what are the dimensions of the rectangle? 52. There are 4 more nickels than dimes in a bank. The total amount of money in the bank is $2.45. How many of each type of coin are in the bank? 53. One month Lori and Jon spent $680 for electric- ity, rent, and telephone. The electric bill was 1 4 of the rent and the rent was $400 more than the phone bill. How much was the electric bill? 54. A hockey team played 64 games one season. It won 15 more games than it tied and lost 10 more games than it won. How many games did it win? lose? tie? 55. Reggie, Jenna, and Achmed are counting calo- ries. For lunch one da , Re ie ate two cookies and a banana, for a total of 260 calories. Jenna had a cup of yogurt and a banana, for a total of 245 calories. Achmed ate a cookie, a cup of yo- gurt, and two bananas, for a total of 415 calories. How many calories are in each item? CUMULATIVE REVIEW: CHAPTERS 1-3 page 209 SYNTHESIS 56. Simplify: (6xa+2yb+2) (-2xa-2yy+1). 57. An automotive dealer discovers that when $1000 is spent on radio advertising, weekly sales increase by $101,000. When $1250 is spent on radio advertising, weekly sales increase by $126,000. Assuming that sales increase according to a linear equation, by what amount would sales increase when $1500 is spend on radio advertising? 58. Given that f(x) = mx + b and that f(5) = -3 when f(-4) = 2, find m and b. page 216 Chapter 4 The Multiplication Principle for inequalities For any real numbers a and b and for any positive number c. a < b is equivalent to ac < bc a > b is equivalent to ac > bc. 4.1 INEQUALITIES AND APPLICATIONS page 219 2. Translate. The record R(t) is to be less than 19.0 sec. we have R(t) < 19.0. We replace R(t) with -0.045t + 19.32 to find the times t that solve the inequality: -0.045t + 19.32 < 19.0. Substituting -o.045t + 19.32 for R(t) 3. Carry out. We solve the inequality: -0.045t + 19.32 < 19.0 -0.045t < -0.32 Adding -19.32 to both sides t > 7.1. Dividing both sides by -0.045, reversing the symbol, and rounding 4. Check. A partial check is to substitute a value for t greater than 7.1. We did that in the Familiarize step. i 5. State. The record will be less than 19.0 sec for races occurring more than t 7.1 years after 1996, or approximately all years after 2003. E x a m p 1 e 8 Earnings plans. On a new job, Rose can be paid in one of two ways: Plan A: A salary of $600 per month, plus a commission of 4% of sales; Plan B: A salary of $800 per month, plus a commission of 6% of sales in excess of $10,000. For what amount of monthly sales is plan A better than plan B, if we assume that sales are always more than $10,000? Solution 1. Familiarize. Listing the given information in a table will be helpful. Plan A: Plan B: Monthly Income Monthly Income $600 salary $800 salary 4% of sales 6% of sales over $10,000 Total: $600 + 4% of sales Total: $800 + 6% of sales over $10,000  Next, suppose that Rose sold a certain amount-say, $12,000-in one month. Which plan would be better? Under plan A, she would earn $600 plus 4% of $12,000, or 600 + 0.04(12,000) _ $l080. Since with plan B commissions are paid only on sales in excess of $10,000, Rose would earn $800 plus 6% of ($12,000 - $10,000), or 800 + 0.06(2000) _ $920. This shows that for monthly sales of $12,000, plan A is better. Similar cal-culations will show that for sales of $30,000 a month, plan B is better. To determine all values for which plan A earns more money, we must solve an inequality that is based on the calculations above. page 213 {x|x < 4}. The set of all x is less than 4. Brackets are used for endpoints. [a,b]= { x|a < x < b} page 214 chapter 4 The Addition Principle for Inequalities the inequalities x > 4 and 4 < x are equivalent. a < b is equivalent to a + c < b + c a > b is equivalent to a + c > b + c. page 219 Plan A A salary of $600 per month plus a commission of 4% of sales. Plan B A salary of $800 per month plus a commission of 6% of sales in excess of $10,000. For what amount of monthly sales is plan A better than plan B if we assume sales are always more than $10,000? Solution Plan A Plan B Monthly Income Monthly Income $600 salary $800 salary 4% of sales 6% of sales over $10,000 Total $600 + 4% of sales Total $8.00 + 6% of sales over $10,000 600 + 0.04(12,000) = $1080. 800 + 0.06(2000) = $920. page 220 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 2. Translate. We let S = the amount of monthly sales, in dollars. Examining the calculations in the Familiarize step, we see that monthly income from plan A is 600 + 0.04S and from plan B is 800 + 0.06(S - 10,000). We want to fmd all values of S for which 3. Carry out. We solve the inequality: 600 + 0.04S > 800 + 0.06(S - 10,000) 600 + 0.045 > 800 + 0.06S - 600 Using the distributive law 600 + 0.04S > 200 + 0.06S Combining like terms 400 > 0.025 Subtracting 200 and 0.04S from both sides 20,000 > S, or S < 20,000. Dividing both sides by 0.02 4. Check. For S = 20,000, the income from plan A is 5. State. For monthly sales of less than $20,000, plan A is better. FOR EXTRA HELP Income from is greater income from plan A than plan B W W 600 + 0.04S > 800 + 0.06(S - 10,000). 600 + 4% ~ 20,000, or $1400. ~ This confirms that for sales totaling $20,000, Rose's pay is the same 800 + 6% ~ (20,000 - 10,000), or $1400. ~ under either plan. The income from plan B is In the Familiarize step, we saw that for sales of $12,000, plan A pays more. Since 12,000 < 20,000, this is a partial check. Since we cannot check all possible values of S, we will stop here. page 220 Determine whether the given numbers are solutions of the inequaliry. 1. x - 1 _> 7; -4, 0, 8, 13 2. 3x + 5 <_ -10; -5, -10, 0, 27 3. t - 6 > 2t - 1; 0, -8, -9, -3 4. 5y - 9 < 3 - y; 2, -3, 0, 3 Graph each inequality, and write the solution set using both set-builder and interval notation. 5. y < 6 6. x > 4 7. x >_ -4 8. t <_ 6 9. t > -3 10. y < -3 ll. x <_ -7 12. x >_ -6 Solve. Then graph. 13. x + 9 > 4 14. x + 5 > 2 15. a + 7 <_ -13 16. a + 9 <- -12 17. x - 5 <_ 7 18. t + 14 >_ 9 19. y - 9 > -18 20. y - 8 > -14 21. y - 20 <_ -6 22. x - 11 <- -2 23. 9t < -81 24. 8x >_ 24 page 221 Exercise Set 4,1 Digital Video Tutor CD 3 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 7 4.1 INEQUALITIESANDAPPLICATIONS ZZ'I 25. 0.5x < 25 26. 0.3x < -18 27. -8y <_ 3.2 28. -9x >_ -8.1 29. 5y <_ 3 -6 4 30. 3 5 -4x _> -8 31. 5y + 13 > 28 32. 2x + 7 < 19 33. -9x + 3x _> -24 34. 5y + 2y <_ -21 35. Let f(x) = 8x - 9 and g(x) = 3x - 11. Find all val- ues of x for which f(x) < g(x). 36. Let f(x) = 2x - 7 and g(x) = 5x - 9. Find all val- ues of x for which f(x) < g(x). 37. Let f(x) = 0.4x + 5 and g(x) = 1.2x - 4. Find all val- ues of x for which g(x) > f(x). 3 1 38. Let f(x) = 8 + 2x and g(x) = 3x - 8. Find all val- ues of x for which g(x) > f(x). Solve. 39. 4(3y - 2) _> 9(2y + 5) 40. 4m + 7 _> 14(m - 3) 41. 5(t - 3) + 4t < 2(7 + 2t) 42. 2(4 + 2x) > 2x + 3(2 - 5x) 43. 5[3m - (m + 4)] > -2(m - 4) 44. 8x - 3(3x + 2) - 5 _> 3(x + 4) - 2x 45. 19 -(2x + 3) <_ 2(x + 3) + x 46. 13 - (2c + 2) _> 2(c + 2) + 3c 47. 1 1 4(8y + 4) - 17 < -2(4y - 8) 48. 1 1 3(6x + 24) - 20 > - 4(12x - 72) 49. 2[8 - 4(3 - x)]-2_>8[2(4x - 3) + 7]- 50 50. 5[3(7 - t) - 4(8 + 2t)] - 20 <_ -6[2(6 + 3t) - 4] 51. Truck rentals. Campus Entertainment rents a truck for $45 plus 20cents per mile. A budget of $75 has been set for the rental. For what mileages will they not exceed the budget? 52. Truck rentals Metro Concerts can rent a truck for either $55 with unlimited mileage or $29 plus 40cents per mile. For what mileages would the unlimited mileage plan save money? 53. Insurance claims. After a serious automobile acci- dent, most insurance companies will replace the damaged car with a new one if repair costs exceed 80% of the NADA, or "blue-book," value of the car. Miguel's car recently sustained $9200 worth of damage but was not replaced. What was the blue-book value of his car? 54. Phone rates. A long-distance telephone call using Down East Calling costs 10 cents for the first minute and 8 cents for each additional minute. The same call, placed on Long Call Systems, cost 15 cents for the first minute and 6 cents for each additional minute. For what length phone calls is Down East Calling less expensive? Phone rates. In Vermont, Uerizon charges customers $13.55 for monthly service plus 2.2cents per minute for local phone calls between 9 A.M. and 9 P.M. weekdays. The charge for off-peak local calls is 0.5cents per minute. Calls are free after the total monthly charges reach $39.40. 55. Assume that only peak local calls were made. For how long must a customer speak on the phone if the $39.40 maximum charge is to apply? 56. Assume that only off-peak calls were made. For how long must a customer speak on the phone if the $39.40 maximum charge is to apply? 57. Checking account rates The Hudson Bank offers two checking-account plans. Their Anywhere plan charges 20cents per check whereas their Acu-checking plan costs $2 per month plus 12cents per check. For what numbers of checks per month will the Acu- checking plan cost less? 58. Moving costs Musclebound Movers charges $85 plus $40 an hour to move households across town. Champion Moving charges $60 an hour for cross town moves. For what lengths of time is Champion more expensive? 59. Wages. Toni can be paid in one of two ways: Plan A: A salary of $400 per month, plus a com- mission of 8% of gross sales; plan B: A salary of $610 per month, plus a com- mission of 5% of gross sales. For what amount of gross sales should Toni select plan A? 60. Wages. Branford can be paid for his masonry work in one of two ways: Plan A: $300 plus $9.00 per hour; Plan B: Straight $12.50 per hour. Suppose that the job takes n hours. For what values of n is plan B better for Branford? page 222 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 61. Insurance benefits. Bayside Insurance offers two plans. Under plan A, Giselle would pay the first $50 of her medical bills and 20% of all bills after that. Under plan B, Giselle would pay the first $250 of bills, but only 10% of the rest. For what amount of medical bills will plan B save Giselle money? (Assume that her bills will exceed $250.) 62. Wedding costs. The Arnold Inn offers two plans for wedding parties. Under plan A, the inn charges $30 for each person in attendance. Under plan B, the inn charges $1300 plus $20 for each person in excess of the first 25 who attend. For what size par- ties will plan B cost less? (Assume that more than 25 guests will attend.) 63. Show business. Slobberbone receives $750 plus 15% of receipts over $750 for playing a club date. If a club charges a $6 cover charge, how many people must attend in order for the band to receive at least $1200? 64. Temperature conversion. The function C(F) =5(F - 32) 9 can be used to find the Celsius temperature C(F) that corresponds to F° Fahrenheit. a) Gold is solid at Celsius temperatures less than 1063°C. Find the Fahrenheit temperatures for which gold is solid. b) Silver is solid at Celsius temperatures less than 960.8°C. Find the Fahrenheit temperatures for which silver is solid. 65. Manufacturing. Ergs, Inc., is planning to make a new kind of radio. Fixed costs will be $90,000, and variable costs will be $15 for the production of each radio. The total-cost function for x radios is C(x) = 90,000 + 15x. The company makes $26 in revenue for each radio sold. The total-revenue function for x radios is R(x) = 26x. (See Section 3.8.) a) When R(x) < C(x), the company loses money. Find the values of x for which the company loses money. b) When R(x) > C(x), the company makes a profit. Find the values of x for which the company makes a profit. 66. Publishing The demand and supply functions for a locally produced poetry book are approximated by D( p) = 2000 - 60p and S( p) = 460 + 94p, where p is the price in dollars (see Section 3.8). a) Find those values of p for which demand exceeds supply. b) Find those values of p for which demand is less than supply. 67. Explain in your own words why the inequality symbol must be reversed when both sides of an inequality are multiplied by a negative number. 68. Why isn't roster notation used to write solutions of inequalities? SKILL MAINTENANCE Find the domain of f. 69. f(x) = 3 x - 2 70. f(x) = x - 5 4x + 12 71. f(x) = 5x 7 - 2x 72. f(x) x + 3 9 - 4x Simplify. 73. 9x - 2(x - 5) 74. 8x + 7(2x - 1) SYNTHESIS 75. A Presto photocopier costs $510 and an Exact Image photocopier costs $590. Write a problem that involves the cost of the copiers, the cost per page of photocopies, and the number of copies for which the Presto machine is the more expensive. machine to own. 76. Explain how the addition principle can be used to avoid ever needing to multiply or divide both sides of an inequality by a negative number. Solve. Assume that a, b, c, d, and m are positive constants. 77. 3ax + 2x _> 5ax - 4; assume a > 1 78. 6by - 4y <_ 7by + 10 79. a(by - 2) > b(2y + 5); assume a > 2 80. c(6x - 4) < d(3 + 2x); assume 3c > d 81. c(2 - 5x) + dx > m(4 + 2x); assume 5c + 2m < d 82. a(3 - 4x) + cx < d(5x + 2); assume c > 4a + 5d 4.1 INEQUALITIES AND APPLICATIONS page 223 Determine whether each statement is true or false. If false, give an example that shows this. 83. For any real numbers a, b, c, and d, if a < b and t c < d,then a - c < b - d. 84. For all real numbers x and y, if x < y, then x2 < y2. 85. Are the inequalities x < 3 and x + 1 < 3 + 1 x x equivalent? Why or why not? 1 86. Are the inequalities x < 3 and 0 * x < 0 - 3 equivalent? Why or why not? Solve. Then graph. 87. x + 5 <_ 5 + x 88. x + 8 < 3 + x 89. x2 > 0 1 90. Assume that the graphs of yl = - 2x + 5, y2 = x - 1, and y3 = 2x - 3 are as shown below. Solve each inequality, referring only to the figure. 1 a) -2x + 5 >x - 1 b) x - 1 <_ 2x - 3 1 c) 2x - 3 _> 2x + 5 91. Using an approach similar to that in the Technol-ogy Connection on p. 217, use a grapher to check your answers to Exercises 13, 37, 53, and 61. CORNER rIll Reduce, Reuse, and Recycle Focus: Inequalities and problem solving Time. 15-20 minutes Group size: 2 ACTIVITY Assume that the amount of solid waste being generated and the amount recycled are both in-creasing linearly. One group member should find a linear function w for which w(t) repre- In the United States, the amount of solid waste sents the number of pounds of waste generated (rubbish) being recycled is slowly catching up to per person per day t years after 1991. The other the amount being generated. In 1991, each per- group member should find a linear function r son generated, on average, 4.3 1b of solid waste for which r(t) represents the number of pounds every day, of which 0.8 1b was recycled. In 2000, recycled per person per day t years after 1991. each person generated, on average, 4.4 1b of - Finally, working together, the group should solid waste, of which 1.3 lb was recycled. determine those years for which the amount (Sources: U.S. Census 2000 and EPA Municipal recycled will meet or exceed the amount Solid Waste Factbook) generated. page 224 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 4. Intersections of Sets and Conjunctions of Sentences • Unions of Sets and Disjunctions of Sentences • Interval Notation and Domains We now consider compound inequalities-that is, sentences like " -2 < x and x < 1" or "x < -3 or x > 3" that are formed using the word and or the word or. Intersections of Sets and Conjunctions of Sentences The intersection of two sets A and B is the set of all elements that are common to both A and B. We denote the intersection of sets A and B as A n B. The intersection of two sets is often pictured as shown here. A n s E X a m p 1 e 1 Find the intersection: {1, 2, 3, 4, 5} n {-2, -1, 0,1, 2, 3}. Solution The numbers 1, 2, and 3 are common to both sets, so the intersec-tion is { 1, 2, 3}. When two or more sentences are joined by the word and to make a com-pound sentence, the new sentence is called a conjunction of the sentences. The following is a conjunction of inequalities: -2 < x and x < 1. A number is a solution of a conjunction if it is a solution of both of the separate parts. For example, -1 is a solution because it is a solution of -2 < x as well as x < 1. Below we show the graph of -2 < x, followed by the graph of x < 1, and finally the graph of the conjunction -2 < x and x < 1. Note that the solution set of a conjunction is the intersection of the solution sets of the individual sentences. {x| -2 < x} -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2, 8 {x|x < 1} -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -8, 1 {x|-2 < x}n{x|x < 1} = {x| -2 < x and x < 1 } -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2, 1 4.2 INTERSECTIONS, UNIONS, AND COMPOUND INEQUALITIES page 229 Example 8 Solve: 3x - 11 < 4 or 4x + 9 _> 1. Solution We solve the individual inequalities separately, retaining the word or: 3x - 11 < 4 or 4x + 9 _> 1 3x < 15 or 4x _> -8 x < 5 or x _> -2. T Keep the word "or." To find the solution set, we first look at the individual graphs. {x| x < 5} E I I I I I I i I I -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 {x| x _> -2} E--i--Tr-~ ~-f-•I I F-+-E-I-~ f-2~ °°) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 {x|x < 5} U {x| x _> -2} I ~-f- ~I-~^--1 F ~ ' > (_oo, m) = IIB _ {x| x < 5 or _> -2} -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Since all numbers are less than 5 or greater than or equal to -2, the two sets fill the entire number line. Thus the solution set is f{8, the set of all real numbers. Interval Notation and Domains In Section 2.2, we saw that if g(x) = (5x - 2)/(3x - 7), then the domain of g = {x| x is a real number and x =/ 7 3}. We can now represent such a set using interval notation: {x| x is a real number and x =/7 = -8 7 7 8 8 is a sideways 8. 3} 3 u 3 -3 -2 -1 0 1 2I 3 4 5 6 7 E x a m p 1 e 9 Use interval notation to write the domain of f if f (x) = x + 2. Solution The expression x + 2 is not a real number when x + 2 is nega-tive. The domain of f is the set of all x-values for which x + 2 >- 0. Since x + 2 _> 0 is equivalent to x >_ -2, we have Domain of f = {x|x _> -2} = [-2, 8). page 230 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING technology connection To visualize the domain of a sum, difference, or product 1. Determine algebraically the domains for yj and yz and of two functions, we have graphed below yj = V-3- x and TRACE each curve to verify your answer. yz= x+1. 2. The Y-VARS option of the VARS key permits us to ac-cess the functions already entered. Use this feature to Yt = (X), v, _ () let y3 = yj + Yz, y4 = yj - Yz, ye = Yz - yn and i o ys = yj - yz. Determine the domains of yj + yz, yj - yz, _ , yz - yi, and yj - yZ algebraically and then check by graphing. Y, -s 1s FOR EXTRA HELP Exercise Set 4,2 Digital Video Tutor CD 3 InterAct Math Math Tutor Center MathXL MyMathLab.com Find each indicated intersection or union. 1. {7,9,11}^{9,11,13} 2. {2,4,8}U{8,9,10} 3. {l,5,10,15} U {5,15,20} 4. {2,5,9,13} nis upside down u {5,8,10} 5. {a,b,c,d,e,f}n{b,d,f} 6. {a, b, c} U {a, c} 7. {r,s,t}U{r,u,t,s,v} 8. {m, n, o, p} n {m, o, p} 9. {3,6,9,12} n {5,10,15} 10. { 1, 5,9} U {4,6,8} 11. {3,5,7} U 0 12. {3,5,7} n 0 has a / through it. Graph and write interval notation for each compound inequality. 13. 3 < x < 8 14. O <_ y <_ 4 15. -6 <_ y <_-2 16. -9 <_ x < -5 17. x < -2 or x > 3 18. x < -5 or x > 1 19. x <_ -l or x > 5 20. x <_ -5 or x > 2 21. -4 <_ -x < 2 22. x > -7 and x < -2 23. x > -2 and x < 4 24. 3 > -x _> -1 25. 5 > a or a > 7 26. t _> 2 or -3 > t 27. x _> 5 or -x _> 4 28. -x < 3 or x < -6 29. 4 > y and y _> -6 30. 6 > -x _> 0 31. x < 7 and x _> 3 32. x _> -3 and x < 3 33. t < 2 or t < 5 34. t > 4 or t > -1 35. x > -1 or x <_ 3 36. 4 > x or x _> -3 37. x _> 5 and x > 7 38. x <_ 5 -4 and x < 1 Solve and graph each solution set. 39. -1 < t + 2 < 7 40. -3 < t + 1 <_ 5 41. 2 < x + 3 and x + 1 <_ 5 42. -1 < x + 2 and x - 4 < 3 43. -7 <_ 2a - 3 and 3a + 1 < 7 page 231 44. -4 <_ 3n + 2 and 2n - 3 <_ 5 Aha' 45. x + 7 <_ -2 or x + 7 _> -3 46. x + 5 < -3 or x + 5 _> 4 47. 2 <_ f(x) <_ 8, where f(x) = 3x - 1 48. 7 _> g(x) _> -2, where g(x) = 3x - 5 49. -21 < f(x) < 0, where f(x) = -2x - 7 50. 4 > g(t) _> 2, where g(t) = -3t - 8 51. f(x) <_ 2 or f(x) _> 8, where f(x) = 3x - 1 52. g(x) <_ -2 or g(x) _> 10, where g(x) = 3x - 5 53. f(x) < -1 or f(x) > 1, where f(x) = 2x - 7 54. g(x) < -7 or g(x) > 7, where g(x) = 3x + 5 55. 6 > 2a - 1 or -4 <_ -3a + 2 56. 3a - 7 > -10 or 5a + 2 <_ 22 57. a + 4 < -l and 3a - 5 < 7 58. 1 - a < -2 and 2a + 1 > 9 59. 3x + 2 < 2 or 4 - 2x < 14 60. 2x - 1 > 5 or 3 - 2x _> 7 61. 2t - 7 <_ 5 or 5 - 2t > 3 62. 5 - 3a <_ 8 or 2a + 1 > 7 For f(x) as given, use interval notation to write the do- main of f. 63. f(x) = 9 x + 7 64. f(x) = 2 x + 3 ___ 65. f (x) = Vx-6 ___ 66. f (x) = Vx-2 x + 3 67. f(x) = 2x - 5 68. f(x) = x - 1 3x + 4 ______ 69. f(x) = V2x + 8 ______ 70. f(x) = V8 - 4x ______ 71. f(x) = V8 - 2x _______ 72. f(x) = V10 - 2x 73. Why can the conjunction 2 < x and x < 5 be rewritten as 2 < x < 5, but the disjunction 2 < x or x < 5 cannot be rewritten as 2 < x < 5? 74. Can the solution set of a disjunction be empty? Why or why not? SKILL MAINTENANCE Graph. 75. y = 5 76. y = -2 77. f(x) = |x| ` 78. g(x) = x - 1 79. y = x -3, y = 5 80. y = x + 2, y = -3 SYNTHESIS 81. What can you conclude about a, b, c, and d, if [a, b] U [c, d] = [a, b]? Why? 82. What can you conclude about a, b, c, and d, if [a, b] n [c, d] = [a, b]? Why? 83. Use the accompanying graph of f(x) = 2x - 5 to solve -7 < 2x - 5 < 7. The line begins in lower left corner at -8 -7 ends in upper right corner between y and 8. y 8 7 6 5 4 3 2 1 -4 -3 -2 - 1 1 2 3 4 5 6 x -1 -2 -3 f(x)= 2x - 5 -4 -5 -6 -7 -8 84. Use the accompanying graph of g(x) = 4 - x to solve 4 - x < -2 or 4 - x > 7. y 8 7 6 5 g(x)= 4 - x 4 3 2 1 -5 -4 -3 -2 - 1 1 2 3 4 5 6 x -1 -2 -3 -4 -5 85. Minimizing tolls A $3.00 toll is charged to cross the bridge from Sanibel Island to mainland Florida. A six-month pass, costing $15.00, reduces the toll to $0.50. A one-year pass, costing $150, al- lows for free crossings. How many crossings per year does it take, on average, for the two six- month passes to be the most economical choice? Assume a constant number of trips per month. 4.2 INTERSECTIONS, UNIONS, AND COMPOUND INEQUALITIES page 231 page 232 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 86. Pressure at sea depth. The function P(d) = 1 + d 33 gives the pressure, in atmospheres (atm), at a depth of d feet in the sea. For what depths d is the pressure at least 1 atm and at most 7 atm? 87. Converting dress sizes. The function f(x) = 2(x + 10) can be used to convert dress sizes x in the United States to dress sizes f(x) in Italy. For what dress sizes in the United States will dress sizes in Italy be between 32 and 46? 88. Solid-waste generation. The function w(t) = O.Olt + 4.3 can be used to estimate the number of pounds of solid waste, w(t), produced daily, on average, by each person in the United States, t years after 1991. For what years will waste production range from 4.5 to 4.75 1b per person per day? 89. Temperatures of liquids. The formula 5 C = 9 (F - 32) can be used to convert Fahrenheit temperatures F to Celsius temperatures C. a) Gold is liquid for Celsius temperatures C such that 1063° <_ C < 2660°. Find a comparable in- equality for Fahrenheit temperatures. b) Silver is liquid for Celsius temperatures C such that 960.8° <_ C < 2180°. Find a comparable inequality for Fahrenheit temperatures. 90. Records in the women's 100-m dash. Florence Griffith loyner set a world record of 10.49 sec in the women's 100-m dash in 1988. The function R(t) = -0.0433t + 10.49 can be used to predict the world record in the women's 100-m dash tyears after 1988. Predict (in terms of an inequality) those years for which world record was between 11.5 and 10.8 sec. (Measure from the middle of 1988.) Solve and graph. 91. 4a - 2 <_ a + 1 <_ 3a + 4 92. 4m - 8 > 6m + 5 or 5m - 8 < -2 93. x - 10 < 5x + 6 <_ x + 10 94. 3x < 4 - 5x < 5 + 3x Determine whether each sentence is true or false for all real numbers a, b, and c. 95. If -b < -a, then a < b. 96. If a <_ c and c <_ b, then b > a. 97. If a < c and b < c, then a < b. 98. If -a < c and -c > b, then a > b. For f(x) as given, use interval notation to write the do-main of f. ______ 99. f(x) = V5 + 2x x - 1 ____ 100. f(x) = V3 - 4x x + 7 101. Let y1 = -1,y2 = 2x + 5 and y3 = 13. Then use the graphs of yl, y2, and y3 to check the solution to Example 2. 102. Let yl = -2x - 5, y2 = -2, y3 = x - 3, and y4 = -10. Then use the graphs of yl, y2, y3, and y4 to check the solution to Example 7. 103. Use a grapher to check your answers to Exercises 33-36 and Exercises 53-56. 104. On many graphers, the TEST key provides access to inequality symbols, while the LOGIC option of that same key accesses the conjunction and and the disjunction or. if yl = x > -2 and y2 = x < 4, Exercise 23 can be checked by form- ing the expression y3 = yl and y2. As in the Tech- nology Connection on p. 217, the interval(s) in the solution set are shown as a horizontal line 1 unit above the x-axis. (Be careful to "deselect" yl and y2 so that only y3 is drawn.) Use this approach to check your answers to Exercises 29 and 60. 4.3 ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES page 233 CORNER IM Saving on Shipping! Focus: Compound inequalities and solution sets Time. 20-30 minutes Group size: 2-3 At present (2001), the U.S. Postal Service charges 21 cents per ounce plus an additional 13-cent delivery fee (1 oz or less costs 34 cents; more than 1 oz, but not more than 2 oz, costs 55 cents; and so on). Rapid Delivery charges $1.05 per pound plus an additional $2.50 delivery fee (up to 16 oz costs $3.55; more than 16 oz, but less than or equal to 32 oz, costs $4.60; and so on). Let x be the weight, in ounces, of an item being mailed.* ACTIVITY One group member should determine the func-tion p, where p(x) represents the cost, in dollars, of mailing x ounces at a post office. Another group member should determine the function r, where r(x) represents the cost, in dollars, of mailing x ounces with Rapid Delivery. The third group member should graph p and r on the same set of axes. Finally, working together, use the graph to determine those weights for which the Postal Service is less expensive. Express your answer using both set-builder and interval notation. *Based on an article by Michael Contino in Mathematics Teacher, May 1995. Absolute-Value Equations and Inequalities 4.3 Equations with Absolute Value • Inequalities with Absolute Value Equations with Absolute Value Recall from Section 1.2 the definition of absolute value. Absolute Value The absolute value of x, denoted |x|, is defined as x, if x _> 0, |x| = -x, if x < 0. (When x is nonnegative, the absolute value of x is x. When x is negative, the absolute value of x is the opposite of x.) page 234 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING To better understand this definition, suppose x is -5. Then |x| = |-5| = 5, and 5 is the opposite of -5. This shows that when x represents a negative num-ber, we have |x| = -x. Since distance is always nonnegative, we can think of a number's absolute value as its distance from zero on a number line. Example 1 Find the solution set: (a) |x| = 4; (b) |x| = 0; (c) |x| = -7. Solution a) We interpret |x| = 4 to mean that the number x is 4 units from zero on a number line. There are two such numbers, 4 and -4. The solution set is {-4, 4}. 4 units 4 units -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 |x| = 4 A second way to visualize this problem is to graph f(x) = |x| (see Section 2.1). We also graph g(x) = 4. The x-values of the points of intersec-tion are the solutions of |x| = 4. b) We interpret |x| = 0 to mean that x is 0 units from zero on a number line. The only number that satisfies this is 0 itself. Thus the solution set is {0}. c) Since distance is always nonnegative, it doesn't make sense to talk about a number that is -7 units from zero. Remember: The absolute value of a number is never negative. |x| = -7 has no solution; the solution set is 0/. Study Tip The absolute-value principle will be used with a variety of replacements for X. Make sure that the principle, as stated here, makes sense before going further. Example 1 leads us to the following principle for solving equations. The Absolute-Value Principle for Equations For any positive number p and any algebraic expression X: a) The solutions of |X| = p are those numbers that satisfy X = -p or X = p. b) The equation |X| = 0 is equivalent to the equation X = 0. c) The equation |X| = -p has no solution. 4.3 ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES page 239 Example 8 Solve |3x - 2| < 4. Then graph. Solution We use part (b) of the principles listed above. In this case, X is 3x - 2 and p is 4: |X| < P |3x - 2| < 4 Replacing X with 3x - 2 and p with 4 -4 < 3x - 2 < 4 The number 3x - 2 must be within 4 units of zero. -2 < 3x < 6 Adding 2 2 1 - 3 < x < 2. Multiplying by 3 2 The solution set is {x| - 3 < x < 2}. In interval notation, the solution set is 2 - 3, 2). The graph is as follows: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 |3x -2| < 4 To solve an inequality like |4x + 2| _> 6 with a grapher, graph the equations y1 = abs(4x + 2) and y2 = 6. Then use INTERSECT to find where yl = y2. The x-values on the graph of yl = |4x + 2| that are on or above the line y = 6 solve the inequality. Given that f (x) = |4x + 2|, find all x for which f(x) _> 6. Solution We have f(x) _> 6, or |4x + 2| _> 6. Substituting To solve, we use part (c) of the principles listed above. In this case, X is 4x + 2 and p is 6: |X| _> p |4x + 2| _> 6 Replacing X with 4x + 2 and p with 6 4x + 2 <_ -6 or 6 <_ 4x + 2 The number 4x + 2 must be at least 6 units from zero. 4x <_ -8 or 4 <_ 4x Adding -2 x <_ -2 or 1 <_ x. Multiplying by 1 4 The solution set is {x | x <_ -2 or x _> 1}. In interval notation, the solution is (-00, -2] U [1,8 is sideways). The graph is as follows: -7 -6 -5 -4 - 3 -2 -1 0 1 2 3 4 5 6 7 |4x + 2| _> 6. page 24O CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING Exercise Set 4.3 FOR EXTRA HELP Digital Video Tutor CD 3 Videotape 8 Solve. 1. |x| = 4 2. |x| = 9 Aha~ 3. |x| = -5 4. |x| = - 3 5. |Y| 7.3 6. |p| = 0 7. |m| = 0 8. |t| = 5.5 9. |5x + 2| = 7 10. |2x - 3| = 4 11. |7x - 2| = -9 12. |3x - 10| = -8 13. |x - 3| = 8 14. |x - 2| = 6 15. |x - 6| = 1 16. |x - 5| = 3 17. |x - 4| = 5 18. |x - 7| = 9 19. |2y| - 5 = 13 20. |5x| - 3 = 37 21. 7 |z| + 2 = 16 22. 5 |q| - 2 = 9 23. 4 - 5x = 3 6 24. 2x - 1 = 5 3 25. |t - 7| + 1 = 4 26. |m + 5| + 9 = 16 27. 3 |2x - 5| - 7 = -1 28. 5 - 2 |3x - 4| = -5 29. Let f(x) = |3x - 4|. Find all x for which f(x) = 8. 30. Let f(x) = |2x - 7|. Find all x for which f(x) = 10. 31. Let f(x) = |x| - 2. Find all x for which f(x) = 6.3. 32. Let f(x) = |x| + 7. Find all x for which f(x) = 18. 33. Let f(x) = 3x - 2. Find all x for which f(x) = 2. 5 34. Let f(x) = 1 - 2x Find all x for which f(x) = 1. 3 Solve 35. |x + 4| = |2x - 7| 36. |3x + 5| = |x - 6| 37. |x - 9| = |x + 6| 38. |x + 4| = |x - 3| 39. |5t + 7| = |4t + 3| 40. |3a - 1| = |2a + 4| Pha'41. |n - 3| = |3 - n| 42. |y - 2| = |2 - Y| 43. |7 - a| = |a + 5| 44. |6 - t| = |t + 7| 45. 1 1 2x - 5 = |4x + 3| 46. 2 7 2 - 3 x| = |4 + 8x| Solve and graph. 47. |a| <_ 7 48. |x| < 2 49. |x| > 8 50. |a| _> 3 51. |t| > 0 52. |t| _> 1.7 53. |x - 3| < 5 54. |x - 1| < 3 55. |x + 2| <_ 6 56. |x + 4| <_ 1 57. |x - 3| + 2 > 7 58. |x - 4| + 5 > 2 Pha~59. |2y - 7| > -5 60. |3y - 4| > 8 61. |3a - 4| + 2 _> 8 62. |2a - 5| + 1 _> 9 63. |y - 3| < 12 64. |p - 2| < 3 65. 9- |x + 4| <_5 66. 12 - |x - 5| <_ 9 67. |4 - 3y| > 8 68. |7 - 2y| < -6 Pha~ 69. |3 - 4x| < -5 70. 7 + |4a - 5| <_ 26 71. 2 - 5x _> = 2 4 3 72. 1 + 3x > 7 5 8 73. |m + 5| + 9 <_ 16 74. |t - 7| + 3 _> 4 75. 25 - 2 |a + 3| > 19 76. 30 - 4 |a + 2| > 12 77. Let f(x) = |2x - 3|. Find all x for which f(x) <_ 4. 78. Let f(x) =|5x + 2|. Find all x for which f(x) < 3. 79. Let f(x) = 2 + |3x - 4|. Find all x for which f(x) > 13. 80. Let f(x) = |2 - 9x|. Find all x for which f(x) _> 25. 81. Let f(x) = 7 + |2x - 1|. Find all x for which f(x) < 16. 82. Let f(x) = 5 + |3x + 2|. Find all x for which f(x) < 19. 83. Explain in your own words why -7 is not a solution of |x| < 5. 84. Explain in your own words why [6, 8issidways) is only part of the solution of |x| _> 6. 4.3 ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES 241 page 241 SKILL MAINTENANCE Solve using substitution or elimination. 85. 2x - 3y = 7, 3x + 2y = -10 86. 3x - 5y = 9, 4x - 3y = 1 87. x = -2 + 3y, x - 2y = 2 88. y = 3 - 4x, 2x - y = -9 Solve graphically. 89. x + 2y = 9, 3x - y = -1 90. 2x + y = 7, -3x - 2y = 10 SYNTHESIS 91. Is it possible for an equation in x of the form |ax + b| = c to have exactly one solution? Why or why not? 92. Explain why the inequality |x + 5| _> 2 can be in- terpreted as "the number x is at least 2 units from -5." 93. From the definition of absolute value, |x| = x only when x _> 0. Solve |3t - 5| = 3t - 5 using this same reasoning. Solve. 94. |x + 2| > x 95. 2 <_ |x - 1| <_5 96. |5t - 3| = 2t + 4 97. t - 2 <_ |t - 3| 98. -3 < x < 3 99. -5 <_ y <_ 5 100. x <_ -6 or 6 <_ x 101. x < -4 or 4 < x 102. x < -8 or 2 < x 103. -5 < x < 1 104. x is less than 2 units from 7. 105. x is less than 1 unit from 5. Write an absolute-value inequality for which the inter- val shown is the solution. 106. < -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 107. < -5 -4 -3 -2 -I 0 1 2 3 4 5 6 7 8 9 108. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 109. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 110. Motion of a spring A weighted spring is bounc- ing up and down so that its distance d above the ground satisfies the inequality |d - 6 ft| <_ 1 ft (see 2 the figure below). Find all possible distances d. 6 ft on left 4 feet in center 8 ft on right 111. Use the accompanying graph of f(x) = |2x - 6| to solve |2x - 6| <_ 4. start at 6 end at 3 and 6,5. y 6 5 4 3 2 1 112. Describe a procedure that could be used to solve any equation of the form g(x) < c graphically. 113. Use a grapher to check the solutions to Examples 3 and 5. 114. Use a grapher to check your answers to Exercises 1, 9, 15, 41, 53, 63, 71, and 95. 115. Isabel is using the following graph to solve |x - 3| < 4. 10 is on all 2 sides. -10 How can you tell that a mistake has been made? page 242 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING Inequalities in 4.4 TWO Variables Graphs of Linear Inequalities • Systems of Linear Inequalities In Section 4.1, we graphed inequalities in one variable on a number line. Now we graph inequalities in two variables on a plane. Graphs of Linear Inequalities When the equals sign in a linear equation is replaced with an inequality sign, a linear inequality is formed. Solutions of linear inequalities are ordered pairs. E x a m p 1 e 1 Determine whether (-3, 2) and (6, -7) are solutions of the inequality 5x - 4y > 13. Solution Below, on the left, we replace x with -3 and y with 2. On the right, we replace x with 6 and y with -7. 5x - 4y > 13 5x - 4y > 13 5(-3) - 4 * 2 ? 13 5(6) - 4(-7) ? 13 -15 -8 30 + 28 -23 13 False 58 13 TRUE Since -23 > 13 is Since 58 > 13 is true, false, (-3, 2) is not (6, -7) is a solution. a solution. The graph of a linear equation is a straight line. The graph of a linear in-equality is a half-plane, bordered by the graph of the related equation. To find an inequality's related equation, we simply replace the inequality sign with an equals sign. Example 2 Graph: y <_ x. Solution We first graph the related equation y = x. Every solution of y = x is an ordered pair, like (3, 3), in which both coordinates are the same. The graph of y = x is shown on the left below. Since the inequality symbol is <_, the line is drawn solid and is part of the graph of y <_ x. dotted lines with dots on graph 1 1, 1 2, 2 3, 3 -3, -3 -2,-2 -1, -1 y 5 4 3 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x -1 -2 -3 -4 -5 dotted lines with dots on graph 2 4, 2 2, -2 3, -4 -3, -5 4.4 INEQUALITIES IN TWO VARIABLES page 243 StudyTip Note that in the graph on the right each ordered pair on the half-plane below y = x contains a y-coordinate that is less than the x-coordinate. All these pairs represent solutions of y <_ x. We check one pair, (4, 2), as follows: y < x 2 4 TRUE 4,2 y <_ x 5, -2 2, -4 -3, -5 It is never too soon to begin reviewing for the final examination. Take a few 2 I 4 TRUE minutes each week to review important problems, formulas, and properties. There is also at least one Connecting the Concepts in each chapter. Spend time reviewing the information in this special feature. It turns out that any point on the same side of y = x as (4,2) is also a solution. if one point in a half-plane is a solu-tion, then all points in that half-plane are solutions. We complete the drawing of the solution set by shading the half-plane below y = x. The complete solution set consists of the shaded half-plane and the line. Note too that for any inequality of the form y <_ f (x) or y < f(x), we shade below the graph of y = f(x). Example 3 Graph: 8x + 3y > 24. Solution First, we sketch the line 8x + 3y = 24. Since the inequality sign is >, points on this line do not represent solutions of the inequality, so the line is drawn dashed. Points representing solutions of 8x + 3y > 24 are in either the half-plane above the line or the half-plane below the line. To determine which, we select a point that is not on the line and determine whether it is a solution of 8x + 3y > 24. We try (-3, 4) as a test point: 8x + 3y > 24 8(-3) + 3 * 4 ? 24 -24 + 12 -12 ~ 24 FALSE Since -12 > 24 is false, (-3,4) is not a solu- tion. no point in s the half-plane contain- ing (-3, 4) is a solution. ' 6 The points in the other This point is ; (-3. 4) _ = 4 half-plane are solu- not a solution. tions, so we shade that z half-plane and obtain the graph shown at right. page 244 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING Steps for Graphing Linear Inequalities 1. Replace the inequality sign with an equals sign and graph this related equation. If the inequality symbol is < or >, draw the line dashed. If the inequality symbol is <_ or >_, draw the line solid. 2. The graph consists of a half-plane on one side of the line and, if the line is solid, the line as well. a) If the inequality is of the form y < f(x) or y <_ f(x), shade below the line. If the inequality is of the form y _> f(x) or y ? f(x), shade above the line. b) If y is not isolated, either use algebra to isolate y and proceed as in part (a) or select a point not on the line. If the point represents a solution of the inequality, shade the half-plane containing the point. If it does not, shade the other half-plane. Example 4 Graph: 6x - 2y < 12. Solution We first graph the related equation, 6x - 2y = 12, as a dashed line. This line passes through the points (2, 0), (0, -6), and (3, 3), and serves as the boundary of the solution set of the inequality. Since y is not isolated, we deter-mine which half-plane to shade by testing a point not on the line. The pair (0, 0) is easy to substitute: 6x - 2y < 12 6 * 0 - 2 * 0 ? 12 0 - 0 0 12 TRUE Since the inequality 0 < 12 is true, the point (0, 0) is a solution, as are all points in the half-plane containing (0, 0). The graph is shown below. 6x - 2y < 12 4.4 INEQUALITIES IN TWO VARIABLES page 249 CONNECTING THE CONCEPTS We have now solved a variety of equations, inequalities, systems of equations, and systems of inequalities. In each case, there are different ways to represent the solution. Below is a list of the different types of problems we have solved, along with illustrations of each type. Ti"", Example Solution Graph Linear equations 2x - 8 = 3(x + 5) A number E * -+ in one variable -23 0 Linear inequalities -3x + 5 > 2 A set of numbers; in one variable an interval 0 1 Linear equations 2x + y = 7 A set of ordered pairs YT in two variables 6~ sl 2 + y = 7 4 2 . 1 4-3-2-11 I ? 3 5 6 Linear inequalities x + Y > 4 4 A set of ordered pairs in two variables System of equations x + y = 3, An ordered pair or a in two variables 5x - y = -27 (possibly empty) set of ordered pairs System of inequalities 6x - 2y <_ 12, A set of ordered in two variables y - 3 <_ 0, pairs x + y _> Keeping in mind how these solutions vary and what their graphs look like will help You as you progress further in this book and in mathematics in general. g '7) , , , , 5x -y = -27 - 5-4-3 2-11 1 2 3N YT (-3' 13):5 . (3, 3) , 5-4-3-2-11 3 4 5 : x , -4 : . page 250 CHAPTER 4, INEQUALITIESAND PRQBLEM SOLVING FOR EXTRA HELP Exercise Set 4,4 Digital Video Tutor CD 3 InterAct Math Videotape 8 ~~s Math Tutor Center MathXL MyMathLab.com Determine whether each ordered pair is a solution of the Graph each system of inequalities. Find the coordinates given inequality. of any vertices formed. 1. (-4, 2); 2x + 3y < -1 2. (3, -6); 4x + 2y <_ -2 3. (8, 14); 2y - 3x _> 9 4. (7, 20); 3x - y > -1 Graph on a plane. 1 5. y > 2 x 6. y > 2 x 7. y _> x - 3 8. y < x + 3 9. y <_x + 4 10. y > x - 2 11. x - y <_ 5 12. x + y < 4 13. 2x + 3y < 6 14. 3x + 4y <_ 12 15. 2y - x <_ 4 16. 2y - 3x > 6 17. 2x - 2y _> 8 + 2y 18. 3x - 2 <_ 5x + y 19. y _> 2 20. x < -5 21. x <_ 7 22. y > -3 23. -2 < y < 6 24. -4 < y < -1 25. -4 <_ x <_ 5 , 26. -3 <_ y <_ 4 27. 0 <_ y <_ 3 28. 0 <_ x <_ 6 Graph each system. 29. y > x, y < - x + 2 30. y < x, y > - x + l 31. y _> x, y _> 2x - 4 32. y _> x, y <_ -x + 4 33. y < -3, x _> -1 34. y _> -3, x _> 1 35. x > -4, y < -2x + 3 36. x < 3, y > -3x + 2 37. y <_ 3 y _> -x + 2 38. y _> -2 y _> x + 3 39. x + y <_ 6, x - y < 4 40. x + y < 1, x - y < 2 41. y + 3x > 0, y + 3x < 2 42. y - 2x _> 1, y - 2 x <_ 3 43. y <_ 2x - 1, y _> -2x + 1 x <_ 44. 2y - x <_ 2, y - 3x _> -4 y _> -1 45. x + 2y <_ 12, 2x + y <_ 12 x _> 0 y _> 0 46. x - y <_ 2, x + 2y _> 8 y <_ 4 47. 8x + 5y <_ 40, x + 2y <_ 8 x _> 0 y _> 0 48. 4y - 3x _> -12, 4y + 3x _> -36, y <_ 0 x <_ 0 49. y - x _> 1, y - x <_ 3, 2 <_ x <_ 5 50. 3x + 4y _> 12, 5x + 6y <_ 30, 1 <_ x <_ 3 51. In Example 7, is the point (4, 0) part of the solution set? Why or why not? 52. When graphing linear inequalities, R makes a habit of always shading above the line when the symbol _> is used. Is this wise? Why or why not? SKILL MAINTENANCE 53. Catering. Sandy's Catering needs to provide lO lb of mixed nuts for a wedding reception. Peanuts cost $2.50 per pound and fancy nuts cost $7 per pound. If $40 has been allocated for nuts, how many pounds of each type should be mixed? 54. Household waste The Hendersons generate two and a half times as much trash as their neighbors, the Savickis. Together, the two households produce 14 bags of trash each month. How much trash does each household produce? 55. Paid admissions. There were 203 tickets sold for a volley ball game. For activity card holders the price was $1.25, and for noncard holders the price was $2. The total amount of money collected was $310. How many of each type of ticket were sold? 56. Paid admissions. There were 200 tickets sold for a women's basketball game. Tickets for students 4.4 INEQUALITIES IN TWO VARIABLES page 251 were $2 each and for adults were $3 each. The total amount collected was $530. How many of each type of ticket were sold? 57. Landscaping. Grass seed is being spread on a tri- angular traffic island. If the grass seed can cover an area of 200 ft2 and the island's base is 16 ft long, how tall a triangle can the seed fill? 58. Interest rate. What rate of interest is required in order for a principal of $320 to earn $17.60 in half a year? SYNTHESIS 59. Explain how a system of linear inequalities could have a solution set containing exactly one pair. 60. Do all systems of linear inequalities have solutions? Why or why not? 61. x + y > 8, x + y <_ -2 62. x + y _> 1, -x + y _> 2, x <_ 4, y _> 0, Y <_ 4, x <_ 2 63. x - 2y <_ 0, -2x + y <_ 2, x <_ 2 y <_ 2 x + y <_ 4 64. Write four systems of four inequalities that de- scribe a 2-unit by 2-unit square that has (0, 0) as one of the vertices. 65. Luggage Size Unless an additional fee is paid, most major airlines will not check any luggage that is more than 62 in. long. The U.S. Postal Service will ship a package only if the sum of the package's length and girth (distance around its midsection) does not exceed 108 in. Concert Productions is ordering several 62-in. long trunks that will be both mailed and checked as luggage. Using w and h for width and height (in inches), respectively, write and graph an inequality that represents all accept- able combinations of width and height. 66. Hockey wins and losses The Skating Stars figure that they need at least 60 points for the season in order to make the playoffs. A win is worth 2 points and a tie is worth 1 point. Graph a system of in- equalities that describes the situation. (Hint: Let w = the number of wins and t = the number of ties.) 67. Elevators. Many elevators have a capacity of 1 metric ton (1000 kg). Suppose that c children, each weighing 35 kg, and a adults, each 75 kg, are on an elevator. Graph a system of inequalities that indicates when the elevator is overloaded. 68. Widths of a basketball floor. Sizes of basketball floors vary due to building sizes and other con- straints such as cost. The length L is to be at most 94 ft and the width W is to be at most 50 ft. Graph a system of inequalities that describes the possible dimensions of a basketball floor. 69. Use a grapher to graph each inequality. a) 3x + 6y > 2 b) x - 5y <_ 10 c) 13x - 25y + 10 <_ 0 d) 2x + 5y > 0 70. Use a grapher to check your answers to Exer- cises 29-42. Then use INTERSECT to determine any point(s) of intersection. page 252 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING tORNER ® The Rule of 85 Focus: Linear inequalities Time: 20-30 minutes Group size: 3 Under a proposed "Rule of 85," full-time faculty in the California State Teachers Retirement Sys-tem (kindergarten through community college) who are a years old with y years of service would have the option of retirement if a + y _> 85. ACTlVITY 1. Decide, as a group, the age range of full-time teachers. Express this age range as an in-equality involving a. 2. Decide, as a group, the number of years someone could teach full-time before retir-ing. Express this answer as a compound inequality involving y. 3. Using the Rule of 85 and the answers to parts (1) and (2) above, write a system of inequalities. Then, using a scale of 5 yr per square, graph the system. To facilitate com-parisons with graphs from other groups, plot a on the horizontal axis and y on the vertical axis. 4. Compare the graphs from all groups. Try to reach consensus on the graph that most clearly illustrates what the status would be of someone who would have the option of retirement under the Rule of 85. 5. If your instructor is agreeable to the idea, at-tempt to represent him or her with a point on your graph. Applicati©ns Using Linear Programming 4.5 Objective Functions and Constraints • Linear Programming There are many real-world situations in which we need to find a greatest value (a maximum) or a least value (a minimum). For example, most businesses would like to know how to make the most profit and how to make their ex-penses the least possible. Some such problems can be solved using systems of inequalities. Objective Functions and Constraints Often a quantity we wish to maximize depends on two or more other quanti-ties. For example, a gardener's profits P might depend on the number of shrubs s and the number of trees t that are planted. If the gardener makes a $5 profit 4.5 APPLICATIONS USING LINEAR PROGRAMMING page 257 Study Tip On lengthy problems, like Example 2, it is helpful to carefully check each step of your work before proceeding to the next step. 3. Carry out. The mathematical manipulation consists of graphing the system and evaluating T at each vertex. The graph is as follows: y x _> 0 25 20 15 x + y <_ 16 10 3x + 6y <_ 60 5 A 5 10 B 20 x y _> 0 We can find the coordinates of each vertex by solving a system of two lin-ear equations. The coordinates of point A are obviously (0, 0). To find the coordinates of point C, we solve the system x + y = 16 1 3x + 6y = 60, (2) as follows: -3x - 3y = -48 Multiplying both sides of equation (1) by -3 3x + 6y 3x + 6y = 60 3y = 12 Adding y = 4. Then we find that x = 12. Thus the coordinates of vertex C are (12,4). Point B is the x-intercept of the line given by x + y = 16, so B is (16, 0). Point D is the y-intercept of 3x + 6y = 60, so D is (0,10). Computing the test score for each ordered pair, we obtain the following: Vertex Score (x, y) T = lOx + 15y A (0, 0) 0 B (16, 0) 160 C (12,4) 180 D (0,10) 150 The greatest score in the table is 180, obtained when 12 multiple-choice and 4 short-answer questions are answered. 4. Check. We can check that T <_ 180 for any other pair in the shaded region. This is left to the student. 5. State. In order to maximize her score, Corinna should answer 12 multiple-choice questions and 4 short-answer questions. page 258 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING Exercise Set 4.5 FOR EXTRA HELP Digital Video Tutor CD 3 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 8 1. F = 2x + 14y, subject to 5x + 3y <_ 34, 3x + 5y <_ 30 x _> 0 y _> 0 2. G = 7x + 8y, subject to 3x + 2y <_ 12, 2y - x <_ 4 x _> 0 y _> 0 3. P = 8x - y + 20, subject to 6x + 8y <_ 48 0 <_ y <_ 4 0 <_ x <_ 7 4. Q = 24x - 3y + 52, subject to 5x + 4y <_ 20, 0 <_ y <_ 4 0 <_ x <_ 3 5. F = 2y - 3x, subject to y <_ 2x + 1 y _> -2x + 3 x <_ 3 6. G = 5x + 2y + 4, subject to y <_ 2x + 1 y _> -x + 3 x <_ 5 Solve 7. Lunch-time profits. Elrod's lunch cart sells burri- tos and chili. To stay in business, Elrod must sell at least 10 orders of chili and 30 burritos each day. Be- cause of limited space, no more than 40 orders of chili or 70 burritos can be made. The total number of orders cannot exceed 90. If profit is $1.65 per chili order and $1.05 per burrito, how many of each item should Elrod sell in order to maximize profit? 8. Milling Johnson Lumber can convert logs into either lumber or plywood. In a given week the mill can turn out 400 units of production of which 100 units of lumber and 150 units of plywood are required by regular customers. The profit on a unit of lumber is $20 and on a unit of plywood is $30. How many units of each type should the mill pro- duce in order to maximize profit? 9. Cycle production Yawaka manufactures motor- cycles and bicycles. To stay in business, the num- ber of bicycles made cannot exceed 3 times the number of motorcycles made. Yawaka lacks the fa- cilities to produce more than 60 motorcycles or more than 120 bicycles. The total production of motorcycles and bicycles cannot exceed 160. The profit on a motorcycle is $1340 and on a bicycle is $200. Find the number of each that should be manufactured in order to maximize profit. 10. Gas mileage. Roschelle owns a car and a moped. She has at most 12 gal of gasoline to be used be- tween the car and the moped. The car's tank holds at most 18 gal and the moped's 3 gal. The mileage for the car is 20 mpg and for the moped is 100 mpg. How many gallons of gasoline should each vehicle use if Roschelle wants to travel as far as possible? What is the maximum number of miles? 11. Test scores. Phil is about to take a test that con- tains matching questions worth 10 points each and essay questions worth 25 points each. He must do at least 3 matching questions, but time restricts doing more than 12. Phil must do at least 4 essays, but time restricts doing more than 15. If no more than 20 questions can be answered, how many of each type should Phil do in order to maximize his score? What is this maximum score? 12. Test scores Edy is about to take a test that con- tains short-answer questions worth 4 points each and word problems worth 7 points each. Edy must do at least 5 short-answer questions, but time re- stricts doing more than 10. She must do at least 3 word problems, but time restricts doing more than 10. Edy can do no more than 18 questions in total. How many of each type of question must Edy do in order to maximize her score? What is this maximum score? Aha! 13. Investing. Rosa is planning to invest up to $40,000 in corporate or municipal bonds, or both. She must invest from $6000 to $22,000 in corporate bonds, and she does not want to invest more than $30,000 in municipal bonds. The interest on corporate bonds is 8% and on municipal bonds is 7 1%. This 2 is simple interest for one year. How much should Rosa invest in each type of bond in order to earn the most interest? What is the maximum interest? 4.5 APPLICATIONS USING LINEAR PROGRAMMING page 259 14. Grape growing. Auggie's vineyard consists of 240 acres upon which he wishes to plant Merlot and Cabernet grapes. Profit per acre of Merlot is $400 and profit per acre of Cabernet is $300. Fur- thermore, the total number of hours of labor avail- able during the harvest season is 3200. Each acre of Merlot requires 20 hr of labor and each acre of income? Cabernet requires 10 hr of labor. Determine how the land should be divided between Merlot and Cabernet in order to maximize profit. 15. Coffee blending. The Coffee Peddler has 1440 1b of Sumatran coffee and 700 1b of Kona coffee. A batch of Hawaiian Blend requires 81b of Kona and 12 1b of Sumatran, and yields a profit of $90. A batch of Classic Blend requires 4 lb of Kona and 16 lb of Sumatran, and yields a $55 profit. How many batches of each kind should be made in order to maximize profit? What is the maximum profit? (Hint: Organize the information in a table.) 16. Investing Jamaal is planning to invest up to $22,000 in City Bank or the Southwick Credit Union, or both. He wants to invest at least $2000 but no more than $14,000 in City Bank. The South- wick Credit Union does not insure more than a $15,000 investment, so he will invest no more than that in the Southwick Credit Union. The interest in City Bank is 6% and in the credit union is 6 1%. This 2 is simple interest for one year. How much should he invest in each bank in order to earn the most in- terest? What is the maximum interest? 17. Textile production. It takes Cosmic Stitching 2 hr of cutting and 4 hr of sewing to make a knit suit. To make a worsted suit, it takes 4 hr of cutting and 2 hr of sewing. At most 20 hr per day are available for cutting and at most 16 hr per day are available for sewing. The profit on a knit suit is $68 and on a worsted suit is $62. How many of each kind of suit should be made in order to maximize profit? 18. Biscuit production. The Hockeypuck Biscuit Factory makes two types of biscuits, Biscuit Jumbos and Mitimite Biscuits. The oven can cook at most 200 biscuits per hour. Jumbos each require 2 oz of flour, Mitimites require 1 oz of flour, and there is at most 1440 oz of flour available. The income from Jumbos is $1.00 and from Mitimites is $0.80. How many of each type of biscuit should be made in order to maximize income? What is the maximum income? 19. Before a student begins work in this section, what three sections of the text would you suggest he or she study? Why? 20. What does the use of the word "constraint" in this section have in common with the use of the word in everyday speech? SKILL MAINTENANCE Evaluate. 21. 5x3 - 4x2 - 7x + 2, for x = -2 3 and 2 are uppercorner numbers. 22. 6t3 - 3t2 + 5t, for t = 2 3 and 2 are uppercorner numbers. Simplify. 23. 3(2x - 5) + 4(x + 5) 24. 4(5t - 7) + 6(t + 8) 25. 6x - 3(x + 2) 26. 8t - 2(3t - 1) SYNTHESIS 27. Explain how Exercises 16 and 18 can be answered by logical reasoning without linear programming. 28. Write a linear programming problem for a class- mate to solve. Devise the problem so that profit must be maximized subject to at least two (non- trivial) constraints. 29. Airplane production Alpha Tours has two types of airplanes, the T3 and the S5, and contracts requir- ing accommodations for a minimum of 2000 first class, 1500 tourist-class, and 2400 economy-class passengers. The T3 costs $30 per mile to operate and can accommodate 40 first-class, 40 tourist- class, and 120 economy-class passengers, whereas the S5 costs $25 per mile to operate and can ac- commodate 80 first-class, 30 tourist-class, and 40 economy-class passengers. How many of each type of airplane should be used in order to mini- mize the operating cost? page 26O CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 30. Airplane production. A new airplane, the T4, is now available, having an operating cost of $37.50 per mile and accommodating 40 first-class, 40 tourist-class, and 80 economy-class passengers. If the T3 of Exercise 29 were replaced with the T4, how many S5's and how many T4's would be needed in order to minimize the operating cost? 31. Furniture production. P J. Edward Furniture Design produces chairs and sofas. The chairs re- quire 20 ft of wood, 1 lb of foam rubber, and 2 sq yd of fabric. The sofas require 100 ft of wood, 50 1b of foam rubber, and 20 sq yd of fabric. The company has 1900 ft of wood, 500 lb of foam rubber, and 240 sq yd of fabric. The chairs can be sold for $80 each and the sofas for $1200 each. How many of each should be produced in order to maximize income? Summary and Review 4 Key Terms Inequality, p. 212 Compound inequality, p. 224 Related equation, p. 242 Solution, p. 212 Intersection, p. 224 Test point, p. 243 Solution set, p. 212 Conjunction, p. 224 Vertices (singular, vertex), Set-builder notation, p. 213 Union, p. 227 P. 248 Interval notation, p. 213 Disjunction, p. 227 Objective function, p. 253 Open interval, p. 213 Absolute value, p. 233 Constraint, p. 253 Closed interval, p. 213 Linear inequality, p. 242 Linear programming, p. 253 Half-open interval, p. 213 Half-plane, p. 242 Feasible region, p. 254 Important Properties and Formulas The Addition Principle for Inequalities For any real numbers a and b, and for any For any real numbers a, b, and c: negative number c, a < b is equivalent to a + c < b + c; a > b is equivalent to a + c > b + c. Similar statements hold for < and >-. a < b is equivalent to ac > bc; a > b is equivalent to ac < bc. Similar statements hold for <_ and >_. Set intersection: A fl B = {x | x is in A and x is in B} The Multiplication Principle for Inequalities For any real numbers a and b, and for any Set union: positive number c, A U B = {x | x is in A or in B, or both} a < b is equivalent to ac < bc; Intersection corresponds to "and"; union a > b is equivalent to ac > bc. corresponds to "or." REVIEW EXERCISES: CHAPTER 4 page 261 |x| = x if x _> 0. |x| = -x if x <_ 0; b) If y is not isolated, either isolate y and proceed as in part (a) or select a test point not on the line. If the point The Absolute-Value Principles for represents a solution of the inequal- Equations and Inequalities iry, shade the half-plane containing For any positive number p and any alge- the point. If it does not, shade the braic expression X: other half-plane. a) The solutions of 1X1 = p are those num- bers that satisfy X = -p or X = p. The Corner Principle b) The solutions of 1X1 < p are those num- Suppose that an objective function F = bers that satisfy -p < X < p, ax + by + c depends on x and y. Suppose c) The solutions of 1X1 > p are those num- also that F is subject to constraints on x bers that satisfy X < -p or p < X. and y, which form a system of linear in- If |X| = 0, then X = 0. If p is negative, then equalities. If F has a minimum or a maxi- X1 = p and |X| < p have no solution, and mum value, it can then be found as any value of Xwill satisfy |X| > p. follows: 1. Graph the system of inequalities and find the vertices. Steps for Graphing Linear inequalities 2. Find the value of the objective function 1. Graph the related equation. Draw the at each vertex. The largest and the line dashed if the inequality symbol is smallest of those values are the maxi- < or > and solid if the inequality sym- mum and the minimum of the function, bol is <- or >-. respectively. 2. The graph consists of a half-plane on 3. The ordered pair at which the maxi- one side of the line and, if the line is mum or minimum occurs indicates the solid, the line as well. choice of (x, y) for which that maximum a) Shade below the line if the inequality or minimum occurs. is of the form y < f(x) or y < f(x). Shade above the line if the inequality is of the form y > f(x) or y >- f(x). Review Exercises Graph each inequality and write the solution set using both set-builder and interval notation. 1. x <_ -2 2. a + 7 <_ -14 3. y - 5 _> -12 4. 4y > -15 5. -0.3y < 9 6. -6x - 5 < 4 7. -1x -1 > 1 - 1x 2 4 2 4 Solve. 8. 0.3y -7 < 2.6y + 15 9. -2(x - 5) _> 6(x + 7) - 12 10. Let f(x) = 3x - 5 and g(x) = 11 - x. Find all values of x for which f(x) <_ g(x). 11. Jessica can choose between two summer jobs. She can work as a checker in a discount store for $8.40 an hour, or she can mow lawns for $12.00 an hour. In order to mow lawns, she must buy a $450 lawn-mower. How many hours of labor will it take Jessica to make more money mowing lawns? page 262 CHAPTER 4 INEQUALITIES AND PROBLEM SOLVING 12. Clay is going to invest $4500, part at 6% and the rest at 7%. What is the most he can invest at 6% and still be guaranteed $300 in interest each year? 13. Find the intersection: {1,2,5,6,9}n{1,3,5,9}. 14. Find the union: {1,2,5,6,9}U{1,3,5,9}. Graph each system of inequalities. Find the coordinates of any vertices formed. 15. x <_ 3 and x > -5 16. x <_ 3 or x > -5 Solve and graph each solution set. 17. -4 < x + 3 <_ 5 18. -15 < -4x -5 < 0 19. 3x < -9 or -5x < -5 20. 2x + 5 < -17 or - 4x + 10 <_ 34 21. 2x + 7 <_ -5 or x + 7 > 15 22. f(x) < -5 or f(x) > 5, where f(x) = 3 - 5x For f(x) as given, use interval notation to write the domain of f. 23. f(x)= x x - 3 _____ 24. f(x)= Vx + 3 ______ 25. f(x) = V8 - 3x Solve. 26. |x| = 4 27. |t| _> 3.5 28. |x - 2| = 7 29. |2x + 5| < 12 30. |3x - 4| _> 15 31. |2x + 5| = |x - 9| 32. |5n + 6| = -8 33. x + 4 <_ 1 8 34. 2|x - 5| - 7 > 3 35. Let f(x) = |3x - 5|. Find all x for which f(x) < 0. 36. Graph x - 2y _> 6 on a plane. Graph each system of inequalities. Find the coordinates of any vertices formed. 37. x + 3y > -1, x + 3y < 4 38. x - 3y <_ 3, x + 3y _> 9 y <_ 6 39. Find the maximum and the minimum values of F = 3x + y + 4 subject to y <_ 2x + 1, x <_ 7 y _> 3. 40. Edsel Computers has two manufacturing plants. The Oregon plant cannot produce more than 60 computers a month, while the Ohio plant can- not produce more than 120 computers a month. The Electronics Outpost sells at least 160 Edsel computers each month. It costs $40 to ship a com- puter to The Electronics Outpost from the Oregon plant and $25 to ship from the Ohio plant. How many computers should be shipped from each plant in order to minimize cost? SYNTHESIS 41. Explain in your own words why |X| = p has two so lutions when p is positive and no solution when p is negative. 42. Explain why the graph of the solution of a system of linear inequalities is the intersection, not the union, of the individual graphs. 43. Solve: |2x + 5| <_ |x + 3|. 44. Classify as true or false: If x < 3, then x2 < 9. If false, give an example showing why. 45. Just-For-Fun manufactures marbles with a 1.1-cm diameter and a ±0.03-cm manufacturing tolerance, or allowable variation in diameter. Write the tolerance as an inequality with absolute value. TEST: CHAPTER 4 page 263 Chapter Test 4 Graph each inequality and write the solution set using both set-builder and interval notation. 1. x - 2 < 10 2. -0.6y < 30 3. -4y -3 _> 5 4. 3a - 5 <_ -2a + 6 5. 4(5 - x) < 2x + 5 6. -8(2x + 3) + 6(4 - 5x) _> 2(1 - 7x) - 4(4 + 6x) 7. Let f(x) = -5x - 1 and g(x) = -9x + 3. Find all values of x for which f(x) > g(x). 8. Lia can rent a van for either $40 with unlimited mileage or $30 with 100 free miles and an extra charge of 15cents for each mile over 100. For what numbers of miles traveled would the unlimited mileage plan save Lia money? 9. A refrigeration repair company charges $40 for the first half-hour of work and $30 for each additional hour. Blue Mountain Camp has budgeted $100 to repair its walk-in cooler. For what lengths of a serv- ice call will the budget not be exceeded? 10. Find the intersection: {1, 3, 5, 7, 9} n {3, 5, 11,13}. 11. Find the union: {1, 3, 5, 7, 9} U {3, 5, 11,13}. 12. Write the domain of f using interval notation if f(x) = V7 - x. Solve and graph each solution set. 13. - 3 < x - 2 < 4 14. -11 <_ -5t - 2 < 0 15. 3x - 2 < 7 or x - 2 > 4 16. -3x > 12 or 4x > -10 17. - 1 <_ 1x - 1 < 1 3 6 4 18. |x| = 9 19. |a| > 3 20. |4x - 1| < 4.5 21. |-5t - 3| _> 10 22. |2 - 5x| = -10 23. g(x) < -3 or g(x) > 3, where g(x) = 4 - 2x 24. Let f(x) = |x + 10| and g(x) = |x - 12| Find all values of x for which f(x) = g(x). Graph the system of inequalities. Find the coordinates of any vertices formed. 25. x + y _> 3, x - y _> 5 26. 2y - x _> -7, 2y + 3x <_ 15, y <_ 0, x <_ 0 27. Find the maximum and the minimum values of F = 5x + 3y subject to x + Y <_ 15 1 <_ x <_ 6, O <_ y <_ 12. 28. Sassy Salon makes $12 on each manicure and $18 on each haircut. A manicure takes 30 minutes and a haircut takes 50 minutes, and there are 5 stylists who each work 6 hours a day. If the salon can schedule 50 appointments a day, how many should be manicures and how many haircuts in order to maximize profit? What is the maximum profit? SYNTHESIS Solve. Write the solution set using interval notation. 29. |2x - 5| <_ 7 and |x - 2| _> 2 30. 7x < 8 - 3x < 6 + 7x 31. Write an absolute-value inequality for which the interval shown is the solution. -16 -l5 -14 -13 -12 -11 -10 5.1 INTRODUCTION TO POLYNOMIALS AND POLYNOMIAL FUNCTIONS page 267 2 3 4 are uppercorner small size numbers. exponents of the variables. 5x2 has degree 2 and 9a3b4 has degree 7. Nonzero constant terms, like -9, can be written -9x° and therefore have de-gree 0. The term 0 itself is said to have no degree. The number 5 is said to be the coefficient of 5x2. Thus the coefficient of 9a3b4 is 9 and the coefficient of -2x is -2. The coefficient of a constant term is just that constant. A polynomial is a monomial or a sum of monomials. Of the expressions listed, 5x2, 9a3b4, 6x2 + 3x + 1, -9, and 5 - 2x are polynomials. In fact, with the exception of 9a3b4, these are all polynomials in one variable. The expression 9a3b4 is a polynomial in two variables. Note that 5 - 2x is the sum of 5 and -2x. 5 and -2x are the terms in the polynomial 5 - 2x. The leading term of a polynomial is the term of highest degree. Its coeffi-cient is called the leading coefficient. The degree of a polynomial is the same as the degree of its leading term. Example 1 For each polynomial given, find the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient. a) 2x3 + 8x2 - 17x - 3 b) 6x2 + 8x2y3 - 17xy - 24xy2Z4 + 2y + 3 Solution (a)2x3 8x2 -17x -3 (b)6x2 8x2y3 -17xy -24xy2Z4 + 2y + 3 Term(a)2x3 8x2 -17x -3 (b)6x2 8x2y3 -17xy -24xy2Z4 + 2y + 3 Degree 3 2 1 0 2 5 2 7 1 0 Leading Term 2x3 -24xy2Z4 Leading Coefficient 2 -24 Degree of Polynomial A polynomial of degree 0 or 1 is called linear. A polynomial in one variable is said to be quadratic if it is of degree 2 and cubic if it is of degree 3. The following are some names for certain kinds of polynomials. (b) 6xZ + 8x2y3 - 17xy - 24xy2z4 + 2y + 3 Definition ~ Exainples Monomial ~ A polynomial of one term ~ 4, -3p, 5x2, -7a2b3, 0, xyz Binomial I A polynomial of two terms ~ 2x + 7, a - 3b, 5x2 + 7y3 Trinomial A polynomial of three terms I - x2 - 7x + 12, 4a2 + 2ab + b2 page 268 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS We generally arrange polynomials in one variable so that the exponents de-crease from left to right. This is called descending order. Some polynomials may be written with exponents increasingfrom left to right, which is ascending order. Generally, if an exercise is written in one kind of order, the answer is written in that same order. E x a m p 1 e 2 Arrange in ascending order: 12 + 2x3 - 7x + x2. Solution 12 +2 x3 - 7x + x2 = 12 - 7x + x2 + 2x3 Polynomials in several variables can be arranged with respect to the powers of one of the variables. Example 3 Arrange in descending powers of x: y4 + 2 - 5x2 + 3x3y + 7xy2. 4 2 3 2s are uppercase small size numbers. Solution y4 + 2 - 5x2 + 3x3y + 7xy2 = 3x3y - 5x2 + 7xy2 + y4 + 2 Polynomial Functions A polynomial function is a function in which ordered pairs are determined by evaluating a polynomial. For example, the function P given by P(x) = 5x7 + 3x5 - 4x2 - 5 is an example of a polynomial function. To evaluate a polynomial function, we substitute a number for the variable just as in Chapter 2. In this text, we limit ourselves to polynomial functions in one variable. E x p m p 1 e 4 For the polynomial function P(x) = -x2 + 4x - 1, find the following: • (a) P(2); (b) P(10); (c) P(-10). Solution a) P(2) = -2 2is upper smallnumber + 4(2) - 1 We square the input before taking its opposite. = -4 + 8 - 1 = 3 b) P(10) = - 10 2 is upper smallnumber + 4(10) - 1 = -100 + 40 - 1 = -61 c) P(-1o) = -(-l0)2 + 4(-10) - 1 = -100 - 40 -1 = -141 Example 5 Veterinary medicine. Gentamicin is an antibiotic frequently used by veteri-narians. The concentration, in micrograms per milliliter (mcg/mL), of Gen-tamicin in a horse's bloodstream thours after injection can be approximated by ; the polynomial function C(t) = -0.005t4 + 0.003t3 + 0.35t2 + 0.5t. (Source: Michele Tulis, DVM, telephone interview) page 273 Chapter 5 Exercise Set 5.1 FOR EXTRA HELP 5.1 INTRODUCTION TO POLYNOMIALS AND POLYNOMIAL FUNCTIONS page 273 Digital Video Tutor CD 3 InterAct Math Math Tutor Center MathXL MyMathlab.com Videotape 9 Determine the degree of each term and the degree of the polynomial. 1. -6x5 - 8x3 + x2 + 3x -4 5 3 2 72are uppercorner small size numbers. 2. t3 - 5t2 + 2t + 7 3. y3 + 2y7 + x2y4 - 8 4. -2u2 + 3v5 - u3 v4 - 7 5. a5 + 4a2b4 + 6ab + 4a - 3 6. 8p6 + 2p4t4 - 7p3t + 5p2 - 14 Arrange in descending order. Then find the leading term and the leading coefficient. 7. 19 - 4y3 + 7y - 6y2 8. 3 - 5y + 6y2 + lly3 - 18y4 9. 5x2 + 3x7 - x + 12 10. 9 - 3x - lOx4 + 7x2 11. a + 5a3 - a7- 19a2 + 8a5 12. a3 - 7 + 11a4 + a9 - 5a2 Arrange in ascending powers of x. 13. 6x - 9 + 3x4 - 5x2 14. -3x4 + 4x -x3 + 9 15. 7x3y + 3xy3 + x2y2 - 5x4 16. 5x2y2 - 9xy + 8x3y2 - 5x4 17. 4ax - 7ab + 4x6 - 7ax2 18. 5xy8 - 3ax5 + 4ax3 - 12a + 5x5 8 5 3 5 are upper small size numbers. Find the specified function values. 19. Find P(4) and P(0): P(x) = 3x2 - 2x + 7. 20. Find Q(3) and Q(-1): Q(x) = -4x3 + 7x2 - 6. 21. Find P(-2) and P 1): P(Y) = 8Y3 - 12y - 5. 3 22. Find Q(-3) and Q(0): Q(Y)= -8Y3 + 7y2 - 4y - 9. Evaluate each polynomial for x = 4. 23. -7x + 5 24. 4x - 13 25. x3 - 5x2 + x 26. 7 - x + 3x2 Evaluate each polynomial function for x = -1. 27. f(x) = -5x3 + 3x2 - 4x - 3 28. g(x) = -4x3 + 2x2 + 5x - 7 Electing officers. For a club consisting of n people, the number of ways in which a president, vice president, and treasurer can be elected can be determined using the function given by p(n) = n3 - 3n2 + 2n. 29. The Southside Rugby Club has 20 members. In how many ways can they elect a president, vice president, and treasurer? 30. The Stage Right drama club has 12 members. In how many ways can a president, vice president, and treasurer be elected? Falling distance The distances(t),in feet,traveled by a body falling freely from rest in t seconds is approximated by the function given by s(t) = 16t2. 31. A paintbrush falls from a scaffold and takes 3 sec to hit the ground. How high is the scaffold? 32. A stone is dropped from the Briar Cliff lookout and takes 5 sec to hit the ground. How high is the cliff? page 274 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Total revenue An electronics firm is marketing a new kind of DVD player. The firm determines that when it sells x DVD players, its total revenue is R(x) = 280x - 0.4x2 dollars. Total cost The electronics firm determines that the total cost in dollars of producing x DVD players is given by C(x) = 5000 + 0.6x2. 2 is small size upper number. 33. What is the total revenue from the sale of 75 DVD players? 34. What is the total revenue from the sale of 100 DVD players? 35. What is the total cost of producing 75 DVD players? 36. What is the total cost of producing 100 DVD players? Daily accidents The number of daily accidents (the average number of accidents per day) involving drivers of age a is approximated by the polynomial function P(a) = 0.4a2 - 40a + 1039. ~ 37. Find the number of daily accidents involving a 20-year-old driver. 38. Find the number of daily accidents involving a 25-year-old driver. NASCAR attendance. Attendance at NASCAR auto races has grown rapidly over the past 10 years. Atten- dance A, in millions, can be approximated by the poly- nomial function given by A(x) = 0.0024x3 - 0.005x2 + 0.31x + 3, where x is the number of years since 1989. Use the fol- lowinggraph for Exercises 39-42. attendance in millions A x 10 9 8 7 6 5 4 3 a red line begins at 3 goes up to 9 and 10 for A. 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 x Years since 1989. 3, 1 3, 2 4,3 5,4 5,5 6,7 6,7 7,9 Source: NASCAR 39. Estimate the attendance at NASCAR races in 1995. 40. Estimate the attendance at NASCAR races in 1999. 41. Approximate A(8). 42. Approximate A(12). Medicine. Ibuprofen is a medication used to relieve pain. The polynomial function M(t) = 0.5t4 + 3.45t3 - 96.65t2 + 347.7t, 0 <_ t <_ 6 can be used to estimate the number of milligrams of ibuprofen in the bloodstream t hours after 400 mg of the medication has been swallowed (Source: Based on data from Dr. P. Carey, Burlington, VT). Use the following graph for exercises 43 - 46. Milligrams in bloodstream M(t) 400 M(t) = 0.5t4 + 3.45t3 - 96.65t2 + 347.7t. 360 0 <_ t <_ 6 320 280 240 200 160 120 80 40 1 2 3 4 5 6 7 The u line goes from 0 to 360 down to 6 annd 7. Time in hours 43. Use the graph above to estimate the number of milligrams of ibuprofen in the bloodstream 2 hr after 400 mg has been swallowed. 44. Use the graph above to estimate the number of milligrams of ibuprofen in the bloodstream 4 hr after 400 mg has been swallowed. 45. Approximate M(5). 46. Approximate M(3). Surface area of a right circular cylinder The surface area of a right circular cylinder is given by the polynomial 2 ttrh + 2 ttr2, where h is the height, r is the radius of the base, and h and r are given in the same units. 5.1 INTRODUCTION TO POLYNOMIALS AND POLYNOMIAL FUNCTIONS page 275 ® 47. A 16-oz beverage can has height 6.3 in. and radius 1.2 in. Find the surface area of the can. (Use a cal-culator with a tt key or use 3.141592654 for tt.) ® 48. A 12-oz beverage can has height 4.7 in. and radius 1.2 in. Find the surface area of the can. (Use a cal-culator with a tt key or use 3.141592654 for tt.) Combine like terms. 49. 5a + 6 - 4 + 2a3 - 6a + 2 50. 6x + 13 - 8 - 7x + 5x2 + 10 51. 3a2b + 4b2 - 9a2b - 7b2 52. 5x2y2 + 4x3 - 8x2y2 - 12x3 53. 9x2 - 3xy + 12y2 + x2 - y2 + 5xy + 4y2 54. a2 - 2ab + b2 + 9a2 + 5ab - 4b2 + a2 Add. 55. (8a + 6b - 3c) + (4a - 2b + 2c) 56. (7x - 5y + 3z) + (9x + 12y - 8z) 57. (a2 - 3b2 + 4c2) + (-5a2 + 2b2 - c2) 58. (x2 - 5y2 - 9z2) + (-6x2 + 9y2 - 2z2). 59. (x2 + 2x - 3xy - 7)+(-3x2 - x + 2xy + 6) 60. (3a2 - 2b + ab + 6)+(-a2 + 5b - 5ab - 2) 61. (8x2y - 3xy2 + 4xy) + (-2x2y - xy2 + xy) 62. (9ab - 3ac + 5bc) + (13ab - 15ac - 8bc) 63. (2r2 + 12r - 11) + (6r2 - 2r + 4) + (r2 - r - 2) 64. (5x2 + 19x - 23) + (-7x2 - llx + 12) + (-x2 -9x + 8) 65. 1 3 1 3 8xY - 5x3Y2 + 4.3y3) + (-3xy - 4x3y2 - 2.9y3) 66. 2 5 4 3 (3xy + 6xy2 + 5.1x2y) + (-5xy + 4xy2 - 3.4x2y) Write two equivalent expressions for the opposite, or ad-ditive inverse, of each polynomial. 67. 5x3 - 7x2 + 3x - 9 68. -8y4 - 18y3 + 4y - 7 69. -12y5 + 4ay4 - 7by2 70. 7ax3y2 - 8by4 - 7abx - 12ay Subtract. 71. (7x - 5) - (-3x + 4) 72. (8y + 2) - (-6y - 5) ` ' 73. (-3x2 + 2x + 9)-(x2 + 5x - 4) 74. (-9y2 + 4y + 8) - (4y2 + 2y - 3) 75. (6a - 2b + c) - (3a + 2b - 2c) 76. (7x - 4y + z) - (4x + 6y - 3z) 77. (3x2 - 2x - x3) - (5x2 - 8x - x3) 2 3 are uppercase numbers. 78. (8y2 - 3y - 4y3) - (3y2 - 9y - 7y3) 79. (5a2 + 4ab - 3b2) - (9a2 - 4ab + 2b2) 80. (7y2 - 14yz - 8z2)-(12y2 - 8yz + 4z2) 81. (6ab - 4a2b + 6ab2) - (3ab2 - l0ab - 12a2b) 82. (lOxy - 4x2y2 - 3y3) - (-9x2y2 + 4y3 - 7xy) 83. 5 1 2 1 3 4 3 2 1 (8x - 4x - 4x - 2) - (-8x + 4x + 2) 84. 5 4 1 2 3 4 3 2 (6Y - 2Y - 7.8Y) - (-5Y + 4Y + 3.4y) Total profit. Total profit is defined as total revenue minus total cost. In Exercises 85 and 86, R(x) and C(x) are the revenue and cost, respectively, from the sale of x futons. 85. If R(x) = 280x - 0.4x2 and C(x) = 5000 + 0.6x2, find the profit from the sale of 70 futons. 86. If R(x) = 280x - 0.7x2 and C(x) = 8000 + 0.5x2, find the profit from the sale of 100 futons. 87. Is the sum of two binomials always a binomial? Why or why not? 88. Ani claims that she can add any two polynomials but finds subtraction difficult. What advice would you offer her? SKILL MAINTENANCE Simplify. 89. 2(x + 3) + 5(x + 2) 90. 7(a + 2) + 3(a + 15) 91. a(a - 1) + 4(a - 1) 92. x(x - 3) + 2(x - 1) 93. x5 * x4 94. a2 * a6 SYNTHESIS 95. Write a problem in which revenue and cost func- tions are given and a profit function, P(x), is re- quired. Devise the problem so that P(0) < 0 and P(100) > 0. page 276 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 96. Write a problem in which revenue and cost func- tions are given and a profit function, P(x), is re- quired. Devise the problem so that P(10) < 0 and P(50) > 0. For P(x) and Q(x) as given, find the following. P(x) = 13x5 - 22x4 - 36x3 + 40x2 - 16x + 75, Q(x) = 42x5 - 37x4 + 50x3 - 28x2 + 34x + 100 97. 2[P(x)] + Q(x) 98. 3[P(x)] - Q(x) 99. 2[Q(x)] - 3[P(x)] 100. 4[P(x)] + 3[Q(x)] 101. Volume of a display. The number of spheres in a triangular pyramid with x layers is given by the function 1 1 1 N(x) = 6 x3 + 2x2 + 3x. The volume of a sphere of radius r is given by the function V(r) = 4 ttr 3. 3 where tt can be approximated as 3.14. Chocolate Heaven has a window display of truffles piled in a triangular pyramid formation 5 layers deep. If the diameter of each truffle is 3 cm, find the volume of chocolate in the display. 102. If one large truffle were to have the same volume as the display of truffles in Exercise 101, what would be its diameter? 103. Find a polynomial function that gives the outside surface area of a box like this one, with an open top and dimensions as shown. x x x - 2 1 feet or 12 inches 104. Develop a formula for the surface area of a right circular cylinder in which h is the height, in cen- timeters, and r is the radius, in meters. (See Exer-cises 47 and 48.) Perform the indicated operation. Assume the exponents are natural numbers. 105. (2x2a + 4xa + 3) + (6x2a + 3xa + 4) 106. (3x6a - 5x5a + 4x3a + 8) - (2x6a + 4x4a + 3x3a + 2x2a) 107. (2x5b + 4x4b + 3x3b + 8) - (X5b + 2x3b + 6x2b + 9xb + 8) 108. Use a grapher to check your answers to Exercises 39, 41, and 43. 109. Use a grapher to check your answers to Exercises 20, 33, and 34. 110. A student who is trying to graph p(x) = 0.05x4 - x2 + 5 gets the following screen. io 10 10 io The colored line looks like a W. How can the student tell at a glance that a mistake has been made? 5.2 MULTIPLICATION OF POLYNOMIALS page 277 C O R N E R How Many Handshakes? Focus: Polynomial functions W Time: 20 minutes Group size: 5 Activity a 0 m Group Size Number of Handshakes 1 2 3 4 5 1. All group members should shake hands with each other. Without "double counting," deter-mine how many handshakes occurred. 2. Complete the table in the next column. 3. Join another group to determine the number of handshakes for a group of size 10. 4. Try to find a function of the form produces all of the values in the table above. H(n) = an2 + bn, for which H(n) is the num- (Hint: Use the table to twice select n. and H(n). ber of different handshakes that are possible Then solve the resulting system of equations for in a group of n people. Make sure that H(n) a and b.) Multiplication of 5•2 Polynomials Multiplying Monomials • Multiplying Monomials and -` ~ Binomials • Multiplying Any Two Polynomials • The Product of Two Binomials: FOIL • Squares of Binomials • Products of Sums and Differences • Function Notation Just like numbers, polynomials can be multiplied. The product of two polyno-mials P(x) and Q(x) is a polynomial R(x) that gives the same value as P(x) - Q(x) for any replacement of x. Multiplying Monomials To multiply monomials, we first multiply their coefficients. Then we multiply the variables using the rules for exponents and the commutative and associa-tive laws. With practice, we can work mentally, writing only the answer. page 278 CHAPTER 5. POLYNOMIALS AND POLYNOMIAL FUNCTIONS Example 1 Multiply and simplify: (a) (-8x4y7) (5x3y2); (b) (-2x2yz5) (-6x5y10z2). Solution 4 7 32 4 3 7 2 are upper small numbeers. a) (-8x4y7) (5x 3y2) = -8 * 5 * x4 * x3 * y7 * y2 Using the associative and commutative laws = -40x4 + 3y7+2 Multiplying coefficients; adding exponents = -40x7y9 7 9 are small numbers. so are 2 5 5 10 2 b) (-2x2yz5)(-6x5y10z2) = (-2) (-6) * x2 * x5 * y * y10 * z5 * z2 = 12x7yllz7 Multiplying coefficients; adding exponents Multiplying Monomials and Binomials The distributive law is the basis for multiplying polynomials other than mono-mials. We first multiply a monomial and a binomial. .Example 2 Multiply: (a) 2x(3x - 5); (b) 3a2b(a2 - b 2). Solution a) 2x(3x - 5) = 2x * 3x - 2x * 5 Using the distributive law = 6x2 - lOx Multiplying monomials b) 3a2b(a2 - b2) = 3a2b * a2 - 3a2b * b2 Using the distributive law = 3a4b - 3a2b3 The distributive law is also used when multiplying two binomials. In this case, however, we begin by distributing a binomial rather than a monomial. With practice, some of the following steps can be combined. Example 3 Multiply: (y3 - 5) (2y3 + 4). Solution (Y3 - 5)(2y3 +`4) = (Y3 - 5)2Y3 + (Y3 - 5)4 "Distributing" the y3 - 5 Using the commuta-tive law for multipli-cation. Try to do this step mentally. = 2Y3(Y3 - 5) + 4(Y3 - 5) = 2y3 * y3 - 2y3 * 5 + 4 * y3 -4 * 5 Using the distributive law (twice) monomials = 2y6 - 10y3 + 4y3 -20 Combining like terms = 2y6 - 6y3 - 20 Multiplying the chapter 5 5.2 MULTIPLICATION OF POLYNOMIALS page 283 technology F-J connection B Solution a) (5y + 4 + 3x) (5y + 4 - 3x) = (5y + 4)2 - (3x)2 Try to be alert for situations like this. = 25y2 + 40y + 16 - 9x2 The check outlined in Technol- We can also multiply (5y + 4 + 3x) (5y + 4 - 3x) using columns, but not ogy Connection B in Section 5.1 as quickly. can be used to check multiplica- tion of polynomials in one vari- 2 2 2 2 are small size numbers. b) (3Xy2 + 4y)(-3xy2 + 4y) = (4y + 3xy2) (4y - 3xy2) Rewriting able. To check Example 3, using = (4y)2 - (3xy2)2 x in place e of y, let ~ = 16y2 - 9x2y4 yl = (x - 5) (2x +4), yz = 2x6 - 6x3 - 20, and c) (a - 5b) (a + 5b) (a2 - 25b2) = (a2 - 25b2) (a2 - 25b2) y3 = yz - yl. Since the multipli- = (a2 - 25b2)2 cation is correct, yz = yl and y3 = 0. Using the G-T mode, ; - = (a2)2 - 2(a2) (25b2) + (25b2)2 available on many graphers, and ~ , Squaring a binomial graphing with a wide line, we = a4 - 50a2b2 + 625b4 can split the window vertically to display both a graph and a table. X I Y3 -2 0 -1 0 0 0 t o z o 3 0 4 0 Function Notation Let's stop for a moment and look back at what we have done in this section. We have shown, for example, that (x - 2)(x + 2)=x2 -4, that is, x2 - 4 and (x - 2) (x + 2) are equivalent expressions. From the viewpoint of functions, if Had we found y3 =/ 0, we would have known that a mis- I f (X) = x2 - 4 take had been made. and 1. Use the procedure above to check Examples 4 and 5. g(x) = (x - 2) (x + 2), then for any given input x, the outputs f(x) and g(x) are identical. Thus the graphs of these functions are identical and we say that f and g represent the same function. Functions like these are graphed in detail in Chapter 8. 3 5 5 2 0 0 1 -3 -3 0 -4 -4 -1 -3 -3 -2 0 0 -3 5 5 Our work with multiplying can be used when evaluating functions. x f(x) g(x) 3 5 5 2 0 0 1 -3 -3 0 -4 -4 -1 -3 -3 -2 0 0 -3 5 5 page 284 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 2 is small size numbers. E x a m p 1 e 11 Given f (x) = x2 - 4x + 5, find and simplify each of the following. a) f(a + 3) b) f(a + h) - f(a) Solution a) To find f(a + 3), we replace x with a + 3. Then we simplify: f(a + 3)=(a + 3)2 - 4(a + 3)+5 =a2 + 6a + 9 - 4a - 12 + 5 =a2 + 2a + 2. b) To fmd f (a + h) and f (a), we replace x with a + h and a, respectively. f(a + h) - f(a) = [(a + h)2 - 4(a + h) + 5] - [a2 - 4a + 5] =a2 + 2ah +h2 - 4a - 4h + 5 - a2 + 4a -5 = 2ah + h2 - 4h Exercise Set 5,2 Digital Video Tutor CD 4 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 9 Multiply. 1. 8a2 small size number * 4a 2. -5x3 * 2x 3. 5x(-4x2Y) 4. -3ab2(2a2b2) 5. (2x3y2) (-5x2y4) 6. (7a2bc4) (-8ab3c2) 7. 7x(3 - x) 8. 3a(a2 - 4a) 9. 5cd(4c2d - 5cd2) 10. a2(2a2 - 5a3) 11. (2x + 5) (3x - 4) 12. (2a + 3b) (4a - b) 13. (m + 2n) (m - 3n) 14. (m - 5) (m + 5) 15. (3y + 8x)(y - 7x) 16. (x + y)(x - 2y) 2 small size number 17. (a2 - 2b2)(a2 - 3b2) 18. (2m2 - n2) (3m2 - 5n2) 19. (x - 4)(x2 + 4x + 16) 20. ( y + 3) (Y2 - 3y + 9) 21. (x + y) (x2 - xy + y2) 22. (a - b)(a2 + ab + b2) 23. (a2 + a - 1) (a2 + 4a - 5) 24. (x2 - 2x + 1) (x2 + x + 2) 25. (3a2b - 2ab + 3b2) (ab - 2b + a) 26. (2x2 + y2 - 2xy) (x2 - 2Y2 - xY) 27. (x - 1) (x - 1) 2 4 28. (b - 1) (b - 1) 3 3 29. (1.2x - 3y) (2.5x + 5y) 30. (40a - 0.24b) (0.3a + l0b) 31. Let P(x) = 3x2 - 5 and Q(x) = 4x2 - 7x + 1. Find P(x) * Q(x). 2 3 2 2small numbers. 32. Let P(x) = x2 - x + 1 and Q(x) = x3 + x2 + 5. Find P(x)~ Q(x). Multiply. 33. (a + 4) (a + 5) 34. (x + 3) (x + 2) 35. (y - 8)(y + 3) 36. (y - 1)(y + 5) 37. (x + 5)2 is small number 38. (y - 7)2 39. (x - 2y)2 40. (2s + 3t)2 41. (2x + 9)(x + 2) 42. (3b + 2)(2b - 5) 43. (l0a - 0.12b)2 44. (lOp2 + 2.3q)2 45. (2x - 3y) (2x + y) 46. (2a - 3b) (2a - b) 5.2 MULTIPLICATION OF POLYNOMIALS page 285 numbers 3 2 2 are small size 47. (2x3 - 3y2)2 48. (3s2 + 4t3)2 49. (a2b2 + 1)2 50. (x2y - xy2)2 51. Let P(x) = 4x - 1. Find P(x) * P(x). 52. Let Q(x) = 3x2 + 1. Find Q(x) * Q(x). 53. Let F(x) = 2x - 1 . Find [F(x)]2. 3 54. Let G(x) = 5x - 1. Find [G(x)]2. 2 Multiply 55. (c + 2) (c - 2) 56. (x - 3) (x + 3) 57. (4x + 1) (4x - 1) 58. (3 - 2x) (3 + 2x) 59. (3m - 2n) (3m + 2n) 60. (3x + 5y) (3x - 5y) 61. (x3 + yz)(x3 - yz) 62. (4a3 + 5ab) (4a3 - 5ab)3 is a small size numbers 63. (-mn + m2) (mn + m2) 64. (-3b + a2) (3b + a2) 65. (x + 1) (x - 1) (x2 + 1) 66. (y - 2) (y + 2) (y2 + 4) 67. (a - b) (a + b) (a2 - b2) 68. (2x - y) (2x + y) (4x2 - y2) Aha! 69. (a + b + 1) (a + b - 1) 70. (m + n + 2) (m + n - 2) 71. (2x + 3y + 4) (2x + 3y - 4) 72. (3a - 2b + c) (3a - 2b - c) 73. Compounding interest. Suppose that P dollars is invested in a savings account at interest rate i, compounded annually, for 2 yr. The amount A in the account after 2 yr is given by A = P(1 + i)2. Find an equivalent expression for A. 74. Compounding interest. Suppose that P dollars is invested in a savings account at interest rate i, compounded semiannually, for 1 yr. The amount A in the account after 1 yr is given by A = P 1 + i 2 2 Find an equivalent expression for A. 75. Given f(x) = x2 + 5, find and simplify. a) f(t - 1) b) f(a + h) - f(a) c) f(a) - f(a - h) 76. Given f(x) = x2 + 7, find and simplify. a) f(p + 1) b) f(a + h) - f(a) c) f(a) - f(a - h) 77. Find two binomials whose product is x2 - 25 and explain how you decided on those two binomials. 78. Find two binomials whose product is x2 - 6x + 9 explain how you decided on those two binomials. SKILL MAINTENANCE Solve. 79. ab + ac = d, for a 80. xy + yz = w, for y 81. mn + m = p, for m 82. rs + s = t, for s 83. Value of coins. There are 50 dimes in a roll of dimes, 40 nickels in a roll of nickels, and 40 quar- ters in a roll of quarters. Kacie has 13 rolls of coins, which have a total value of $89. There are three more rolls of dimes than nickels. How many of each type of roll does she have? 84. Wages. Takako worked a total of 17 days last month at her father's restaurant. She earned $50 a day during the week and $60 a day during the weekend. Last month Takako earned $940. How many weekdays did she work? SYNTHESIS 85. We have seen that (a - b) (a + b) = a2 - b2. Ex- plain how this result can be used to develop a fast way of multiplying 95 * 105. 86. A student incorrectly claims that since 2x2 * 2x2 = 4x4, it follows that 5x5 * 5x5 = 25x25. What mistake is the student making? Multiply. Assume that variables in exponents represent natural numbers. 87. [(-xayb)4]a 88. (zn2)n3(Z4n3)n2 89. (axb2Y) (1a3xb)2 2 90. (axby)w + z 91. y3zn(y3nz3 - 4yz2n) page 286 CHAPTER 5 POIYNOMIALS AND POLYNOMIAL FUNCTIONS 92. [(a + b) (a - b)] [5 - (a + b)] [5 + (a + b)] Aha 93. (a-b + c - d)(a + b + c + d) 94. 2x + 1y + 1 (2x - 1y - 1 3 3 3 3 95. (4x2 + 2xy + y2) (4x2 - 2xy + y2) 96. (x2 - 3x + 5)(x2 + 3x + 5) 97. (xa + yb) (xa - yb) (x2a + y2b) 98. (x - 1) (x2 + x + 1) (x3 + 1) 99. (xa-b)a+b 100. (Mx+y)x+y 101. (x - a) (x - b) (x - c) *** (x - z) 102. Draw rectangles similar to those on p. 280 to show that (x + 2)(x + 5) = x2 + 7x + 10. 103. Use a grapher to determine whether each of the following is an identity. a) (x - 1)2 = x2 - 1 b) (x - 2)(x + 3)= x2 + x -6 c) (x - 1)3 = x3 - 3x2 + 3x - i d) (x + l)4 = x4 + 1 e) (x + 1)4 = x4 + 4x3 + 8x2 + 4x + 1 104. Use a grapher to check your answers to Exercises 23, 35, and 65. Consider the following dialogue: CORNER Algebra and Number Tricks Focus: Polynomial multiplication Time: 15-20 minutes Group size: 2 Cal: Okay. The constant term is 16. Jinny: Then the other term is -4x and the num-ber you chose is 4. Cal: You're right! How did you do it? ACTI Vl TY Jinny: Cal, let me do a number trick with you. Think of a number between 1 and 7. I'll have O you perform some manipulations to this num-ber, you'll tell me the result, and I'll tell you your t~ number. 1. Each group member should follow Jinny's in-structions. Then determine how Jinny deter-mined Cal's number and the other term. 2. Suppose that, at the end, Cal told Jinny the x-term. How would Jinny have determined ~` '1 Cal's number and the other term? 3. Would Jinny's "trick" work with any real number? Why do you think she specified numbers between 1 and 7? 4. Each group member should create a new number "trick" and perform it on the other Cal: How did you know I had an x2? I thought group member. Be sure to include a vari- this was rigged! able so that both members can gain prac Jinny: It is. Now, divide by 4 and tell me either tice with polynomials. your constant term or your x-term. I'll tell you the other term and the number you chose. e( Cal: Okay. I've thought of a number. Jinny: Good. Write it down so I can't see it, double it, and then subtract x from the result. Cal: Hey, this is algebra! V Jir2ny: I know Now square your binomial and subtract xZ. Common Factors and ~• FdCtU~'IfI~ ~ly ~~"OIIpIfig Terms with Common Factors • Factoring by Grouping Factoring is the reverse of multiplication. To factor an expression means to write an equivalent expression that is a product. Skill at factoring will assist us when working with polynomial ,functions and solving polynomial equations later in this chapter. CONNECTING THE CONCEPTS 5.3 COMMON FACTORS AND FAC70RING BY GROUPING page 287 Despite all the equals signs in Sections 5.1 and multiply polynomials. Not until Section 5.8 will 5.2, we have not solved any equations since we return to solving equations. At that point, Chapter 4. We have concentrated instead on however, we will study a new type of equation writing equivalent expressions by adding, sub- that cannot be solved without understanding tracting, and multiplying polynomials. We how to factor. As we have seen before, mathe- found that we could not have multiplied poly- matics consistently builds on learned concepts. nomials had we not first learned how to add or The more mastery you develop in factoring, the subtract them. In a similar manner, we will find better prepared you will be to solve the equa that our work with factoring polynomials in tions of Section 5.8. Sections 5.3-5.7 relies heavily on our ability to Terms with Common Factors When factoring, we look for factors common to every term in an expression and then use the distributive law Example 1 Factor out a common factor: 4y2 - 8. 4y2 - 8 = 4 * y2 - 4 * 2 Noting that 4 is a common factor = 4(y2 - 2) Using the distributive law In some cases, there is more than one common factor. In 5x4 + 20x3, for instance, 5 is a common factor, x3 is a common factor, and 5x3 is a common factor. If there is more than one common factor, we factor out the largest, or greatest, common factor, that is, the common factor with the largest coefficient and the highest degree. In 5x4 + 20x3, the largest common factor is 5x3. page 288 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Example 2 Factor out a common factor. a) 5x4 + 20x3 b) 12x2y - 20x3y c) lOp6q2 - 4p5q3 - 2p4q4 a) 5x4 + 20x3 = 5x3 (x + 4) Try to write your answer directly. Multiply mentally to check your answer. b) 12x2y - 20x3y = 4x2y(3 - 5x) Check: 4x2y * 3 = 12x2y and 4x2y(-5x) = -20x3y, so 4x2y(3 - 5x) = 12x2y - 20x3y. c) lOP6q2 - 4P5q3 - 2P4q4 = 2P4q2(5P2 - 2Pq - q2) The check is left to the student. The polynomials in Examples 1 and 2 cannot be factored further unless (in the cases of Examples 1 and 2c) square roots or (in the case of Example 2b) frac-tions are used. In both examples, we have factored completely over the set of integers. The factors used are said to be prime polynomials over the set of integers. When a factor contains more than one term, it is usually desirable for the leading coefficient to be positive. To achieve this may require factoring out a common factor with a negative coefficient. Example 3 Factor out a common factor with a negative coefficient. a) -4x - 24 b) -2x3 + 6x2 - lOx Solution a) -4x - 24 = -4(x + 6) b) -2x3 + 6x2 - lOx = -2x(x2 - 3x + 5) Example 4 Height of a thrown object. Suppose that a baseball is thrown upward with an initial velocity of 64 ft/sec. Its height in feet, h(t), after t seconds is given by h(t) = -16t2 + 64t. Find an equivalent expression for h(t) by factoring out a common factor. h(t) = -16t2 + 64t. 5.3 COMMON FACTORS AND FACTORING BY GROUPING page 287 Common Factors and 5. FIC~OrIn~ ~3~ ~rol.lpln~ Terms with Common Factors • Factoring by Grouping Factoring is the reverse of multiplication. To factor an expression means to write an equivalent expression that is a product. Skill at factoring will assist us when working with polynomial functions and solving polynomial equations later in this chapter. C O N N E C T 1 N G T H E CON C E P T S Despite all the equals signs in Sections 5.1 and multiply polynomials. Not until Section 5.8 will 5.2, we have not solved any equations since - we return to solving equations. At that point, Chapter 4. We have concentrated instead on however, we will study a new type of equation writing equivalent expressions by adding, sub- that cannot be solved without understanding tracting, and multiplying polynomials. We how to factor. As we have seen before, mathe- found that we could not have multiplied poly- matics consistently builds on learned concepts. nomials had we not first learned how to add or The more mastery you develop in factoring, the subtract them. In a similar manner, we will find better prepared you will be to solve the equa that our work with factoring polynomials in tions of Section 5.8. Sections 5.3-5.7 relies heavily on our ability to Terms with Common Factors When factoring, we look for factors common to every term in an expression and then use the distributive law Example 1 Factor out a common factor: 4y2 - 8. Solution 4y2 - 8 = 4 * y2 - 4 * 2 Noting that 4 is a common factor = 4(y2 - 2) Using the distributive law In some cases, there is more than one common factor. In 5x4 + 20x3, for instance, 5 is a common factor, x3 is a common factor, and 5x3 is a common factor. If there is more than one common factor, we factor out the largest, or greatest, common factor, that is, the common factor with the largest coefficient and the highest degree. In 5x4 + 20x3, the largest common factor is 5x3. 5.3 COMMON FACTORS AND FACTORING BY GROUPING page 289 To check Example 4 with a table, Note that we can obtain function values using either expression for h(t), since let yl = -16x2 + 64x and factoring forms equivalent expressions. For example, y2 = -16x(x - 4). Then com- pare values of yl and yZ. x y1 y2 0 0 0 1 48 48 2 64 64 3 48 48 4 0 0 5 -80 -80 6 -192 -192 x = 0 1. How can y3 = y2 - yl and a table be used as a check? OTBL = 1 h(t) = -16t2 + 64t = -16t(t-4) Solution We factor out -16t as follows: Factoring by Grouping h(1) = -16 * 1 2is smaller number + 64 * 1 = 48 h(1) = -16 * 1 1-4 = 48 The largest common factor is sometimes a binomial. Example 5 Factor: (a - b)(x + 5) + (a - b)(x - y2). Solution Here the largest common factor is the binomial a - b: (a-b)(x+5)+(a-b)(x-y2) (a-b)[(x+5)+ a - b (x-y2)] : =(a-b)+ [x+5- xy2]. Often, in order to identify a common binomial factor, we must regroup into two groups of two terms each. Example 6 Factor: (a) y3 + 3y2 + 4y + 12; (b) 4x3 - 15 + 20x2 - 3x. Solution a) y3 + 3yZ + 4y + 12 = ( y3 + 3y2) + (4y + 12) Each grouping has a common factor. = y2( y + 3) + 4( y + 3) Factoring out a common factor from each binomial =( y + 3) ( y2 + 4) Factoring out y + 3 technology connection h(t) = -16t2 + 64t = -16t(t - 4). Check: -16t ~ t = -16tZ and -16t(-4) = 64t. and h(1) = -16 ~ 1(1 - 4) = 48. Using the factorization In Example 4, we could have evaluated -16t2 + 64t and -16t(t - 4) using any value for t. The results should always match. Thus a quick partial check of any factorization is to evaluate the factorization and the original polynomial for one or two convenient replacements. The check in Example 4 becomes foolproof if three replacements are used. In general, an nth-degree factoriza-tion is correct if it checks for n + 1 different replacements. page 290 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCfIONS b) When we try grouping 4x3 - 15 + 20x2 - 3x as (4x3 - 15) + (20x2 - 3x), we are unable to factor 4x3 - 15. When this happens, we can rearrange the polynomial and try a different grouping: 4x3 - 15 + 20x2 - 3x = 4x3 + 20x2 - 3x - 15 Using the commu-tative law to re-arrange the terms = 4x2(x + 5) - 3(x + 5) = (x + 5) (4x2 - 3). In Example 8 of Section 1.3 (see p. 25), we saw that b - a, -(a - b), and -1(a - b) are equivalent. Remembering this can help anytime we wish to reverse sub-traction (see the third step below). Example 7 Factor: ax - bx + by - ay. ax - bx + by - ay = (ax - bx) + (by - ay) Grouping = x(a - b) + y(b - a) Factoring each binomial = x(a - b) + y(-1) (a - b) Factoring out -1 to reverse b - a = x(a - b) - y(a - b) = (a - b) (x - y). Factoring out a - b Simplifying We can always check our factoring by multiplying: Check: (a - b)(x - y) = ax - ay - bx + by = ax - bx + by - ay. Some polynomials with four terms, like x3 + x2 + 3x - 3, are prime. Not only is there no common monomial factor, but no matter how we group terms, there is no common binomial factor: x3 + x2 + 3x - 3 = x2(x + 1) + 3(x - 1); No common factor x3 + 3x + x2 - 3 = x(x2 + 3) + (x2 - 3); No common factor x3 - 3 + x2 + 3x = (x3 - 3) + x(x + 3). No common factor 5.3 COMMON FACTORS AND FACTORING BY GROUPING page 291 Exercise Set 5.3 Factor. 1. 2t2 + 8t 2. 3y2 + 6y 3. y2 - 5y 4. x2 + 9x 5. y3 + 9y2 6. x3 + 8x2 7. 15x2 - 5x4 8. 8y2 + 4y4 9. 4x2y - 12xy2 10. 5x2y3 + 15x3y2 11. 3y2 - 3y - 9 12. 5x2 - 5x + 15 13. 6ab - 4ad + 12ac 14. 8xy + lOxz - 14xw 15. 9x3y6z2 - 12x4y4z4 + 15x2y5z3 16. 14a4b3c5 + 21a3b5c4 - 35a4b4c3 Factor out a factor with a negative coefficient. 17. -5x + 35 18. -5x - 40 19. -6y - 72 20. -8t + 72 21. -2x2 + 4x - 12 22. -2x2 + 12x + 40 23. 3y - 24x 24. 7x - 56y 25. 7s - 14t 26. 5r - lOs 27. -x2 + 5x - 9 28. -p3 - 4p2 + 11 29. -a4 + 2a3 - 13a 30. -m3 - m2 + m - 2 31. a(b - 5) + c(b - 5) 32. r(t - 3) - s(t - 3) 33. (x + 7) (x - 1) + (x + 7) (x - 2) 34. (a + 5) (a - 2) + (a + 5) (a + 1) 35. a2(x - y) + 5( y - x) 36. 5x2(x - 6) + 2(6 - x) 37. ac + ad + bc + bd 38. xy + xz + wy + wz 39. b3 - b2 + 2b - 2 40. y3 - y2 + 3y - 3 41. a3 - 3a2 + 6 - 2a 42. t3 + 6t2 - 2t - 12 43. 72x3 - 36x2 + 24x 44. 12a4 - 21a3 - 9a2 45. x6 - x5 - x3 + x4 46. y4 - y3 - y + y2 47. 2y4 + 6y2 + 5y2 + 15 48. 2xy - x2y - 6 + 3x FOR EXTRA HElP ®• Digital Video Tutor CD 4 InterAct Math Math Tutor Center MathXL MyMathlab.com Videotape 9 49. Height of a baseball. A baseball is popped up with an upward velocity of 72 ft/sec. Its height in feet, h(t), after t seconds is given by h(t) = -16t2 + 72t. a) Find an equivalent expression for h(t) by factor-ing out a common factor with a negative coefficient. b) Perform a partial check of part a by evaluating both expressions for h(t) at t = 3. 50. Height of a rocket. A model rocket is launched up ward with a initial velocity of 96 ft/sec. Its height in feet h(t) after t seconds is given by h(t) = -16t2 + 96t. a) Find a equivalent expression for h(t) by factor- ing out a common factor with a negative coefficient. b) Check your factoring by evaluating both expres-sions for h(t) at t = 1. 51. Airline routes. When an airline links n cities so that from any one city it is possible to fly directly to each of the other cities, the total number of direct routes is given by R(n) = n2 - n. Find an equivalent expression for R(n) by factoring Factor. out a common factor. 52. Surface area of a silo. A silo is a structure that is shaped like a right circular cylinder with a half sphere on top. The surface area of a silo of height h and radius r (including the area of the base) is given by the polynomial 2 ttrh + ttr2. Find an equivalent expression by factoring out a common factor. page 292 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCfIONS 53. Total profit When x hundred CD players are sold, Rolics Electronics collects a profit of P(x), where P(x) = x2 - 3x and P(x) is in thousands of dollars. Find an equiva- lent expression by factoring out a common factor. 54. Total profit After tweeks of production, Claw Foot, Inc., is making a profit of P(t) = t2 - 5t from sales of their surfboards. Find an equivalent expression by factoring out a common factor. 55. Total revenue. Urban Sounds is marketing a new MP3 player. The firm determines that when it sells x units, the total revenue R is given by the polyno- mial function R(x) = 280x - 0.4x2 dollars. Find an equivalent expression for R(x) by factoring out 0.4x. 56. Total cost. Urban Sounds determines that the total cost C of producing x MP3 players is given by the polynomial function. C(x) = 0.18x + 0.6x2 Find an equivalent expression for G(x) by factoring out 0.6x. 57. Counting spheres in a pile. The number N of spheres in a triangular pile like the one shown here is a polynomial function given by N(x)= 1 3 1 2 1 6x + 2 x + 3x, where x is the number of layers and N(x) is the number of spheres. Find an equivalent expression for N(x) by factoring out 1. 6 58. Number of games in a league. If there are n teams in a league and each team plays every other team once, we can find the total number of games played by using the polynomial function 1 2 1 f(n) = 2n - 2n. Find an equivalent expression by factoring out 1. 2 59. High fives When a team of n players all give each other high-fives, a total of H(n) hand slaps occurs, where H(n)= 1 1 2n2 - 2n. 1 Find an equivalent expression by factoring out 2 n. 60. Number of diagonals The number of diagonals of a polygon having n sides is given by the polynomial function 1 3 P(n) = 2 n2 - 2 n. Find a equivalent expression for P(n) by factoring out 1. 2 61. Under what conditions would it be easier to evalu-ate a polynomial after it has been factored? ® 62. Explain in your own words why -(a - b) = b - a. SKILL MAINTENANCE ° Simplify. 63. 2(-3) + 4(-5) 64. -7(-2) + 5(-3) 65. 4(-6) - 3(2) 66. 5(-3) + 2(-2) 67. Geometry. The perimeter of a triangle is 174. The lengths of the three sides are consecutive even numbers. What are the lengths of the sides of the triangle? 68. Manufacturing. In a factory, there are three ma- chines A, B, and C. When all three are running, they produce 222 suitcases per day. If A and B work but C does not, they produce 159 suitcases per day. If B and C work but A does not, they produce 147 suit- cases. What is the daily production of each machine? SYNTHESIS 69. Is it true that if a polynomial's coefficients and ex- ponents are all prime numbers, then the polyno- mial itself is prime? Why or why not? page 293 70. Following Example 4, we stated that checking the factorization of a second-degree polynomial by making a single replacement is only a partial check. Write an incorrect factorization and explain how evaluating both the polynomial and the fac- torization might not catch the mistake. Complete each of the following. 71. x5y4 + ____ = x3Y(___+ xY5) 72. a3b7 - ____ = ___(ab4 - c2) Factor 73. rx2 - rx + 5r + sx2 - sx + 5s 74. 3a2 + 6a + 30 + 7a2b + 14ab + 70b 75. a4x4 + a4x2 + 5a4 + a2x4 + a2x2 + 5a2 + 5x4 + 5x2 + 25 (Hint: Use three groups of three.) Factor out the smallest power of x in each of the following. 76. xl/2 + 5x3/2 77. xl/3 - 7x4/3 78. x3/4 + x1/2 - x1/4 79. x1/3 - 5x1/2 + 3x3/4 Factor. Assume that all exponents are natural numbers. 80. 2x3a + 8xa + 4x2a 81. 3an+1 + 6an - 15an+2 82. 4xa+b + 7xa-b 83. 7y2a+b - 5ya+b + 3ya+2b 84. Use the TABLE feature of a grapher to check your an- swers to Fxercises 17, 29, and 33. 85. Use a grapher to show that (x2 - 3x + 2)4 = x8 + 81x4 + 16 is not an identity. 5.4 FACTORINGTRINOMIALS page 293 Factaring Trinomials Study Tip 5.4 Factoring Trinomials of the Type x2 + bx + c • Factoring Trinomials of the Type ax2 + bx + c, a =/ 1 Our study of the factoring of trinomials begins with trinomials of the type x2 + bx + c. We then move on to the type ax2 + bx + c, where a =/ 1. Spending extra time studying this section will save you time Factoring Trinomials of tfle Type X2 + bx + C when you work in Sections 5.5-5.s. When trying to factor trinomials of the type x2 + bx + c, we can use a trial-and-error procedure. Constant Term Positive Recall the FOIL method of multiplying two binomials: F O I L (x + 3)(x + 5)= x2 + 5x + 3x + 15 = x2 + 8x + 15. Because the leading coefficient in each binomial is 1, the leading coefficient in the product is also 1.,To factor x2 + 8x + 15, we think of FOIL: The first term, page 294 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS x2, is the product of the First terms of two binomial factors, so the first term in each binomial must be x. The challenge is to find two numbers p and q such that x2 + 8x + 15 = (x + p)(x + q). = x2 + qx + px + pq. Note that the Outer and Inner products, qx and px, can be written as (p + q)x. The Last product, pq, will be a constant. The numbers p and q must be se-lected so that their product is 15 and their sum is 8. In this case, we know from above that these numbers are 3 and 5. The factorization is (x + 3) (x + 5), or (x + 5) (x + 3). Using a commutative law In general, to factor x2 + ( p + q)x + pq, we use FOIL in reverse: x2 + ( p + q)x + pq = (x + p) (x + q). Example 1 Factor: x2 + 9x + 8. Solution We think of FOIL in reverse. The first term of each factor is x. We are looking for numbers p and q such that x2 + 9x + 8 = (x + p)(x + q) = x2 + (p + q) x + pq. we search for factors of 8 whose sum is 9. Pair of Factors Sum of Factors 2, 4 6 1, 8 9 The factorization is (x + 1) (x + 8). The student should check by multi-plying to confirm that the product is the original trinomial. When factoring trinomials with a leading coefficient of 1, it suffices to consider all pairs of factors along with their sums, as we did above. At times, however, you may be tempted to form factors without calculating any sums. It is essential that you check any attempt made in this manner! For example, if we attempt the factorization x2 + 9x + 8?= (x+2)(x+4), a check reveals that (x + 2)(x + 4) = x2 + 6x + 8 =/ x2 + 9x + 8. This type of trial-and-error procedure becomes easier to use with time. As you gain experience, you will find that many trials can be performed mentally. 6 9 ~- The numbers we need are 1 and 8. 5.4 FACTORING TRINOMIALS page 295 When the constant term of a trinomial is positive, the constant terms in both binomial factors must have the same sign. This ensures a positive prod-uct. The sign used is that of the trinomial's middle term. Example 2 Factor: y2 - 9y + 20. Solution Since the constant term is positive and the coefficient of the middle term is negative, we look for a factorization of 20 in which both factors are negative. Their sum must be -9. Pair of Factors Sum of Factors -1, -20 -21 -2, -10 -12 -4, -5 -9< The factorization is (y - 4) (y - 5). Constant Term Negative When the constant term of a trinomial is negative, we look for one negative fac-tor and one positive factor. The sum of the factors must still be the coefficient of the middle term. The numbers we need are -4 and -5. Example 3 Factor: x3 - x2 - 30x. Solution Always look first for a common factor! This time there is one, x. We factor it out: x3 - x2 - 30x = x(x2 - x - 30). Now we consider x2 - x - 30. We need a factorization of -30 in which one fac-tor is positive, the other factor is negative, and the sum of the factors is -1. Since the sum is to be negative, the negative factor must have the greater absolute value. we need only consider the following pairs of factors. Pair of Factors Sum of Factors 1,-30 -29 3,-10 -7 5, -6 -1 <_ The numbers we need are 5 and -6. The factorization of x2 - x - 30 is (x + 5) (x - 6). Don't forget to include the factor that was factored out earlier! In this case, the factorization of the original trinomial is x(x + 5) (x - 6). page 296 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Example 4 Factor: 2x2 + 34x - 220. Solution Always look first for a common factor! This time we can factor out 2: technology connection I 2x2 + 34x - 220 = 2(x2 + 17x - 110). The method described in Tech-nology Connection B in Section 5.1 can be used to check Example 4: Let yl = 2x2 + 34x - 220, y2 = 2(x - 5) (x + 22), and Y3 =Y2 -Yl. 1. How should the graphs of yl and y2 compare? 2. What should the graph of y3 look like? 3. Check Example 3 with a grapher. 4. Use graphs to show that (2x + 5) (x - 3) is not a fac-torization of 2x2 + x - 15. We next look for a factorization of -110 in which one factor is positive, the other factor is negative, and the sum of the factors is 17. Since the sum is to be positive, we examine only pairs of factors in which the positive term has the larger absolute value. Pair of Factors Sum of Factors -1 110 109 -2, 55 53 -5, 22 17 The numbers we need are -5 and 22. The factorization of x2 + 17x - 110 is (x - 5) (x + 22). The factorization of the original trinomial, 2x2 + 34x - 220, is 2(x - 5) (x + 22). Some polynomials are not factorable using integers. Example 5. Factor: x2 - x - 7. Solution There are no factors of -7 whose sum is -1. This trinomial is not factorable into binomials with integer coefficients. Although x2 - x - 7 can be factored using more advanced techniques, for our purposes the polynomial is prime. Tips for Factoring x2 + bx + c 1. If necessary, rewrite the trinomial in descending order. Search for factors of c that add up to b. Remember the following: • If c is positive, the signs of the factors are the same as the sign of b. • If c is negative, one factor is positive and the other is negative. • If the sum of the two factors is the opposite of b, changing the signs of both factors will give the desired factors whose sum is b. 2. Check the result by multiplying the binomials. These tips still apply when a trinomial has more than one variable. chapter 5 5.4 FACfORING TRINOMIALS page 300 page 301 3. Try to factor -24 so that the sum of the factors is 10: -24 = 12(-2) and 12 + (-2) = 10. 4. Split lOx using the results of step (3): lOx = 12x - 2x. 5. Finally, factor by grouping: 3x2 + lOx - 8 = 3x2 + 12x - 2x - 8 = 3x(x + 4) - 2(x + 4)~ Factoring by = (x + 4)(3x - 2). grouping Exercise Set 5,4 page 301 FOR EXTRA HELP Digital Video Tutor CD 4 InterAd Math Math Tutor Center MathXL MyMathLab.com Videotape 9 a. 2 is uppercase small size number. 1. x2 + 8x + 12 2. x2 + 6x + 5 3. t2 + 8t - 15 4. y2 + 12y + 27 5. x2 - 27 - 6x 6. t2 - 15 -2t 7. 2n2 - 20n + 50 8. 2a2 - 16a + 32 9. a3 - a2 - 72a 10. x3 + 3x2 - 54x 11. 14x + x2 + 45 12. 12y + y2 + 32 13. y2 + 2y - 63 14. p2 - 3p - 40 15. t2 - 14t + 45 16. a2 - l1a + 28 17. 3x + x2 - 10 18. x + x2 - 6 19. 3x2 + 15x + 18 20. 5y2 + 40y + 35 21. 56 + x - x2 22. 32 + 4y - y2 23. 32y + 4y2 - y3 24. 56x + x2 - x3 25. x4 + 11x3 - 80x2 26. y4 + 5y3 - 84y2 27. x2 + 12x + 13 28. x2 - 3x + 7 29. p2 - 5pq - 24q2 30. x2 + 12xy + 27y2 31. y2 + 8yz + 16z2 32. x2 - 14xy + 49y2 33. p4 - 80p3 + 79p2 34. x4 - 50x3 + 49x2 35. x6 + 7x5 - 18x4 36. x6 + 2x5 - 63x4 37. 6x2 - 5x - 25 38. 3x2 - 16x - 12 39. 10y3 - 12y - 7y2 40. 6x3 - 15x - x2 41. 24a2 - 14a + 2 42. 3a2 - l0a + 8 43. 35y2 + 34y + 8 44. 9a2 + 18a + 8 45. 4t + lOt2 - 6 46. 8x + 30x2 - 6 47. 8x2 - 16 - 28x 48. 18x2 - 24 - 6x 49. a6 + a5 - 6a4 50. t8 + 5t7 - 14t6 51. 14x4 - 19x3 - 3x2 52. 70x4 - 68x3 + 16x2 53. 12a2 - 4a - 16 54. 12a2 - 14a - 20 55. 9x2 + 15x + 4 56. 6y2 + 7y + 2 Aha 57. 4x2 + 15x + 9 58. 2y2 + 7y + 6 59. -8t2 - 8t + 30 60. -36a2 + 21a - 3 61. 18xy3 + 3xy2 - lOxy 62. 3x3y2 - 5x2y2 - 2xy2 63. 24x2 - 2 - 47x 64. 15y2 - 10 - 47y 65. 63x3 + 111x2 + 36x 66. 50y3 + 115y2 + 60y 67. 48x4 + 4x3 - 30x2 68. 40y4 + 4y3 - 12y2 69. 12a2 - 17ab + 6b2 70. 20p2 - 23pq + 6q2 71. 2x2 + xy - 6y2 72. 8m2 - 6mn - 9n2 73. 6x2 - 29xy + 28y2 74. lOp2 + 7pq - 12q2 75. 9x2 - 30xy + 25y2 76. 4p2 + 12pq + 9q2 77. 9x2y2 + 5xy - 4 78. 7a2b2 + 13ab + 6 page 3O2 CHAFTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 79. How can one conclude that x2 + 5x + 200 is a prime polynomial without performing any trials? 80. How can one conclude that x2 - 59x + 6 is a prime polynomial without performing any trials? SKILL MAINTENANCE Factor 81. lOx3 - 35x2 + 5x 82. 12t3 - 40t2 - 8t Simplify 83. (5a4)3 84. (-2x2)5 85. If g(x) = -5x2 - 7x, find g(-3). 86. Height of a rocket. A model rocket is launched upward with an initial velocity of 96 ft/sec from a height of 880 ft. Its height in feet, h(t), after t sec- onds is given by the other factor. h(t) = -16t2 + 96t + 880. What is its height after 0 sec, 1 sec, 3 sec, 8 sec, and 10 sec? SYNTHESIS 87. Describe in your own words an approach that can be used to factor any "nonprime" trinomial of the form ax2 + bx + c. 88. Suppose (rx + p) (sx - q) = ax2 - bx + c is true. Explain how this can be used to factor ax2 + bx + c. Factor. Assume that variables in exponents represent positive integers. 89. 2a4b6 - 3a2b3 - 20ab2 90. 5x8y6 + 35x4y3 + 60 91. x2 - 4 3 25 + 5x 92. y2 - 8 2 49 + 7Y 93. y2 + 0.4y - 0.05 94. 4x2a - 4xa - 3 95. x2a + 5xa - 24 96. x2 + ax + bx + ab 97. bdx2 + adx + bcx + ac 98. 2ar2 + 4asr + as2 - asr 99. a2p2a + a2pa - 2a2 Aha 100. (x+3)2 -2(x+3)- 35 101. 6(x - 7)2 + 13(x - 7) - 5 102. Find all integers m for which x2 + mx + 75 can be factored. 103. Find all integers q for which x2 + qx - 32 can be be factored. 104. One factor of x2 - 345x - 7300 is x + 20. Find the other factor. 105. To better understand factoring ax2 + bx + c by grouping, suppose that ax2 + bx + c = (mx + r)(nx + s). Show that if P = ms and Q = rn, then P + Q = b and PQ = ac. 106. Use the TABLE feature to check your answers to Exercises 11, 65, and 93. 107. Let yl = 3x2 + lOx - 8, y2 = (x + 4) (3x - 2), y3 = y2 - yl to check Example 9 graphically. 108. Explain how the following graph of y = x2 + 3x - 2 - (x - 2)(x + 1) can be used to show that x2 + 3x - 2 =/ (x-2)(x+l). io io 5.5 FACTORING PERFECT-SQUARE TRINOMIALS AND DIFFERENCES OF SQUARES page 303 5.5 Factoring Perfect- Square Trinomials Perfect-Square Trinomials • Differences of Squares • and Differences More Factoring by Grouping of squares We now introduce a faster way to factor trinomials that are squares of bino-mials. A method for factoring differences of squares is also developed. Perfect-Square Trinomials Consider the trinomial x2 + 6x + 9. To factor it, we can proceed as in Section 5.4 and look for factors of 9 that add to 6. These factors are 3 and 3 and the factorization is x2 + 6x + 9 = (x + 3)(x + 3)=(x + 3)2. Note that the result is the square of a binomial. Because of this, we call x2 + 6x + 9 a perfect-square trinomial. Although trial and error can be used to factor a perfect-square trinomial, a faster procedure can be used if we rec-ognize when a trinomial is a perfect square. To Recognize a Perfect-Square Trinomial • Two of the terms must be squares, such as A2 and B2. * There must be no minus sign before A2 or B2. • The remaining term is twice the product of A and B, 2AB, or its opposite, -2AB. Example 1 3 Determine whether each polynomial is a perfect-square trinomial. a) x2 + lOx + 25 b) 4x + 16 + 3x2 c) 100y2 + 81 - 180y Solution a) • Two of the terms in x2 + 10x + 25 are squares: x2 and 25. • There is no minus sign before either x2 or 25. • The remaining term, lOx, is twice the product of the square roots, x and 5. x2 + lOx + 25 is a perfect square. page 3O4 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS b) In 4x + 16 + 3x2, only one term, 16, is a square (3x2 is not a square because 3 is not a perfect-square integer and 4x is not a square because x is not a square) . 4x + 16 + 3x2 is not a perfect square. c) It can help to first write the polynomial in descending order: 100y2 - 180y + 81. • Two of the terms, 100y2 and 81, are squares. • There is no minus sign before either 100y2 or 81. • If the product of the square roots, l0y and 9, is doubled, we get the oppo-site of the remaining term: 2(l0y) (9) = 180y (the opposite of -180y). 100y2 + 81 - 180y is a perfect-square trinomial. To factor a perfect-square trinomial, we reuse the patterns that we learned in Section 5.2. Factoring a Perfect-Square Trinomiol A2 + 2AB + B2 = (A + B)2; A2 - 2AB + B2 = (A - B)2 Example 2 ~ Factor. a) x2 - lOx + 25 b) 16y2 + 49 + 56y c) -20xy + 4y2 + 25x2 Solution a) x2 - lOx + 25 = (x - 5)2 We find the square terms and write the square roots with a minus sign between them. Note the sign! b) 16y2 + 49 + 56y = 16y2 + 56y + 49 Using a commutative law We find the square terms and = (4y + 7)2 write the square roots with a plus sign between them. c) -20xy + 4y2 + 25x2 = 4y2 - 20xy + 25x2 Writing descending order with respect to y = (2y - 5x)2 This square can also be expressed as 25x2 - 20xy + 4y2 = (5x - 2y)2. As always, any factorization can be checked by multiplying: ( 5x - 2y)2 = (5x - 2y)(5x - 2y) = 25x2 - 20xy + 4y2. 5.5 FACTORING PERFECT-SQUARE TRINOMIALS AND DIFFERENCES OF SQUARES page 307 b) Grouping a2 - b2 + 8b - 16 into two groups of two terms does not yield a common binomial factor, so we look for a perfect-square trinomial. In this case, the perfect-square trinomial is being subtracted from a2: a2 - b2 + 8b - 16 = a2 - (b2 - 8b + 16) Factoring out -1 and rewriting as subtraction = a2 - (b - 4)2 Factoring the perfect-square trinomial = (a + (b - 4)) (a - (b - 4)) Factoring a differ-ference of squares = (a + b - 4)(a - b + 4) Removing parentheses Exercise Set 5.5 ~Y~y. Videotape 10 Factor completely. 1. x2 - 8x + 16 2. t2 + 6t + 9 3. a2 + 16a + 64 4. a2 - 14a + 49 5. 2a2 + 8a + 8 6. 4a2 - 16a + 16 7. y2 + 36 - 12y 8. y2 + 36 + 12y 9. 24a2 + a3 + 144a 10. -18y2 + y3 + 81y 11. 32x2 + 48x + 18 12. 2x2 - 40x + 200 13. 64 + 25y2 - 80y 14. 1 - 8d + 16d2 15. a3 - 10a2 + 25a 16. y3 + 8y2 + 16y 17. 0.25x2 + 0.30x + 0.09 18. 0.04x2 - 0.28x + 0.49 19. p2 - 2pq + q2 20. m2 + 2mn + n2 21. 25a2 + 30ab + 9b2 22. 49p2 - 84pq + 36q2 23. 4t2 - 8tr + 4r2 24. 5a2 - l0ab + 5b2 25. x2 - 16 26. y2 - 100 27. p2 - 49 28. m2 - 64 29. a2b2 - 81 30. p2q2 - 25 31. 6x2 - 6y2 32. 8x2 - 8y2 33. 7xy4 - 7xz4 34. 25ab4 - 25az4 35. 4a3 - 49a 36. 9x4 - 25x2 37. 3x8 - 3y8 38. 9a4 - a2b2 39. 9a4 - 25a2b4 40. 16x6 - 121x2y4 41. 1 25-x2 42. 1 15-y2 43. (a + b)2 - 9 44. (p + q)2 - 25 45. x2 - 6x + 9 - y2 46. a2 - 8a + 16 - b2 47. m2 - 2mn + n2 - 25 48. x2 + 2xy + y2 - 9 49. 36 - (x + y)2 50. 49 - (a + b)2 51. r2 - 2r + 1 - 4s2 52. c2 + 4cd + 4d2 - 9p2 Pha'~ 53. 16 - a2 - 2ab - b2 54. 9 - x2 - 2xy - y2 55. m3 - 7m2 -4m + 28 56. x3 + 8x2 - x - 8 57. a3 - ab2 - 2a2 + 2b2 58. p2q - 25q + 3p2 - 75 59. Are the product and power rules for exponents (see Section 1.6) important when factoring differences of squares? Why or why not? 60. Describe a procedure that could be used to find a polynomial with four terms that can be factored as a difference of two squares. fQR EXTRA HELP Digital Video Tutor CD 4 InterAct Math Math Tutor Center MathXL MvMathLab.com page 308 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS SKILL MAINTENANCE Simplify. 61. (2a4b5)3 62. (5x2y4)3 63. (x + y)3 64. (a + 1)3 Solve. 65. x - y + z = 6, 2x + y - z = 0, x + 2y + z = 3 66. |5-7x| _>9 67. |5-7x| <_ 9 68. 5 - 7x > -9 + 12x SYNTHESIS 69. Without finding the entire factorization, determine the number of factors of x256 - 1. Explain how you arrived at your answer. 70. Under what conditions can a sum of two squares be factored? Factor completely. Assume that variables in exponents represent positive integers. 8 10 1 2 71. -27r2- 9rs-6s2+3rs 72. 1 2 4 36x8 + 9x4 + 9 73. 0.09x8 + 0.48x4 + 0.64 74. a2 + 2ab + b2 - c2 + 6c - 9 75. r2 - 8r - 25 - s2 - 10s + 16 76. x 2a - y2 77. x4a - y 2b 78. 4y4a + 20y2a + 20y2a + 100 79. 25y 2a - (x 2b - 2xb + 1) 80. 8(a - 3)2 - 64(a - 3) + 128 81. 3(x + 1)2 + 12(x + 1) + 12 82. 5c100 - 80dl00 83. 9x2n - 6xn + 1 84. c2w+l + 2Cw + 1 + C 85. If P(x) = x2, use factoring to simplify P(a + h) - P(a). 86. If P(x) = x4, use factoring to simplify P(a + h) - P(a). 87. Volume of carpeting The volume of a carpet that is rolled up can be estimated by the polynomial ttR2h - ttr2h. a) Factor the polynomial. b) Use both the original and the factored forms to find the volume of a roll for which R = 50 cm, r=10cm,and h = 4m.Use 3.14 for tt. 88. Use a grapher to check your answers to Exercises 1, 35, and 55 graphically by examining yl = the origi-nal polynomial, y2 = the factored polynomial, and Y3 = Y2 - Yl. 89. Check your answers to Exercises 1, 35, and 55 by using tables of values (see Exercise 88). 5.6 FACTORING SUMS OR DIFFERENCES OF CUBES page 311 To factor (4yz)3 - (5xz)3, it is essential to remember the pattern used in Example 1: A3 - B3 = (A - B ) ( A2 + A * B + B 2) (4y2)3 - (5x2)3 = (4y2 - 5x2) ((4 y2)2 + 4y2 * 5x2 + (5x2y2) = (4y2 - 5X2) (16y4 + 20x2y2 + 25x4). 128y7 - 250x6y = 2y(4y2 - 5X2) (16y4 + 20x2y2 + 25x4). d) We have r6 - s6 = (r 3)y 2 - (S 3)2 = (r3 + s3) (r3 - s3) Factoring a difference of two squares = (r + s) (r 2 - rs + s2) (r - s) (r2 + rs + s2). Factoring the sum and differ-ence of two cubes In Example 2(d), suppose we first factored r6 - s6 as a difference of two cubes: (r2)3 - (s2)3 = (r2 - s2) (r4 + r2s2 + s4) = (r + s) (r - s) (r4 + r2s2 + s4). In this case, we might have missed some factors; r4 + r2s2 + s4 can be factored as (r 2 - rs + s2) (r2 + rs + s2), but we probably would never have suspected that such a factorization exists. Given a choice, it is generally better to factor as a difference of squares before factoring as a sum or difference of cubes. Try to remember the following: Useful Factoring Facts Sum of cubes: A3 + B3 = (A + B) (A2 - AB + B2); Difference of cubes: A3 - B3 = (A - B) (A2 + AB + B2); Difference of squares: A2 - B2 = (A + B) (A - B); Sum of squares: A2 + B2 cannot be factored using real numbers if the largest common factor has been removed. page 312 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNC710NS Exercise Set 5.6 Factor completely. 1. t3 + 27 27 + 3 =30 2. x3 + 64. The 3 after letters are uppercase small size numbers. 3. x3-8 8 -3 =5 4. z3 - 1 5. m3 - 64 64 -3 61 6. x3 - 27 27 -3 =24 7. 8a3 + 1 8. 27x3 + 1 27 x3 81 +1 =82 9. 27 - 8t3 3s are small size numbers 10. 64 - 125x3 11. 8x3 + 27 12. 27y3 + 64 13. y3 - z3 14. x3 - y3 15. x3 + 1 27 16. a3 + 1 8 17. 2y3 - 128 18. 8t3 - 8 19. 8a3 + 1000 20. 54x3 + 2 21. rs3 + 64r 22. ab3 + 125a 23. 2y3 - 54z3 24. 5x3 - 40z3 25. y3 + 0.125 26. x3 + 0.001 27. 125c6 - 8d6 6 4is small size numbers 28. 64x6 - 8t6 29. 3z5 - 3z2 5 2 is small size numbers 30. 2y4 - 128y 31. t6 + 1 32. z6 - 1 33. p6 - q6 6 is small size numbers 34. t6 + 64y6 35. a9 + b12c15 36. xl2 - y3zl2 all 3 sets of numbers are small size uppercorner numbers. 37. How could you use factoring to convince someone that x3 + y3 =/ (x + y)3? 38. Is the following statement true or false and why? If A3 and B3 have a common factor, then A and B have a common factor. SKILL MAINTENANCE 39. Height of a baseball. A baseball is thrown upward with an initial velocity of 80 ft/sec from a 224-ft- high cliff. Its height in feet, h(t), after t seconds is given by h(t) = -16t2 + 80t + 224. What is the height of the ball after 0 sec, 1 sec, 3 sec, 4 sec, and 6 sec? 40. The width of a rectangle is 7 ft less than its length. If the width is increased by 2 ft, the perimeter is then 66 ft. What is the area of the original rectangle? 41. If f(x) = 7 - x2, fmd f(-3). 42. Find the slope and the y-intercept of the line given by 4x - 3y = 8. Solve 43. 3x - 5 = 0 44. 2x + 7 = 0 SYNTHESIS 45. Explain how the geometric model below can be used to verify the formula for factoring a3 - b3. 46. Explain how the formula for factoring a difference of two cubes can be used to factor x3 + 8. Factor 47. x6a - y3b 48. 2x3a + 16y3b Aha~ 49. (x + 5)3 + (x - 5)3 50. 1 1 16x3a + 2y6a29b 51. 5x3y6 - 5 8 52. x3 - (x + y)3 FOR EXTRA HELP Digital Video Tutor CD 4 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 10 53. x6a - (x2a + 1)3 54. (x2a - 1)3 - x6a 55. t4 - 8t3 - t + 8 56. If P(x) = x3, use factoring to simplify P(a + h) - P(a). 57. If Q(x) = x6, use factoring to simplify Q(a + h) - Q(a). page 315 2 after 3x is small size numbers upper case Example 8 A. Factor out the largest common factor: 4(3x2 - lOx - 8). B. The trinomial factor is not a square. We factor using trial and error: 12x2 - 40x - 32 = 4(x - 4) (3x + 2). C. We cannot factor further. D. Check: 4(x - 4) (3x + 2) = 4(3x2 + 2x - 12x - 8) = 4(3x2 - lOx - 8) = 12x2 - 40x - 32. A. There is no common factor (other than 1 or -1). B. There are four terms. We try grouping to find a common binomial factor: 3x + 12 + ax2 + 4ax = 3(x + 4) + ax(x + 4) Factoring two grouped binomials = (x + 4) (3 + ax). Removing the com-mon binomial factor C. We cannot factor further. D. Check: (x + 4) (3 + ax) = 3x + ax2 + 12 + 4ax = 3x + 12 + ax2 + 4ax. Factor y2 - 9a2 + 12y + 36. Solution A. There is no common factor (other than 1 or -1). B. There are four terms. We try grouping to remove a common binomial factor, but find none. Next, we try grouping as a difference of squares: (y2 + 12y + 36) - 9a2 Grouping = (y + 6)2 - (3a)2 Rewriting as a difference of squares = (y + 6 + 3a)(y + 6 - 3a) C. No factor with more than one term can be factored further. D. The check is left to the student. A. There is no common factor (other than 1 or -1). B. There are four terms. We try grouping to remove a common binomial factor: x3 - xy2 + x2y - y3 = x(x2 - y2) + y(xZ ~ y2) Factoring two grouped binomials 5.7 FACTORING: A GENERAL STRATEGY 31 S x3 - xy2 + x2y - y3 =x (x2 - y2) + y(x2 - y2). Removing the common binomial factor = (x2 - y2) (x + y). Factoring the difference of squares page 316 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS C. The factor x2 - y2 can be factored further: x3 - xy2 + x2y - y3 = (x + y) (x - y) (x + y), or (x + y)2(x - y). No factor can be factored further, so we have factored completely. D. The check is left to the student. Exercise Set FOR EXTRA HELP Digital Video Tutor CD 4 InterAd Math Math Tutor Center MathXL MyMathLab.com Videotape 10 Factorcompletely. 4 2 are uppercase small numbers 1. 5m4 - 20 2. x2 - 144 3. a2 - 81 4. 2a2 - l1a + 12 5. 8x2 + 18x - 5 6. 2xy2 - 50x 7. a2 + 25 - l0a 8. p2 + 64 + 16p 9. 3x2 + 15x - 252 10. 2y2 + l0y - 132 2 or 3 after 2y or 2a x y are uppercase small numbers. 11. 9x2 - 25y2 12. 16a2 - 81b2 13. t6 + 1 14. 64t6 - 1 15. x2 + 6x - y2 + 9 16. t2 + lOt - p2 + 25 17. 343x3 + 27y3 18. 128a3 + 250b3 19. 8m3 + m6 - 20 20. -37x2 + x4 + 36 21. ac + cd - ab - bd 22. xw - yw + xz - yz 23. 4c2 - 4cd + d2 24. 70b2 - 3ab - a2 25. 24 + 9t2 + 8t + 3t3 26. 4a - 14 + 2a3 - 7a2 27. 2x3 + 6x2 - 8x - 24 28. 3x3 + 6x2 - 27x - 54 29. 54a3 - 16b3 30. 54x3 - 250y3 31. 36y2 - 35 + 12y 32. 2b - 28a2b + l0ab 33. a8 - b8 34. 2x4 - 32 35. a3b - 16ab3 36. x3y - 25xy3 Aha~ 37. (a - 3) (a + 7) + (a - 3) (a - 1) 38. x2(x + 3) - 4(x + 3) 39. 7a4 - 14a3 + 21a2 - 7a 40. a3 - ab2 + a2b - b3 41. 42ab + 27a2b2 + 8 42. -23xy + 20x2y2 + 6 43. p - 64p4 44. 125a - 8a4 Aha'45. a2 - b2 - 6b - 9 46. m2 - n2 - 8n - 16 47. Emily has factored a polynomial as (a - b) (x - y), while Jorge has factored the same polynomial as (b- a)(y - x). Can they both be correct? Why or why not? 48. In your own words, outline a procedure that can be used to factor any polynomial. SKILL MAINTENANCE Solve. 49. 5x - 9 = 0 50. 7x + 13 = 0 Graph 51. g(x) = 3x - 7 52. f(x) = -2x + 8 53. Exam scores There are 75 questions on a college entrance examination. Two points are awarded for each correct answer, and one half point is deducted for each incorrect answer. Ralph scored 100 on the exam. How many correct and how many incorrect answers did Ralph have if all questions were answered? 54. Perimeter. A pentagon with all five sides the same size has the same perimeter as an octagon in which a11 eight sides are the same size. One side of the pentagon is 2 less than three times the length of one side of the octagon. Find the perimeters. page 317 55. Explain how one could construct a polynomial that is a difference of squares that contains a sum of two cubes and a difference of two cubes as factors. 56. Explain how one could construct a polynomial with four terms that can be factored by grouping three terms together. Factor completely. 57. 60x2 - 97xy2 + 30y4 58. 28a3 - 25a2bc + 3ab2c2 59. -16 + 17(5 - y2) - (5 - y2)2 Aha~ 60. (x - P)2 - p2 61. a4 - 50a2b2 + 49b4 62. (y - 1)4 - (Y - 1)2 63. 27x6s + 64y3t 64. x6 - 2x5 + x4 - x2 + 2x - 1 65. 4x2 + 4xy + y2 - r2 + 6rs - 9s2 66. (1 - x)3 - (x - 1)6 67. 24t2a - 6 68. a2w+1 + 2aw+1 + a 69. x27 1000 - 1 70. a - by8 + b - ay8 71. 3(x + 1)2 - 9(x + 1) - 12 72. 3a2 + 3b2 -3c2 - 3d2 + 6ab - 6cd 73. 3(a + 2)2 + 30(a + 2) + 75 74. (m - 1)3 - (m + 1)3 2 2 8 75. If (x + x) = 6, find x3 + x3. 5.8 APPLICATIONS OF POLYNOMIAL EQUATIONS page 317 Polynomial EqUdt10115 The Principle of Zero Products • Polynomial Functions and Granhs • Problem Solving Whenever two polynomials are set equal to each other, we have a polynomial equation. Some examples of polynomial equations are 4x3 + x2 + 5x = 6x - 3, x2 - x = 6 and 3y4 + 2y2 + 2 = 0. The degree of a polynomial equation is the same as the highest degree of any term in the equation. from top to bottom, the degree of each equation listed above is 3, 2, and 4. A second-degree polynomial equation in one variable is usually called a quadratic equation. Of the equations listed above, only x2 - x = 6 is a quadratic equation. Polynomial equations, and quadratic equations in particular, occur fre-quently in applications, so the ability to solve them is an important skill. One way of solving certain polynomial equations involves factoring. 5.8 page 318 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS The Principle of Zero Products When we multiply two or more numbers, the product is 0 if any one of those numbers (factors) is 0. Conversely, if a product is 0, then at least one of the fac-tors must be 0. This property of 0 gives us a new principle for solving equations. The Principle of Zero Products For any real numbers a and b: If ab = 0,then a = 0 or b = 0. If a = O or b = O,then ab = 0. To solve an equation using the principle of zero products, we first write it in standard form: with 0 on one side of the equation and the leading coefficient positive. Example 1 Solve: x2 - x = 6. Solution To apply the principle of zero products, we need 0 on one side of the equation. we subtract 6 from both sides: x2 - x - 6 = 0. Getting 0 on one side In order to express the polynomial as a product, we factor: (x - 3) (x + 2) = 0. Factoring Since (x - 3) (x + 2) is 0, the principle of zero products says that at least one factor is 0. x - 3 = 0 or x + 2 = 0. Using the principle of zero products Each of these linear equations is then solved separately: x=3 or x = -2. We check as follows: x2 -x = 6 x2 - x = 6 32 - 3?6 (-2)2 -(-2)? 6 9-3 4 + 2 6 ~ 6 TRUE 6 6 TRUE Both 3 and -2 are solutions. The solution set is {3, -2}. To Use the Principle of Zero Products 1. Obtain a 0 on one side of the equation using the addition principle. 2. Factor the nonzero side of the equation. 3. Set each factor that is not a constant equal to 0. 4. Solve the resulting equations. page 321 the denominator x2 + 2x - 15 is 0. To make sure these values are excluded we solve x2 + 2x - 15 = 0 Setting the denominator equal to 0 (x - 3)(x + 5) = 0 Factoring x - 3 = 0 or x + 5 = 0 x = 3 or x = -5 These are the values to exclude. could graph the function given by g(x) = x2 - x - 6 and look for values of x for which g(x) = 0. Graph is y 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 a u line starts upper left corner goes down up to right upper corner -2 3 y = 6. g(x) = x2 - x - 6 page 322 height is h h(t) = -15t2 + 75t + 10. t-shirt 70 ft above ground level. How long was the tee shirt in the air? the value of t for which h(t) = 70. -15t2 + 75t + 10 =70. Solve the quadratic equation -15t2 + 75t + 10 = 70 -15t2 + 75t - 60 = 0 Subtracting 70 from both sides. -15(t2 - 5t + 4) = 0 -15(t - 4)(t - 1) = 0 Factoring. t- 4 = 0 or t - 1 = 0 t = 4 or t = 1. We have h(4) = - 15 * 4 2 is uppercasesmall + 75 * 4 + 10 = -240 + 300 + 10 = 70ft h(1) = - 15 * 1 2 is upper + 75 * 1 + 10 = - 15 + 75 + 10 = 70 ft. 5.8 APPLICATIONS OF POLYNOMIAL EQUATIONS page 323 5. State. The tee shirt was in the air for 4 sec before being caught 70 ft above ground level. The following problem involves the Pythagorean theorem, which relates the lengths of the sides of a right triangle. A right triangle has a 90°, or right, angle, which is denoted in the triangle by the symbol ~ or ~. The longest side, opposite the 90° angle, is called the hypotenuse. The other sides, called legs, form the two sides of the right angle. The Pythagorean Theorem In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then 6 The symbol f denotes a 90° angle. a2 + b2 = c2 Example 6 Carpentry. In order to build a deck at a right angle to her house, Lucinda decide to plant a stake in the ground a precise distance from the back wall of their house. This stake will combine with two marks on the house to form a right triangle. From a course in geometry, Lucinda remembers that there are three consecutive integers that can work as sides of a right triangle. Find the measurements of that triangle. Solution 1. Familiarize. Recall that x, x + 1, and x + 2 can be used to represent three unknown consecutive integers. Since x + 2 is the largest number, it must represent the hypotenuse. The legs serve as the sides of the right angle, so one leg must be formed by the marks on the house. We make a drawing in which x = the distance between the marks on the house, x + 1 = the length of the other leg, and x + 2 = the length of the hypotenuse. page 324 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 2. Translate. Applying the PythagoreaWtheorem, we translate as follows: a2 + b2 = c2 x2 + (x + 1)2 = (x + 2)2. 3. Carry out. We solve the equation as follows: x2 + (x2 + 2x + 1) = x2 + 4x + 4 Squaring the binomials 2x2 + 2x + 1 = x2 + 4x + 4 Combining like terms x2 - 2x - 3 = 0 Subtracting x2 + 4x + 4 from both sides (x - 3)(x + 1) = 0 Factoring x - 3 = 0 or x + 1 = 0 Using the principle of zero products 4. Check. The integer -1 cannot be a length of a side because it is negative. For x = 3, we have x + 1 = 4, and x + 2 = 5. Since 3 the 2s are small upper numbers. + 4 2 = 52, the lengths 3, 4, and 5 determine a right triangle. 3, 4, and 5 check. 5. State. Lucinda should use a triangle with sides having a ratio of 3 :4: 5. if the marks on the house are 3 yd apart, they should locate the stake at the point in the yard that is precisely 4 yd from one mark and 5 yd from the other mark: Example 7 Display of a sports card. A valuable sports card is 4 cm wide and 5 cm long. The card is to be sandwiched by two pieces of Lucite, each of which is 5 1 2 times the area of the card. Determine the dimensions of the Lucite that will ensure a uniform border around the card. 1. Familiarize. We make a a + 2x drawing and label it, using x to represent the width of the border, in centimeters. Since the border extends uni-formly around the entire card, the length of the Lucite must be 5 + 2x and the width must be 4 + 2x. • 54. - APPLICATIONS OF POLYNOMIAL EQUATIONS page 325 2. Translate. We rephrase the information given and translate as follows: Area of Lucite , is 5 1 times area of card. 2 (5 + 2x)(4 + 2x) = 5 1 - 5 * 4 2 3. Carry out. We solve the equation: (5 + 2x)(4 + 2x)= 5 1 * 5 * 4 2 20 + lOx + 8x + 4x2 = 110 Multiplying 4x2 + 18x - 90 = 0 Finding standard form 2(2x2 + 9x - 45) = 0 Factoring 2(2x + 15) (x - 3) = 0} 2x + 15 = 0 or x - 3 = 0 Principle of zero products x= - 7 1 or x = 3. 2 4. Check. We check 3 in the original problem. (Note that -7 1 is 2 not a solu-tion because measurements cannot be negative.) If the border is 3 cm wide, the Lucite will have a length of 5 + 2 * 3, or 11 cm, and a width of 4 + 2 * 3, or 10 cm. The area of the Lucite is 11 * 10, or 110 cm2. Since the area of the card is 20 cm 2 and 110 cm2 is 5 1 times 20 cm2, the number 3 checks. 2 5. State. Each piece of Lucite should bell cm long and 10 cm wide. FOR EXTRA HELP Exercise Set 5, Digital Video Tutor CD 4 InterAct Math Math Tutor Center MathXL MyMathLab.com Solve. 1. x2 - 4x = 45 2. t2 - 3t = 28 3. a2 + 1 = 2a 4. r2 + 16 = 8r 5. x2 + 12x + 36 = 0 6. y2 + 16y + 64 = 0 7. 9x + x2 + 20 = 0 8. 8y + y2 + 15 = 0 9. x2 - 8x = 0 10. t2 - 9t = 0 11. a3 - 3a2 = 40a 12. x3 - 2x2 = 63x 3 2 after x are small uppercase corner numbers. Aha~ 13. x2 - 16 = 0 14. r2 - 9 = 0 15. (t - 6) (t + 6) = 45 16. (a - 4) (a + 4) = 20 17. 3x2 - 8x + 4 = 0 18. 9x2 - 15x + 4 = 0 19. 4t3 + 11t2 + 6t= 0 20. 8y3 + lOy2 + 3y = 0 21. (y - 3)(y + 2)=14 22. (z + 4)(z - 2)= -5 23. x(5 + 12x) = 28 24. a(1 + 21a) = 10 25. a2- 1 = 0 64 26. x2 - 1 = 0 25 27. t4 - 26t2 + 25 = 0 28. t4 - 13t2 + 36 = 0 29. Let f(x) = x2 + 12x + 40. Find a such that f(a) = 8. 30. Let f(x) = x2 + 14x + 50. Find a such that f(a) = 5. 31. Let g(x) = 2x2 + 5x. Find a such that g(a) = 12. 32. Let g(x) = 2x2 - 15x. Find a such that g(a) = -7. 33. Let h(x) = 12x + x2. Find a such that h(a) = -27. 34. Let h(x) = 4x - x2. Find a such that h(a) = -32. Find the domain of the function f given by each of the following. 35. f(x) = 3 x2 - 4x - 5 36. f(x) = 2 x2 - 7x + 6 page 326 CHAPTER 5 . .F04YNOMIAIS.AND POLYNOMIAL FUNCTIONS 37. f(x) = x 6x2 - 54 2 3 after x is uppercorner small size number. 38. f(x) = 2x 5x2 - 20 39. f(x) = x - 5 9x - 18x2 40. f(x) = 1 + x 3x - 15x2 41. f(x) = 7 5x3 - 35x2 + 50x 42. f(x) = 3 2x3 - 2x2 - 12x Solve. 43. The square of a number plus the number is 132. What is the number? 44. The square of a number plus the number is 156. What is the number? 45. A photo is 5 cm longer than it is wide. Find the length and the width if the area is 84 cm2. 46. An envelope is 4 cm longer than it is wide. The area is 96 cm2. Find the length and the width. 47. Geometry. If each of the sides of a square is lengthened by 4 m, the area becomes 49 m2. Find the length of a side of the original square. 48. Geometry. If each of the sides of a square is lengthened by 6 cm, the area becomes 144 cm 2. Find the length of a side of the original square. 49. Framing a picture. A picture frame measures 12 cm by 20 cm, and 84 cm2 of picture shows. Find the width of the frame. 50. Framing a picture. A picture frame measures 14 cm by 20 cm, and 160 cm2 of picture shows. Find the width of the frame. 51. Landscaping. A rectangular lawn measures 60 ft by 80 ft. Part of the lawn is torn up to install a side-walk of uniform width around it. The area of the new lawn is 2400 ft2. How wide is the sidewalk? 52. Landscaping. A rectangular garden is 30 ft by 40 ft. Part of the garden is removed in order to in-stall a walkway of uniform width around it. The area of the new garden is one-half the area of the old garden. How wide is the walkway? 53. 54. Tent design Three consecutive even integers are such that the 6x2 _ 54 ' f(x) 5x Z - 20 9x - 18x 54. Three consecutive even integers are such that the square of the square of the first plus the square of the third is 136. Find the three integers. 55. Tent design. The triangular entrance to a tent is 2 ft taller than it is wide. The area of the entrance is 12 ft2. Find the height and the base. Area =12 ft2 56. Antenna wires. A wire is stretched from the ground to the top of an antenna tower, as shown. The wire is 20 ft long. The height of the tower is 4 ft greater than the distance d from the tower's base to the bottom of the wire. Find the distance d and the height of the tower. 57. Sailing. A triangular sail is 9 m taller than it is wide. The area is 56 m2. Find the height and the base of the sail. 20 ft. Area = 56 m2 5.8 APPLICATIONS OF POLYNOMIAL EQUATIONS page 327 58. Parking lot design. A rectangular parking lot is 50 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 250 ft diagonally. 59. Ladder location. The foot of an extension ladder is 9 ft from a wall. The height that the ladder reaches on the wall and the length of the ladder are consecu- tive integers. How long is the ladder? 60. Ladder location. The foot of an extension ladder is 10 ft from a wall. The ladder is 2 ft longer than the height that it reaches on the wall. How far up the wall does the ladder reach? 61. Garden design. Ignacio is planning a garden that is 25 m longer than it is wide. The garden will have an area of 7500 m2. What will its dimensions be? 62. Garden design. A flower bed is to be 3 m longer than it is wide. The flower bed will have an area of 108 m2. What will its dimensions be? 63. Cabinet making Dovetail Woodworking deter- mines that the revenue R, in thousands of dollars, from the sale of x sets of cabinets is given by R(x) = 2x2 + x. If the cost C, in thousands of dol- lars, of producing x sets of cabinets is given by - C(x) = x2 - 2x + 10, how many sets must be pro- duced and sold in order for the company to break even? 64. Camcorder production. Suppose that the cost of making x video cameras is C(x) = 1 x2 + 2x + 1, 9 where C(x) is in thousands of dollars. If the revenue from the sale of x video cameras is given by 5 R(x) = 36 x2 + 2x, where R(x) is in thousands of dollars, how many cameras must be sold in order for the firm to break even? 65. Prize tee shirts. Using the model in Example 5, de- termine how long a tee shirt has been airborne if it is caught on the way up by a fan 100 ft above ground level. 66. Prize tee shirts. Using the model in Example 5, de- termine how long a tee shirt has been airborne if it is caught on the way down by a fan 10 ft above ground level. 67. Fireworks displays. Fireworks are typically launched from a mortar with an upward velocity (initial speed) of about 64 ft/sec. The height h(t), in feet, of a "weeping willow" display, t seconds after having been launched from an 80-ft-high rooftop, is given by h(t) = -16t2 + 64t + 80. After how long will the cardboard shell from the fireworks reach the ground? 68. Safety flares. Suppose that a flare is launched up- ward with an initial velocity of 80 ft/sec from a height of 224 ft. Its height in feet, h(t), after t sec onds is given by h(t) = - 16t2 + 80t + 224. After how long will the flare reach the ground? 69. Suppose that you are given a detailed graph of y = p(x), where p(x) is some polynomial in x. How could the graph be used to help solve the equation p(x) = 0? 70. Can the number of solutions of a quadratic equa- tion exceed two? Why or why not? 3.28 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS page 328 SKILL MAINTENANCE Simplify. 71. 5 - 10 * 3 -4 + 11 * 4 72. 2 * 3 - 5 * 2 7 - 3 2is upper small number 73. Driving At noon, two cars start from the same location going in opposite directions at different speeds. After 7 hr, they are 651 mi apart. If one car is traveling 15 mph slower than the other car, what are their respective speeds? 74. Television sales At the beginning of the month, J.C.'s Appliances had 150 televisions in stock. Dur- ing the month, they sold 45% of their conventional televisions and 60% of their surround-sound televi- sions. If they sold a total of 78 televisions, how many of each type did they sell? Solve. 75. 2x - 14 + 9x > -8x + 16 + lOx 76. x + y = 0, z - y = -2, x - z = 6 SYNTHESIS 77. Explain how one could write a quadratic equation that has -3 and 5 as solutions. 78. If the graph of f(x) = ax2 + bx + c has no x-intercepts, what can you conclude about the equation ax2 + bx + c = 0? Solve. 79. (8x + 11) (12x2 - 5x - 2) = 0 80. (x + 1)3 = (x - 1)3 + 26 81. (x - 2)3 = x3 - 2 82. Use the following graph of g(x) = -x2 - 2x + 3 to solve -x2 - 2x + 3 = 0 and to solve -x2 - 2x + 3 _> -5. y 5 g(x)= -x2 - 2x + 3 3 2 1 -5 -4 -3 -2 -1 2 3 4 5 x -1 -2 -3 -4 -5 The u starts in lower left corner goes up and down into lower right corner. -2 3 or 4 2 -5 83. Find a polynomial function f for which f(2) = 0, f(-1) = 0, f(3) = 0, and f(0) = 30. 84. Find a polynomial function g for which g(-3) = 0, g(1) = 0, g(5) = 0, and g(0) = 45. 85. Use the following graph of f(x) = x2 - 2x - 3 to solve x2 - 2x - 3 = 0 and to solve x2 - 2x - 3 < 5. y 5 4 3 2 1 -5 -4 -3 -2 1 2 4 5 x -3 -4 -5 f(x) = x2 - 2x - 3 The u starts in upper left corner below y above 5 goes down to -3 -4 up to after 2 up to above 5. in upper right corner. 86. Box construction. A rectangular piece of tin is twice as long as it is wide. Squares 2 cm on a side are cut out of each corner, and the ends are turned up to make a box whose volume is 480 cm3. What are the dimensions of the piece of tin? 2x 2cm 2cm 87. Highway fatalities The function given by n(t) = 0.015t2 - 0.9t + 28 can be used to estimate the number of deaths n(t) per 100,000 drivers, for drivers t years above age 20 (Source: based on in- formation in the Statistical Abstract of the United States, 2000). What age group of drivers has a fatal- ity rate of 14.5 deaths per 100,000 drivers? 88. Navigation A tugboat and a freighter leave the same port at the same time at right angles. The freighter travels 7 km/h slower than the tugboat. After 4 hr, they are 68 km apart. Find the speed of each boat. 89. Skydiving. During the first 13 sec of a jump, a sky- diver falls approximately 11.12t2 feet in t seconds. A small heavy object (with less wind resistance) falls about 15.4t2 feet in t seconds. Suppose that a skydiver jumps from 30,000 ft, and 1 sec later a camera falls out of the airplane. How long will it take the camera to catch up to the skydiver? page 329 90. Use the TABLE feature of a grapher to check that -5 and 3 are not in the domain of F as shown in Example 4. 91. Use the TABLE feature of a grapher to check your answers to Exercises 37, 39, and 41. In Exercises 92-96, use a grapher to find any real- number solutions that exist accurate to two decimal places. 92. x2 - 2x - 8 = 0 (Check by factoring.) 93. x2 + 3x - 4 = 0 (Check by factoring.) 94. -x2 + 13.80x = 47.61 95. -x2 + 3.63x + 34.34 = x2 96. x3 - 3.48x2 + x = 3.48 97. Mary Louise is attempting to solve x3 + 20x2 + 4x + 80 = 0 with a grapher. Unfortu- nately, when she graphs yl = x3 + 20x2 + 4x + 80 in a standard [-10,10, -10, 10] window, she sees no graph at all, let alone any x-intercept. Can this problem be solved graphically? If so, how? If not, why not? Summary and Review 5 Key Terms Important Properties and Farmulas Factaring Formulas To Factor ax2 + bx + c Using FO1L A2 + 2AB + B2 = (A + B)2; 1. Factor out the largest common factor, if A2 - 2AB + B2 = (A - B)2; one exists. Here we assume none does. A2 - B2 = (A + B) (A - B); 2• Find two First terms whose product is ax2: A3 + B3 = (A + B) (A2 - AB + B2); A3-B3=(A-B)(A2 + AB + B2) ( x+ )( x+ )= ax2 + bx + c. FoIL SUMMARY AND REVIEW: CHAPTER 5 329 page 330 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 3. Find two Last terms whose product is c: ( x+ `)( x+ )=ax2 + bx + c. FOIL To Factor a Polynomial 4. Repeat steps (2) and (3) until a combi-nation is found for which the sum of the Outer and Inner products is bx: (x+ )( x +)= ax2 + bx + c. Foil is bx. To Factor ax2 + bx + c Using Grouping 1. Make sure that any common factors have been factored out. 2. Multiply the leading coefficient a and the constant c. 3. Try to factor the product ac so that the sum of the factors is b. That is, find inte-gers p and q so that pq = ac and p + q = b. 4. Split the middle term. That is, write bx as px + qx. 5. Factor by grouping. A. Always factor out the largest common factor. B. Once any common factor has been fac-tored out, look at the number of terms. Two terms: Try factoring as a difference of squares first. Next, try factoring as a sum or a difference of cubes. Do not try to factor a sum of squares. Three terms: Try factoring as a perfect-square trinomial. Next, try trial and error, using the FOIL method or the grouping method. Four or more terms: Try factoring by grouping and factoring out a common binomial factor. Next, try grouping into a difference of squares, one of which is a trinomial. C. Always factor completely. If a factor with more than one term can itself be fac-tored further, do so. D. Check the factorization by multiplying. The Principle of Zero Products For any real numbers a and b: If ab = O,then a = 0 or b = 0. If a = O or b = O,then ab = 0. Review Exercises 1. Given the polynomial 2xy6 - 7x8y3 + 2x3 - 3, determine the degree of each term and the degree of the polynomial. 2. Given the polynomial 4x - 5x3 + 2x2 - 7, 3. Arrange in ascending powers of x: 3x6y - 7x8y3 + 2x3 - 3x2. 4. Find P(0) and P(-1): P(x) = x3 - x2 + 4x. 5. Evaluate the polynomial function for x = -2: arrange in descending order and determine the leading term and the leading coefficient. P(x) = 4 - 2x - x2. REVIEW EXERCISES: CHAPTER 5 page 331 Combine like terms. 2 3s after letters are small size numbers uppercase. 6. 6 - 4a + a2 - 2a3 - 10 + a 7. 4x2y - 3xy2 - 5x2y + xy2 Add 8. (-6x3 - 4x2 + 3x + 1) + (5x3 + 2x + 6x2 + 1) 9. (3x4 + 3x3 - 8x + 9) + (-6x4 + 4x + 7 + 3x) 10. (-9xy2 - xy + 6x2y) + (-5x2y - xy + 4xy2) Subtract 11. (3x - 5) - (-6x + 2) 12. (4a - b + 3c) - (6a - 7b - 4c) 13. (8x2 - 4xy + y2) - (2x2 + 3xy - 2y2) Multiply. 14. (3x2y) (-6xy3) 15. (x4 - 2x2 + 3) (x4 + x2 - 1) 16. (4ab + 3c) (2ab - c) 17. (2x + 5y) (2x - 5y) 18. (2x - 5y)2 19. (x + 3) (2x - 1) 20. (x2 + 4y3)2 21. (x - 5)(x2 + 5x + 25) 22. (x - 1)(x - 1) 3 5 Factor. 23. 6x2 + 5x 24. 9y4 - 3y2 25. 15x4 - 18x3 + 21x2 - 9x 26. a2 - 12a + 27 27. 3m2 + 14m + 8 28. 25x2 + 20x + 4 29. 4y2 - 16 30. 5x2 + x3 - 14x 31. ax + 2bx - ay - 2by 32. 3y3 + 6y2 - 5y - 10 33. a4 - 81 34. 4x4 + 4x2 + 20 35. 27x3 - 8 36. 0.064b3 - 0.125c3 37. y5 + y 38. 2z8 - 16z6 39. 54x6y - 2y 40. 36x2 - 120x + 100 41. 6t2 + 17pt + 5p2 42. x3 + 2x2 - 9x - 18 43. a3 - 2ab + b2 - 4t2 Sole 44. x2 - 20x = -100 45. 6b2 - 13b + 6 = 0 46. 8y2 = 14y 47. r2 = 16 48. a3 = 4a2 + 21 a 49. (y - 1) (y - 4 ) = 10 50. Let f(x) = x2 - 7x - 40. Find a such that f(a) = 4. 51. Find the domain of the function f given by f(x) = x - 3 3x2 + 19x - 14 52. The area of a square is 5 more than four times the length of a side. What is the length of a side of the square? 53. The sum of the squares of three consecutive odd numbers is 83. Find the numbers. 54. A photograph is 3 in. longer than it is wide. When a 2-in. border is placed around the photograph, the total area of the photograph and the border is 108 in2. Find the dimensions of the photograph. 55. Tim is designing a rectangular garden with a width of 8 ft. The path that leads diagonally across the garden is 2 ft longer than the length of the garden. How long is the path? SYNTHESIS 56. Explain how to find the roots of a polynomial func- tion from its graph. 57. Explain in your own words why there must be a 0 on one side of an equation before you can use the principle of zero products. Factor 58. 128x6 - 2y6 59. (x - 1)3 - (x + 1)3 Multiply 60. [a - (b - 1)][(b - 1)2 + a(b - 1) + a2] 61. (zn2)n3(z4n3)n2 62. Solve: (x + 1)3 = x2(x + 1). page 332 CHAPTER 5 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Chapter Test 5 Given the polynomial 3xy3 - 4x2y + 5x5y4 - 2x4y. 1. Determine the degree of the polynomial. 2. Arrange in descending powers of x. 3. Determine the leading term of the polynomial 8a - 2 + a2 - 4a3. 4. Given P(x) = 2x3 + 3x2 - x + 4, find P(0) and P(-2). 5. Given P(x) = x2 - 5x, find and simplify P(a + h) - P(a). 6. Combine like terms: 5xy - 2xy2 - 2xy + 5xy2. Add. 7. (-6x3 + 3x2 - 4y) + (3x3 - 2y - 7y2) 8. (5m3 - 4m2n - 6mn2 - 3n3) + (9mn2 - 4n3 + 2m3 + 6m2n) Subtract 9. (9a - 4b) - (3a + 4b) 10. (6y2 - 2y - 5y3) - (4y2 - 7y - 6y3) Multiply 11. (-4x2y) (-16xy2) 12. (6a - 5b)(2a + b) 13. (x - y) (x2 - xy - y2) 14. (2x3 + 5)2 15. (4y - 9)2 16. (x - 2y) (x + 2y) Factor. 17. 15x2 - 5x4 18. y3 + 5y2 - 4y - 20 19. p2 - 12p - 28 20. 12m2 + 20m + 3 21. 9y2 - 25 22. 3r3 - 3 23. 9x2 + 25 - 30x 24. x8 - y8 25. y2 + 8y + 16 - 100t2 26. 20a2 - 5b2 27. 24x2 - 46x + 10 28. 16a7b + 54ab7 29. 4y4x + 36yx2 + 8y2x3 - 16xy Solve. 30. x2 - 18 = 3x 31. 5y2 = 125 32. 2x3 + 21x = -17x2 33. 9x2 + 3x = 0 34. Let f(x) = 3x2 - 15x + 11. Find a such that f(a) = 11. 35. Find the domain of the function f given by 3 - x f(x) = x2 + 2x + 1 36. A photograph is 3 cm longer than it is wide. Its area is 40 cm2. Find its length and its width. 37. To celebrate a town's centennial, fireworks are launched over a lake off a dam 36 ft above the water. The height of a display, t seconds after it has been launched, is given by h(t) = -16t2 + 64t + 36 After how long will the shell from the fireworks reach the water? y h(t) = -16t2 + 64t + 36 36 t The line starts at 36 goes up halfway up to y down to t or line. SYNTHESIS 38. a) Multiply: (x2 + x + 1) (x3 - x2 + 1). b) Factor x5 + x + 1. 39. Factor: 6x2n - 7xn - 20. 6.1 RATIONAL EXPRESSIONS AND FUNCTIONS: MULTIPLYING AND DIVIDING page 339 Solution a) x + 2 . x2 - 4 = (x + 2) (x2 - 4) Multiplying the numerators x - 3 x2 + x - 2 = (x - 3) (x2 + x - 2) and also the denominators = (x + 2) (x - 2) (x + 2) Factoring the numerator (x - 3) (x + 2) (x - 1) and the denominator and finding common factors (x + 2) x + 2 a line crosses out x + 2 (x - 2) (x - 3) x + 2 a line crosses out x + 2 (x - 1) 1: x + 2 = 1 x + 2 = x + 2 (x - 2) x - 3 (x - 1) b) 1 - a3 a5 (1 + a + a2)as Factoring a difference of cubes and a a2 a2 - 1 (a + 1) difference of squares = -1(a - 1) (1 + a + a2)a5 Important! Factoring out -1 reverses a2(a - 1) (a + 1) the subtraction. (a - 1)a2 a - 1 a line crosses out. * a3(-1)(1 + a + a2) Rewriting a5 as a2 * a3; removing a (a - 1)a2 a line crosses out a - 1. (a + 1) factor equal to 1: (a - 1)a2 = 1 (a - 1)a2 = -a3(1 + a + a2) Simplifying. a + 1 As in Example 5, there is no need for us to multiply out the numerator or the denominator of the final result. Dividing and Simplifying Two expressions are reciprocals of each other if their product is 1. As in arith-metic, to find the reciprocal of a rational expression, we interchange numera-tor and denominator. x x2 + 3 The reciprocal of x2 + 3 is x 1 The reciprocal of y - 8 is Y - 8. Quotients of Rational Expressions For any rational expressions A/B and C/D, with B, C, D =/ 0, A divide C = A * D B D B * C (To divide two rational expressions, multiply by the reciprocal of the divisor. We often say that we "invert and multiply.") (x + 2) fx-~-Z (x - 2) Removing a factor equal (x-3)(x-1) to 1:x+2=1 (x + 2) (x - 2) = Simplifying (x - 3) (x - 1) page 340 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Example 7 Divide. Simplify by removing a factor equal to 1 if possible. a) x - 2 divide x + 5 b) a2 - 1 divide a2 - 2a + 1 x + l x - 3 a - 1, a + l StudyTip The procedures covered in this chapter are by their nature rather long. As is often the case in mathematics, it may help to write out each step as you do the problems. If you have difficulty, consider starting over with a new sheet of paper. Don't squeeze your work into a small amount of space. When using lined paper, consider using two spaces at a time, writing the fraction bar on a line of the paper. Solution a) x - 2 divide x + 5 = x - 2 divide x - 3 Multiplying by the reciprocal of x + 1 divide x - 3 = x + 1 divide x + 5 the divisor = (x - 2) (x - 3) Multiplying the numerators and (x + 1) (x + 5) the denominators b) a2 - 1 divide a2 - 2a + 1 = a2 - 1 divide a + 1 Multiplying by the re-ciprocal of the divisor a - 1 a + 1 a - 1 a2 - 2a + 1 (a + 1) (a + 1) (a - 1) (a - 1) Simplifying Exercise Set 6.1 FOR EXTRA HELP Digital Video Tutor CD 4 InterACt Math Math Tutor Center ^RathXl MyMathLab.com Videotape 11 Photo developing. Rik usually takes 3 hr more than 1. How long will it take them, working together, to Pearl does to process a day's orders at Liberty Place complete a day's orders if Pearl can process the Photo. If Pearl takes t hr to process a day's orders, the orders alone in 5 hr? function given by 2. How long will it take them, working together, to H(t) = t2 + 3t complete a day's orders if Pearl can process the 2t + 3 ' orders alone in 7 hr? can be used to determine how long it would take if they worked together. - (a2 - 1) (a + 1) Multiplying the (a - 1) (a2 - 2a + 1) numerators and the denominators (a + 1) (a - 1) (a + 1) Factoring the (a - 1) (a - 1) (a - 1) numerator and the denominator - (a + 1) f,a---n (a + 1) Removing a factor (a - 1) (n---I~(a - 1) equal to 1: a - 1 = 1 a-1 For each rational function, find the function values , xz + 9x + 8 tz - 8t - 9 33. x2 _ 3x - 4 34~ tz + 5t + 4 indicated, provided the value exists. page 341 3. v(t) = 4t2 - 5t + 2, v(0), v(-2), v(7) t + 3 4. f(x) = 5x2 + 4x - 12; f(0),f(-1)f(3) 6 - x 5. 2x3 - 9 g(x) = x2 - 4x + 4 g(0) g(2) g(-1) 6. r(t) = t2 - 5t + 4; r(1) r(2) r(-3) t2 - 9 Multiply to obtain equivalent expressions. Do not sim-plify. Assume that all denominators are nonzero. 7. 4x * x - 3 4x x + 2 8. 3 - a2 * -1 a - 7 -1 9. t - 2 * -1 t + 3 -1 10. x - 4 * x - 5 x + 5 x - 5 Simplify by removing a factor equal to 1. 11. 15x 5x2 12. 7a3 21a 13. 18t3 27t7 14. 8y5 4y9 15. 2a - 10 2 16. 3a + 12 3 17. 15 25a - 30 18. 21 6x - 9 19. 3x - 12 3x + 15 20. 4y - 20 4y + 12 21. 5x + 20 x2 + 4x 22. 3x + 21 x2 + 7x 23. 3a - 1 2 - 6a 24. 6 - 5a 10a - 12 25. 8t - 16 t2 - 4 26. t2 - 9 5t + 15 27. 2t - 1 1 - 4t2 28. 3a - 2 4 - 9a2 29. 12 - 6x 5x - 10 30. 21 - 7x 3x - 9 31. a2 - 25 a2 + 10a + 25 32. a2 - 16 a2 - 8a + 16 6.1 RATIONAL EXPRESSIONSAND FUNCTIONS: MULTIPLYING AND DIVIDING page 341 33. x2 + 9x + 8 x2 - 3x - 4 34. t2 - 8t - 9 t2 + 5t + 4 35. 16 - t2 t2 - 8t + 16 36. 25 - p2 p2 + 10p + 25 Multiply and, if possible, simplify. 37. 5a3 * 7b3 3b 10a7 38. 25a * 3b5 9b8 5a2 39. 8x - 16 * x3 5x 5x - 10 40. 5t3 * 6t - 12 4t - 8 * 10t 41. y2 - 16 * y + 3 4y + 12 y - 4 42. m2 - n2 * m + n 4m + 4n m - n 43. x2 - 16 * x2 - 4x x2 x2 - x - 12 44. y2 + 10y + 25 * y2 + 3y y2 - 9 y + 5 45. 7a - 14 * 5a2 + 6a + 1 4 - a2 35a + 7 46. a2 - 1 * 15a - 6 2 - 5a a2 + 5a - 6 47. t3 - 4t * t4 - t t - t4 4t - t3 48. x2 - 6x + 9 * x6 - 9x4 12 - 4x x3 - 3x2 49. x2 - 2x - 35 * 4x3 - 9x 2x3 - 3x2 7x - 49 50. y2 - 10y + 9 * 1 - y2 y2 - 1 y2 - 5y -36 51. c3 + 8 * c6 - 4c5 + 4c4 c5 - 4c3 * c2 - 2c + 4 52. x3 - 27 * x5 - 6x4 + 9x3 x4 - 9x2 x2 + 3x + 9 53. a3 - b3 * a2 + 2ab + b2 3a2 + 9ab + 6b2 * a2 - b2 54. x3 + y3 * x2 - y2 x2 + 2xy - 3y2 * 3x2 + 6xy + 3y2 55. 4x2 - 9y2 * 4x2 + 6xy + 9y2 8x3 - 27y3 4x2 + 12xy + 9y2 page 342 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 56. 3x2 - 3y2 * 6x2 + 5xy - 6y2 27x3 - 8y3 * 6x2 + 12xy + 6y2 Divide and if possible simplify. 57. 9x5 divide 3x 8y2 16y9 58. 16a7 divide 8a3 3b5 6b 59. 5x + 10 divide x + 2 x8 x3 60. 3y + 15 - y + 5 y7 y2 61. x2 - 4 divide x5 - 2x4 x3 x + 4 62. y2 - 9 divide y5 + 3y4 y2 y + 2 63. 25x2 - 4 divide 2 - 5x x2 - 9 x + 3 64. 4a2 - 1 divide 2a - 1 a2 - 4 2 - a 65. 5y - 5x divide x2 - y2 15y3 3x + 3y 66. x2 - y2 divide 3y - 3x 4x + 4y 12x2 67. x2 - l6 divide 3x - 12 x2 - 10x + 25 x2 - 3x - 10 68. y2 - 36 divide 3y - 18 y2 - 8y + 16 ~ y2 - y - 12 69. y3+3y divide y2+5y-14 y2-9 divide y2+4y-21 70. a3+4a divide a2+8a+15 a2 - 16 a2 +a-20 71. x3 - 64 divide x2 - 16 x3 + 64 x2 - 4x + 16 72. 8y3 - 27 divide 4y2 - 9 64y3 - 1 16y2 + 4y + 1 73. 8a3 + b3 divide 8a2 - 4ab + 2b2 2a2 + 3ab + b2 4a2 + 4ab + b2 74. x3 + 8y3 divide x3 - 2x2y + 4xy2 2x2 + 5xy + 2y2 8x2 - 2y2 75. Is it possible to understand how to simplify ra-tional expressions without first understanding how to multiply rational expressions? Why or why not? 76. Nancy incorrectly simplifies x + 2 as x x + 2 = x is crossed out + 2 = 1 + 2 = 3. x x She insists this is correct because it checks when x is replaced with 1. Explain her misconception. SKILL MAINTENANCE Simplify 77. 3 - 8 10 15 78. 3 - 7 8 10 79. 2 * 5 - 5 * 1 3 7 7 6 80. 4 * 1 - 3 * 2 7 5 10 7 81. (8x3-5x2 + 6x + 2)-(4x3 + 2x2 - 3x + 7) 82. (6t4 + 9t3 - t2 +4t)-(8t4 - 2t3 - 6t + 3) SYNTHESIS 83. Tony incorrectly argues that since a2 - 4 = a2 + -4 = a + 2 a - 2 a -2 is correct it follows that x2 + 9 = x2 + 9 = x + 9. x + 1 x 1 Explain his misconception. 84. Explain why the graphs of f(x) = 5x and 5x2 g(x) = x differ. 85. Let 2x + 3 g(x) = 4x - 1 Determine each of the following. a) g(x + h) b) g(2x - 2) * g(x) c) g(1 x + 1) * g(x) 2 86. Graph the function given by f(x) = x2 - 9 x - 3. (Hint: Determine the domain of f and simplify.) 6.2 RATIONAL EXPRESSIONS AND FUNCTIONS: ADDING AND SUBTRACTING page 343 Perform the indicated operations and simplify. 87. r2 - 4s2 2s r + 2s divide (r + 2s) * r - 2s 88. d2 - d * d - 2 divide 5d d2 - 6d + 8 d2 + 5d d2 - 9d + 20 89. 6t2 - 26t + 30 * 5t2 - 9t - 15 divide 5t2 - 9t -15 8t2 - 15t - 21 * 6t2 - 14t - 20 divide 6t2 - 14t - 20 Simplify. 90. x(x + 1) - 2(x + 3) (x + 3) (x + 1) (x + 2) 91. m2 - t2 m2 + t2 + m + t + 2mt 92. a3 - 2a2 + 2a - 4 a3 - 2a2 - 3a + 6 93. x3 + x2 - y3 - y2 x2 - 2xy + y2 94. u6 + u6 + 2u3u3 u3 - u3 + u2v - uv2 95. x5 - x3 + x2 - 1 - (x3 - 1)(x + 1)2 (x2 - 1)2 96. Let f(x) = 4 4x2 + 8x + 4 x2 - 1 and g(x) = x3 - 1 Find each of the following. a) (f*g)(x) b) (f/g)(x) c) (g/f)(x) 97. Use a grapher to check Example 6. Use the method described on p. 338. 98. Use a grapher to check your answers to Exercises 27, 45, and 71. Use the method described on p. 338. 99. Use a grapher to show that x2 - 16 =/ x - 8 x + 2 100. To check Example 4, Kara lets = 7x2 + 21x x + 3 yl 14x and y2 = 2 Since the graphs of y1 and y2 appear to be identi-cal, Kara believes that the domains of the func-tions described by yl and y2 are the same. How could you convince Kara otherwise? Rational Expressions 6•2 and Functions: Adding when Uenominators Are the same • when denominators and Subtracting Are ~ifferent Rational expressions are added in much the same way as the fractions of arithmetic. When Denominators Are the Same Addition and Subtraction with Like Denominators To add or subtract when denominators are the same, add or sub-tract the numerators and keep the same denominator. A B = A + B A B = A - B C + C C and C - C = C , where C =/ 0. page 344 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Example 1 Solution Add 3 + x + 4 x x 3 + x + 4 = 3 + x + 4 = x + 7 x x x x Because x is not a factor of the numerator and the denominator, the result cannot be simplified. Z _ Y xY - YZ Example 2 Add: 4x2 - 5xy + 2xy - y2 x2 - y2 + x2 - y2 Solution 4x2 - 5xy + 2xy - y2 = 4x2 - 3xy - y2 Adding the numerators and x2 - y2 + x2 - y2 = x2 - y2 combining like terms. The denominator is unchanged. =(x - y) (4x + y) Factoring the numerator (x - y) (x + y) and the denominator and looking for common factors =(x - y is crossed out) (4x + y) (x - y is crossed out) (x + y) Removing a factor equal to 1 x - y = 1 x - y Simplifying Recall that a fraction bar is a grouping symbol. The next example shows that when a numerator is subtracted, care must be taken to subtract, or change the sign of, each term in that polynomial. Example 3 M technology connection y)= 4x + 5 _ x - 2 x + 3 x + 3 and Y2 = 3x + 7x x + 3 on the same set of axes. Since the equations are equivalent, one curve (it has two branches) should appear. Equivalently, you can show that y3 = y2 - yl is 0 for all x not equal to -3. The TABLE or TRACE feature can assist in either type of check. If f(x) = 4x + 5 _ x - 2 x + 3 x + 3 find a simplified form of f (x). Solution f(x) = 4x + 5 - (x - 2) = x + 3 x + 3 f(x) = 4x + 5 _ x - 2 x + 3 _ x + 3 The parentheses remind us to subtract both terms. = 4x + 5 - (x - 2) x + 3 = 4x + 5 - x + 2 x + 3 = 3x + 7 x + 3 Example 8 311 - Example 9 Subtract: 5x - 7 x _ 2y 2y-x. 6.2 RATIONAL EXPRESSIONS AND FUNCTIONS: ADDING AND SUBTRACTING page 347 add subtract 2y + 1 y + 3 y2 - 7y +6 y2 - 5y - 6 Solution 2y + 1 - y + 3 The LCD is = (Y - 6) (Y - 1) (Y - 6) (Y + 1) (y - 6) (y - 1) (y + 1). = 2y + 1 * y + l - y + 3 * y - 1 (y - 6)(y - 1)y + l (y - 6)(y + l)y - 1 Multiplying by 1 to get the LCD in each expression - =(2y + 1) (y + l) - (y + 3) (y - 1) (y - 6)(y - 1)(y + 1) = 2y2 + 3y + 1 - (y2 + 2y - 3) The parentheses are important. (y - 6)(y - 1)(y + 1) = 2y2 + 3y + 1 - y2 - 2y + 3 (y - 6)(y - 1)(y + 1) = y2 + y + 4 We leave the denominator (y - 6) (y - 1) (y + 1) in factored form. Example 8 Add: 3 1 8a + -8a. Solution 3 + 1 = 3 + -1 . 1 When denominators are opposites, we 8a -8a 8a -1 -8a multiply one rational expression by -1/-1 to get the LCD. = 3 + -1 = 2 8a 8a 8a = 2 * 1 = 1 Simplifying by removing a factor equal 2 * 4a 4a to 1: 2 =1 2 Example 9 Subtract 5x - 3y - 7 x - 2y - 2y - x Solution 5x _ 3y - 7 = 5x _ -1 * 3y - 7 Note that x - 2y and x - 2y _ 2y - x = x - 2y _ -1 * 2y - x 2y - x are opposites. = 5x _ 7 - 3y Performing the multiplication. x - 2y x - 2y Note: -1(2y- x) _ -2y + x = 5x -(7 - 3y) = x - 2y. x - 2y Subtracting. The parentheses = 5x - 7 + 3y are important. x - 2y page 348 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS In Example 9, you may have noticed that when 3y - 7 is multiplied by -1 and subtracted, the result is -7 + 3y, which is equivalent to the original 3y - 7. instead of multiplying the numerator by -1 and then subtracting, we could have simply added 3y - 7 to 5x, as in the following: 5x _ 3y - 7 = 5x + (-1). 3y - 7 Rewriting subtraction x - 2y _ 2y - x = x - 2y 2y - x as addition = 5x + 1 . 3y - 7 Writing -1 1 as x - 2y -1 2y - x -1 = 5x + 3y - 7 The opposite of 2y - x x - 2y + x - 2y is x - 2y. = 5x + 3y - 7 This checks with the x - 2y ' answer to Example 9. Example 1 0 Perform the indicated operations and simplify: 2x + 5 _ 1 x2 - 4 + 2 - x _ 2 + x Solution We have Studytip Because there are many steps to these problems, we recommend that you check each step as you go. 2x + 5 - 1 = 2x 5 1 x2 - 4 + 2 - x _ 2 + x = (x - 2)(x + 2) 2 - x - 2 + x Factoring = 2x + -1 . 5 _1 Multiplying by -1 (x - 2)(x + 2) + -1 (2 - x) x + 2 -1 since 2 - x is the opposite of x - 2 = 2x + -5 _ 1 The LCD is (x -2 )(x + 2) + x - 2 x + 2 (x - 2)(x + 2). = 2x + -5 , x + 2 - 1 , x - 2 Multiplying (x - 2)(x + 2) + x - 2 x + 2 x + 2 x - 2 by ltoget = 2x - 5(x+2) - (x-2) = 2x - 5x - 10 - x + 2 the LCD (x - 2)(x + 2) = (x - 2)(x + 2) = -4x - 8 = -4(x + 2) (x - 2)(x + 2) (x - 2)(x + 2) = -4(x + 2is crossed out) Removing a factor equal to 1: x + 2 = 1 (x - 2)(x + 2is crossed out) x + 2 = -4 or 4 x - 2 x - 2 4 4 4 Another correct answer is 2 - x. It is found by writing -x - 2 as -(x -2) and then using the distributive law to remove parentheses. and then, for x ~ -2 and x ~ 2, we have f = g. Note that whereas the domain of f includes all real numbers except -2 or 2, the domain of g excludes only 2. This is illustrated in the graphs below. Methods for drawing such graphs by hand are discussed in more advanced courses. The graphs are for visualization only. A computer-generated visualization of Example 10 Whenever rational expressions are simplified, a quick partial check is to evaluate both the original and the simplified expressions for a convenient choice of x. For instance, to check Example 10, if x = 1, we have 6.2 RATIONALEXPRESSIONSAND FUNCTION$: ADDINGAND SUBTRACTING page 349 Our work in Example 10 indicates that if 2x 5 1 f(x)= x2 - 4 + 2 - x 2 + x and -4 g(x) = x - 2 y 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 f(x) = 2x + 5 - 1 x2 - 4 2 - x 2 + 1 The line in graph 1 starts at above -6 0.5 goes to -2, 1 in upper left right corners. Lower right corner starts ar -6, 3 goes to 6, -1. 2 3 through 5 6 graph 2 y 1 5 1 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 g(x)= -4 x - 2 in graph 2 line starts at -6 0.5 goes to -1,1 2 6. in upper left right corners. Line starts at -6, 2.5 after 1 goes to 6 Since both functions include the pair (1, 4), our algebra was probably correct. Although this is only a partial check (on rare occasions, an incorrect answer might "check"), because it is so easy to perform, it is nonetheless very useful. Further evaluation provides a more definitive check. if x = 1 we have f(1) = 2 * 1 + 5 - 1 1 2- 4 2 - 1 2 + 1 = 2 + 5 - 1 = 5 - 3 = 4 -3 1 3 3 and g(1) = -4 = -4 = 4 1 - 2 -1 page 350 CHAPTER 6 ~: : RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Exercise Set 6,2 FOR EXTRA HELP Digital Video Tutor CD 4 InterAd Math Math Tutor Center MathXL MyMathLab-~om Videotape 11 Perform the indicated operations. Simplify when 31. 4 + x + 2 + 3 possible. 1. 5 + 7 3a 3a 2. 3 + 5 2y 2y 3. 1 _ 5 4a2b 4a2b 4. 5 _ 4 3m2n2 - 3m2n2 5. a - 5b + a + 7b a + b + a + b 6. x - 3y x + 5y x + y + x + y 7. 4y + 2 y - 3 y - 2 - y - 2 8. 3t + 2 t - 2 t - 4 -t - 4 9. 3x - 4 + 3 - 2x x2 - 5x + 4 + x2 -5x + 4 10. 5x - 4 + 5 - 4x x2 - 6x - 7 + x2 -6x - 7 11. 3a - 2 _ 4a - 7 a2 - 25 a2 - 25 12. 2a - 5 _ 3a - 8 a2 - 9 a2 - 9 13. a2 + b2 a - b b - a 14. s2 + r2 r - s s - r 15. 7 _ 8 x -x 16. 2 _ 5 a -a 17. x - 7 _ x - 1 x2 - 16 16 - x2 18. y - 4 _ 9 - 2Y y2 - 25 25 - y2 19. t2 + 3 + 7 t4 + 16+ 16 - t4 20. y2 - 5 + 4 y4 - 81 + 81 - y4 21. m - 3n _ 2n m3 - n3 n3 - m3 22. r - 6s _ 5s r3 - s3 s3 - r3 23. a + 2 + a - 2 a - 4 + a + 3 24. a + 3 + a - 2 a - 5 a + 4 25. 4 + x - 3 x + i 26. 3 + y + 2 y - 5 27. 4xy + x - y x2 - y2 x + y 28. 5ab + a + b a2 - b2 a - b 29. 8 + 3x + 2 2x2 - 7x + 5 2x2 - x - 10 30. 7 9y + 2 3y2 + y - 4 + 3y2 - 2y - 8 31. 4 + x + 2 + 3 x + 1 x2 - 1 x - 1 32. -2 + 5 y + 3 y + 2 y - 2 + y2 - 4 33. x + 6 _ x - 2 5x + 10 4x + 8 34. a + 3 _ a - 1 5a + 25 3a + 15 35. 5ab _ a - b a2 - b2 a + b 36. 6xy _ x + y x2 - y2 x - y 37. x _ 4 x2 + 9x + 20 x2 + 7x + 12 38. x _ 5 x2 + 11x + 30 x2 + 9x + 20 39. 3y _ 2y y2 - 7y + 10 y2 - 8y + 15 40. 5x _ 3x x2 - 6x + 8 x2 - x - 12 41. 2x + 1 + 5x2 - 5xy x - y x2 - 2xy + y2 42. 2 - 3a 3a2 + 3ab a - b + a2 - b2 43. 3y + 2 + 7 y2 + 5y - 24 + y2 + 4y - 32 44. 3x + 2 + 2x x2 - 7x + 10 + x2 - 8x + 15 45. a - 3 _ 3a - 2 a2 - 16 a2 + 2a - 24 46. t + 4 _ 3t - 1 t2 - 9 t2 + 2t - 3 47. 2 + -2 a2 - 5a + 4 + a2 - 4 page 351 48. 3 -3 a2 - 7a + 6 + a2 - 9 49. 5 + t 8 t + 2 - t2 - 4 50. 2 + t _ 18 t - 3 t2 - 9 51. 2y -6 - y + y2 + 2 y2 - 9 y - 1 y2 + 2y - 3 52. x - 1 _ x + x2 + 2 x2 - 1 x - 2 + x2 - x - 2 53. 5y _ 2y + 5y 1 - 4y2 2y + 1 4y2 - 1 54. 4x + 3x _ 4 x2 - 1 1 - x x - 1 55. 2 _ 4 + 2 x2 - 5x + 6 x2 - 2x - 3 x2 + 4x + 3 56. 1 _ 2 _ 1 t2 + 5t + 6 t2 + 3t + 2 t2 + 5t + 6 57. Janine found that the sum of two rational expres-sions was (3 - x)/(x - 5). The answer given at the back of the book is (x - 3)/(5 - x). Is Janine's an swer incorrect? Why or why not? 58. When two rational expressions are added or sub-tracted, should the numerator of the result be fac-tored? Why or why not? SKILL MAINTENANCE Simplify. Use only positive exponents in your answer. 59. 15x-7y12z4 35x-2y6z-3 60. 21a-4b6c8 27a-2b-5c 61. 34s9t-40r30 lOS-3t20r-10 62. Find an equation for the line that passes through the point (-2, 3) and is perpendicular to the line f(x)=-4 5x+7. 63. Value of coins. There are 50 dimes in a roll of dimes, 40 nickels in a roll of nickels, and 40 quar-ters in a roll of quarters. Robert has a total of 12 rolls of coins with a total value of $70.00. If he has 3 more rolls of nickels than dimes, how many of each roll of coins does he have? 6.2 RATIONAL EXPRESSIONSAND EUNCTIONS: ADDINGAND SUBTRACfING 351 64. Audiotapes. Anna wants to buy tapes for her work at the campus radio station. She needs some 30-min tapes and some 60-min tapes. If she buys 12 tapes with a total recording time of 10 hr, how many tapes of each length did she buy? SYNTHESIS 65. Many students make the mistake of always multi-plying denominators when looking for a common denominator. Use Example 7 to explain why this approach can yield results that are more difficult to simplify. 66. Is the sum of two rational expressions always a rational expression? Why or why not? 67. Prescription drugs. After visiting her doctor, Corinna went to the pharmacy for a two-week sup- ply of Zyrtec®, a 20-day supply of Albuterol®, and a 30-day supply of Pepcid. Corinna refills each pre- scription as soon as her supply runs out. How long will it be until she can refill all three prescriptions on the same day? 68. Astronomy The earth, Jupiter, Saturn, and Uranus all revolve around the sun. The earth takes 1 yr, Jupiter 12 yr, Saturn 30 yr, and Uranus 84 yr. How frequently do these four planets line up with each other? 69. Music. To duplicate a common African polyrhythm, a drummer needs to play sextuplets (6 beats per measure) on a t-t while simulta neously playing quarter notes (4 beats per meas-ure) on a bass drum. Into how many equally sized parts must a measure be divided, in order to pre-cisely execute this rhythm? 352 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNC710N5 70. Home appliances. Refrigerators last an average of 20 yr, clothes washers about 14 yr, and dishwashers about 10 yr (Source: U.S. Department of Energy). In 1980, Westgate College bought new refrigerators for its dormitories. In 1990, the college bought new clothes washers and dishwashers. Predict the year in which the college will need to replace all three types of appliances at once. Find the LCM. 71. x8 - x4, x5 - x2, x5 - x3, x5 + x2 72. 2a3 + 2a2b + 2ab2, a6 - b6, 2b2 + ab - 3a2, 2a2b + 4ab2 + 2b3 73. The LCM of two expressions is 8a4b7. One of the expressions is 2a3b7. List all the possibilities for the other expression. 74. Determine the domain and the range of the func- tion graphed below. y 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 The line passes from 2 -6 goes upto 3, -2 4 5 7 upper left corner. Line begins -5 1 passes through 2 3 lower right corner. page 352 If x3 x2 f(x) x2 - 4 and g(x) = x2 + 3x - 10. find each of the following. 75. ( f + g) (x) 76. ( f - g) (x) 77. ( f . g) (x) 78. ( f/g) (x) Perform the indicated operations and simplify. 79. 5(x - 3)-1 + 4(x + 3)-1 - 2(x + 3)-2 80. 4(y - 1)(2y - 5)-1 + 5(2y + 3)(5 - 2y)-1 + (y - 4)(2y - 5)-1 81. x + 4 * x 2 6x2 - 20x * (x2 - x - 20 + x + 4) 82. x2 - 7x + 12 . 3x + 2 + 7 x2 - x - 29/3 x2 +5x - 24 + x2 + 4x - 32 83. 8t5 2t _ 3t 2t2 - lOt + 12 divide (t2 - 8t + 15 t2 - 7t + 10 84. 9t3 t + 4 _ 3t - 1 3t3 - 12t2 + 9t divide (t2 - 9 t2 + 2t - 3) 85. Use a grapher to check your answers to Exercises 9, 25, and 40. Use the method discussed on p. 344. 86. Let f(x) = 2 + x - 3 x + 1 Use algebra, together with a grapher, to determine the domain and the range of f. Complex Rational 6•3 Expressions Multiplying by 1 • Dividing Two Rational Expressions A complex rational expression is a rational expression that contains rational expressions within its numerator and/or its denominator. Here are some ex-amples: x - y 7x _ 4 r + r2 x + y 3 x 6 + 144 2x - y 5x + 8 r 3x + y 6 + 3 12 The rational expres-sions within each complex rational expression are red. 6.3 COMPLEX RATIONAL EXPRESSIONS page 353 Complex rational expressions arise in a variety of real-world applications. For example, the complex rational expression on the far right at the bottom of p. 352 is used when calculating the size of certain loan payments. Two methods are used to simplify complex rational expressions. Method 1: Multiplying by 1 One method of simplifying a complex rational expression is to multiply the en-tire expression by 1. To write 1, we use the LCD of the rational expressions within the complex rational expression. Example 1 t Simplify: 1 1 a3b + b 1 1 a2b2 - b2 Solution The denominators within the complex rational expression are a3b, b, a2b2, and b2. The LCD is a3b2. We multiply by 1, using (a3b2)/(a3b2): 1 1 1 1 a3b + b = a3b + b a3b2 1 _ 1 1 1 a2b2 b2 = a2b2 b2 a3b2 (1 + 1 /a3b2 Multiplying the numerator and a3b b the denominator. Remember to use parentheses. 1 1 (a2b2 - b2)a3b2 = 1 1 a3b * a3b2 + b * a3b2 Using the distributive law to carry out the multiplications 1 1 a2b2 . a3b2 _ b2 . a3b2 a3b is crossed out . b + b . a3b =a3b is crossed out b a2b2 is crossed out . a _ b2 . a3 a2b2 is crossed out b2 = b + a3b a - a3 = b(1 + a3) a(1 - a2) = b(1 + a)is crossed ou(1 - a + a2) a(1 + a)is crossed out)(1 - a) = b(1 - a + a2) a(1 - a) 15 Removing factors that equal 1. Multiplying by 1, using the LCD Study this carefully. Y 3 awa-T?.a b+a3b a - a3 Simplifying b(1 + a3) a(1 - a2) Factoring b~~a (1 - a + a z) Factoring further and identifying aLl-l--al (1 - a) a factor that equals 1 _b(1-a+az) a( I - a) ' Simplifying page 354 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Using Multiplication by 1 to Simplify a Complex Rational Expression 1. Find the LCD of all rational expressions within the complex rational expression. 2. Multiply the complex rational expression by 1, writing 1 as the LCD divided by itself. 3. Distribute and simplify so that the numerator and the denomi-nator of the complex rational expression are polynomials. 4. Factor and, if possible, simplify. Note that use of the LCD when multiplying by a form of 1 clears the nu-merator and the denominator of the complex rational expression of all rational expressions. Example 2 Simplify: + x-1 xZ-1 Solution In this case, to find the LCD, we have to factor first: 3 1 3 1 2x-2 x + 1 _ 2(x - 1) x + 1 The LCD is 1 x 1 x 2(x - 1) (x + 1). x-l+xZ-1 x-1+(x-1)(x+l) 31--_ 1 k~'~ 2(x-1)(x+i) _ 2(x - 1) x + 1 Multiplying by 1, + x . 2(x - 1) (x + 1) using the LCD x-1~(x-1)(x+l)~~ 2(x31). 2(x-I)(x+l)-x+l x11. 2(x-1)(x+l)+(x-1)(x+1)- 2(x-I)(x+l)-< Using the distributive law ~y 3(x + 1) - 2(x - 1) ~~2(x+l)+~ ~ ~2x _ 3(x + 1) - 2(x - 1) Simplifying 2(x + 1) + 2x Removing factors that equal 1 6.3 COMPLEX RATIONAL EXPRESSIONS page 357 If negative exponents occur, we first find an equivalent expression using positive exponents and then proceed as in the preceding examples. Rewriting with positive exponents. We continue, using method 2. _1 _b _1 _a Finding a common _ a ~ b + b ~ a denominator 1 b3 1 a3 Finding a common a3~b3+b3~a3 denominator _b _a ab + ab - bs as a3b3 + a3b3 b+a ab Adding in the numerator b3 + a3 a3b3 Adding in the denominator _ b + a , a3b3 , Multiplying by the reciprocal ab b3 + a3 of the divisor _ (b + a) ~ ab ~ aZbZ Factoring and looking for ab(b + a) (b2 - ab + a2) common factors _ ~b~~ ~ l~ ~ aZb2 Removing a factor equal to 1: y~~h-I~j (bz - ab + a2) (b + a)ab - 1 a2b2 (b + a)ab b2-ab+a2 There is no one method that is best to use. For expressions like 3x+1 _3 __2 x-5 x x 2-x ~r 1 5 + x+3 x+l x+l the second method is probably easier to use since it is little or no work to write the expression as a quotient of two rational expressions. page 358 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS On the other hand, expressions like 3 _ 4 5 2 a2b bc3 or a2 - b2 + a2 + 2ab + b2 1 + 2 1 + 4 b3c ac4 a - b + a + b require fewer steps if we use the first method. Either method can be used with any complex rational expression. FOR EXTRA HELP Exercise Set 6.3 Simplify. If possible, use a second method or evaluation asacheci. 1. 7 + 1 ____a 1 _ 3 a 2. 1 + 2 y____ 1 _ 3 y 3. x - x-1 x + x-1 6 + 7 x y 4. y + y-1 y - y-1 5 + 2 z y 5. x y 7 _ 6 x y x2 - y2 xy 6. z y 4 _ 1 z y a2 - b2 7. xy x-y y 8. ab a - b b 9. y 2y - y x 10. 1 _ 2 3x _ 4 x 9x 11. a-1 + b-1 a2 - b2 ab 1 1 12. x-1 + y-1 x2 - y2 xy 1 1 13. x + h x h 14. a - h a h x2 - x - 12 15. x2 - 2x - 15 x2 + 8x + 12 x2 - 5x - 14 16. a2 - 4 a2 + 3a + 2 a2 - 5a - 6 a2 - 6a - 7 17. 1 + 3 x - 2 x - 1 2 5 x - 1 + x - 2 18. 2 1 y - 3 + y + 1 3 4 y + 1 + y - 3 19. a(a + 3)-1 - 2(a - 1)-1 a(a + 3)-1 - (a - 1)-1 20. a(a + 2)-1 - 3(a - 3)-1 a(a + 2)-1 - (a - 3)-1 21. x _ 1 x2 + 3x - 4 x2 + 3x - 4 x 3 x2 + 6x + 8 + x2 + 6x + 8 x 6 22. x2 + 5x - 6 + x2 + 5x - 6 x _ 2 x2 - 5x + 4 x2 - 5x + 4 23. 2 1 a2 - 1 + a + 1 3 2 a2 - 1 + a - 1 24. 3 2 a2 - 9 + a + 3 4 1 a2 - 9 + a + 3 25. 5 _ 3 x2 - 4 x - 2 4 _ 2 x2 - 4 x + 2 26. 4 _ 3 x2 - 1 x + 1 5 _ 2 x2 - 1 x - 1 27. y + 5 y2 - 4 4 - y2 y2 25 y2 - 4 + 4 - y2 28. y 3 y2 - 1 + 1 - y2 y2 9 y2 - 1 + 1 - y2 Digital Video Tutor CD 5 Inter~ct Math Math Tutor Center ^.1a1hXt MyMathlab.com Videotape 11 page 359 29. y2 _ y y2 - 9 y + 3 y 1 y2 - 9 y - 3 30. y2 _ y y2 - 25 y - 5 y 1 y2 - 25 y + 5 31. a + 4 a + 3 5a a 3 2a + 6 + a 32. a 5 a + 2 + a a + 1 2a + 4 + 3a 33. 1 + 1 x2 - 3x + 2 + x2 - 4 1 1 x2 + 4x + 4 + x2 - 4 34. 1 + 1 x2 + 3x + 2 + x2 - 1 1 1 x2 - 1 + x2 - 4x + 3 35. 3 + 3 a2 - 4a + 3 + a2 - 5a + 6 3 + 3 a2 - 3a + 2 + a2 + 3a - 10 36. 1 _ 2 a2 + 7a + 10 a2 - 7a + 12 2 _ 1 a2 - a - 6 a2 + a - 20 37. y _ 2y y2 - 4 y2 + y - 6 2y _ y y2 + y - 6 y2 - 4 38. y _ 3y y2 - 1 y2 + 5y + 4 3y _ y y2 - 1 y2 - 4y + 3 39. 3 _ 1 x2 + 2x - 3 x2 - 3x - 10 3 _ 1 x2 - 6x + 5 x2 + 5x + 6 40. 1 + 1 a2 + 7a + 12 + a2 + a - 6 1 + 1 a2 + 2a - 8 + a2 + 5a + 4 41. Michael incorrectly simplifies a + b-1 as a + c a + c-1 as a + b What mistake is he making and how could you convince him that this is incorrect? 42. To simplify a complex rational expression in which the sum of two fractions is divided by the difference of the same two fractions, which method is easier? Why? SKILL MAINTENANCE Solve. 43. 2(3x - 1) + 5(4x - 3) = 3(2x + 1) 44. 5(2x + 3) - 3(4x + 1) = 2(3x - 5) 45. Solve for y: t = r. s + y 46. Solve: |2x - 3| = 7. 47. Framing Andrea has two rectangular frames. The first frame is 3 cm shorter, and 4 cm narrower, than the second frame. If the perimeter of the second frame is 1 cm less than twice the perimeter of the first, what is the perimeter of each frame? 48. Earnings. Antonio received $28 in tips on Monday, $22 in tips on Tuesday, and $36 in tips on Wednesday. How much will Antonio need to re ceive in tips on Thursday if his average for the four days is to be $30? SYNTHESIS 49. Lisa claims that she can simplify complex rational expressions without knowing how to add rational expressions. Is this possible? Why or why not? 50. In arithmetic, we are taught that a divide c = a * d b divide d = b * c (to divide by a fraction, we invert and multiply). Use method 1 to explain why we do this. Simplify. 51. 5x-2 + 10x-1y-1 + 5y-2 3x-2 - 3y-2 52. (a2 - ab + b2)-1(a2b-1 + b2a-1) x (a-2 - b -2) (a-2 + 2a-1b-1 + b-2)-1 page 360 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 53. Astronomy When two galaxies are moving in op- posite directions at velocities vl and v2, an observer in one of the galaxies would see the other galaxy receding at speed v1 + v2 1 + v1v2 c2 where c is the speed of light. Determine the ob- served speed if vl and v2 are both one-fourth the speed of light. Find and simplify f(x + h) - f(x) h for each rational function f in Exercises 54-57. 54. f(x) = 2 x2 55. f(x) = 3 x 56. f(x) = x 1 - x 57. f(x) = 2x 1 + x 58. If 3 + 1 F(x) = x 2 - 8 x2 find the domain of F. 59. If _ 1 x x2 - 1 G(x)= 1 _ 1 9 x2 - 16 find the domain of G. 60. Find the reciprocal of y if y = x2 + x + 1 + 1 + 8 x x2 61. For f(x) = 2 , find f( f(a)). 2 + x 62. For g(x) = x + 3 x - 1, find g(g(a)). 63. Let x + 3 + 1 4 x - 3 f(x) =x + 3 - 1 x -3 Find a simplified form of f(x) and specify the do- main of f. 64. Use a grapher to check your answers to Exercises 3, 17, 31, and 59. 65. Use a grapher to check your answers to Exercises 1, 10, 35, and 58. 66. Use algebra to determine the domain of the func- tion given by 1 x - 2 f(x) = x _ 5 x - 2 x - 2 Then explain how a grapher could be used to check your answer. 67. Financial planning Alexis wishes to invest a por- tion of each month's pay in an account that pays 7.5% interest. If he wants to have $30,000 in the ac- count after 10 yr, the amount invested each month is given by 0.075 30,000 * 12 1 + 0.075 120 - 1 12 Find the amount of Alexis' monthly investment. page 361 Focus: Complex rational expressions Time: 10-15 minutes Group size: 2-3 Consider the steps in Examples 2 and 3 for sim-plifying a complex rational expression by each of the two methods. Then, work as a group to simplify 5 _ 1 x + 1 x 2 4 x2 + x subject to the following conditions. 1. The group should predict which method will more easily simplify this expression. 6.4 RATIONAL EQUATIONS page 361 2. Using the method selected in part (1), one group member should perform the first step in the simplification and then pass the prob-lem on to another member of the group. That person then checks the work, performs the next step, and passes the problem on to an-other group member. If a mistake is found, the problem should be passed to the person who made the mistake for repair. This process continues until, eventually, the sim-plification is complete. 3. At the same time that part (2) is being per-formed, another group member should per-form the first step of the solution using the method not selected in part (1). He or she should then pass the problem to another group member and so on, just as in part (2). 4. What method was easier? Why? Compare your responses with those of other groups. 6.4 Rational Equations Solving Rational Equations • Rational Equations and Graphs As we mentioned in the Connecting the Con- in which rational expressions appeared. There cepts feature of Chapter 5, it is not unusual to we used least common denominators to add or learn a skill that enables us to write equivalent subtract. Here in Section 6.4, we return to the expressions and then to put that skill to work task of solving equations, but this time the solving a new type of equation. That is precisely equations contain rational expressions and what we are now doing: Sections 6.1-6.3 have the LCD is used as part of the multiplication been devoted to writing equivalent expressions principle. page 362 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS StudyTip Make sure that all the examples make sense to you before attempting any exercises. This will prove to be a most efficient use of time. Solving Rational Equations In Sections 6.1-6.3, we learned how to simplify expressions. We now learn to solve a new type of equation. A rational equation is an equation that contains one or more rational expressions. Here are some examples: 2 _ 5 = 1 a - 1 = 4 6 3 6 t* a - 5 a2 - 25 x3 + x = 5. As you will see in Section 6.5, equations of this type occur frequently in appli-cations. To solve rational equations, recall that one way to clear fractions from I an equation is to multiply both sides of the equation by the LCD. To Solve a Rational Equation Multiply both sides of the equation by the LCD. This is called clear-ing fractions and produces an equation similar to those we have already solved. Recall that division by 0 is undefined. Note, too, that variables usually appear in at least one denominator of a rational equation. certain num-bers can often be ruled out as possible solutions before we even attempt to solve a given rational equation. Example 1 Solve: x + 4 + x + 8 = 2. 3x 5x Solution Because the left side of this equation is undefined when x is 0, we state at the outset that x =/ 0. Next, we multiply both sides of the equation by the LCD, 3 * 5 * x, or 15x x + 4 + x + 8 Multiplying by the ( 3x + 5x) = 15x * 2 LCD to clear fractions 15x * x + 4 + 15x * x + 8 = 15x * 2 Using the distributive 3x 5x law. 5 * 3x * (x + 4) + 3 * 5x * (x + 8) = 30x 3x + 5x 5(x + 4) + 3(x + 8) = 30x Locating factors equal 5(x + 4) + 3(x + 8) = 30x Removing factors equal to 1: 3x = 5x 3x = 1;5x = 1 5x + 20 + 3x + 24 = 30x 8x + 44 = 30x 44 = 22x 2 = x. This should check since x =/ 0. Using the distributive law. page 363 Check: x + 4 + x + 8 = 2 3x 5x 2 + 4 + 2 + 8 ?2 3 * 2 + 5 * 2 6 +10 6 10 2 2 TRUE The number 2 is the solution. Note that when we clear fractions, all denominators "disappear." This leaves an equation without rational expressions, which we know how to solve. Solution To ensure that neither denominator is 0, we state at the outset the restriction that x ~ 5. Then we proceed as before, multiplying both sides by the LCD, x - 5: In this case, it is important to remember that, because of the restriction above, 5 cannot be a solution. A check confirms the necessity of that restriction. Check: This equation has no solution. To see why 5 is not a solution of Example 2, note that the multiplication principle for equations requires that we multiply both sides by a nonzero num-ber. When both sides of an equation are multiplied by an expression contain ing variables, it is possible that certain replacements will make that expression equal to 0. it is safe to say that if a solution of Solve x - 1 = 4 x - 5 = x - 5 x - 5 * x - 1 = x - 5 * 4 x - 5 x - 5 x - 1 = 4 x = 5. But recall that x =/ 5. Check. x - 1 = 4 x - 5 x - 5 5 - 1 ? 4 5 - 5 5 - 5 4 4 0 0 Division by 0 is undefined. x - 1 = 4 x - 5 x - 5 6.4 RATIONAL EQUATIONS page 363 exists, then it is also a solution of x - 1 = 4. We cannot conclude that every solution of x - 1 = 4 is a solution of the original equation. page 364 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Caution! When solving rational equations, do not forget to list any restrictions as part of the first step. Solution Note that x =A 3. Since the LCD is x - 3, we multiply both sides by xZ 9 (x-3)- _ x-3-(x-3).x-3 x2 = 9 Simplifying xZ - 9 = 0 Getting 0 on one side (x - 3)(x + 3) = 0 Factoring x = 3 or x = -3. Using the principle of zero products Although 3 is a solution of xz = 9, it must be rejected as a solution of the ra-tional equation. You should perform a check to confirm that -3 is a solution despite the fact that 3 is not. Example 4 ' Solve: x + 5 + x 1 5 x2 16 25 Solution To find all restrictions and to assist in finding the LCD, we factor: 2 + 1 - 16 Factoring xz -25 x+5 x-5 (x+5)(x-5)* Note that x =A -5 and x =A 5. We multiply by the LCD, (x + 5) (x - 5), and then use the distributive law: (x + 5) (x - 5)( 2 + 1 I = (x + 5)(x - 5)- 16 x+5 x-5/ (x+5)(x-5) (x+5)(x-5)x+5+(x+5)(x-5)x 1 5 =(x+5)(x-5)- 16 (x + 5)(x - 5) 2(x - 5) + (x + 5) = 16 2x- 10+x+5= 16 3x-5=16 3x = 21 x=7. A check will confirm that the solution is 7. 6.4 RATIONAL EQUATIONS page 365 Rational equations often appear when we are working with functions. 6 Example 5 Let f (x) = x + 6 x. Find all values of a for which f(a) = 5. Solution Since f(a) = a + 6 a, the problem asks that we find all values of a a for which a + 6 = 5 a First note that a ~ 0. To solve for a, we multiply both sides of the equation by the LCD, a: a a + 6 = 5 * a Multiplying a a * a + a * 6 = 5a a a2 + 6 = 5a Simplifying a2 - 5a + 6 = 0 Getting 0 on one side (a - 3)(a - 2) = 0 Factoring a = 3 or a = 2. Using the principle of zero products Check: 6 f(3)=3 + 3 =3 + 2 = 5; 6 f(2)= 2 + 2 = 2 + 3 = 5. The solutions are 2 and 3. For a = 2 or a = 3, we have f(a) = 5. Using the distributive law a Multiplying both sides by a. parentheses are important. There are several ways in which Example 5 can be checked. One way is to confirm that the graphs of yl = x + 6/x and y2 = 5 intersect at x = 2 and x = 3. You can also use a table to check that yl = y2 when x is 2 and again when x is 3. Use a grapher to check Examples 1-3. ' Rational Equations and Graphs One way to visualize the solution to Example 5 is to make a graph. This can be done by graphing 6 f(x) = x + x with a computer, with a calculator, or by hand. We then inspect the graph for any x-values that are paired with 5. (Note that no y-value is paired with 0, since 0 is not in the domain of f.) It appears from the graph that f(x) = 5 when x = 2 or x = 3. Although making a graph is not the fastest or most precise method of solving a rational equation, it provides visualization and is useful when prob-lems are too difficult to solve algebraically. page 366 CHAPTER 6 RATIONAI EXPRESSIONS, EQUATIONS, AND FUNCTIONS y 8 f(x)= x + 6 x 6 4 2 -8 -6 -4 -2 2 4 6 8 -2 -4 The line begins at after -8 ends at just below -4 -9. in lower left corner. The line begins at y 9 goes to 5 goes upto y. in upper right corner. A computer-generated visuatization of Example 5 Exercise Set ~,,4 FOR EXTRA HELP Digital Video Tutor CD S InterAct Math Math Tutor Center MathXL MyMathLab.com Videoiape 11 Solve. 1. 4 1 x 5 + 3 = 9 2. 7 2 x 8 + 5 = 20 3. x - x 3 - 4 =12 4. y - y = 15 5 3 5. 1 _ 1 = 5 3 x 6 6. 5 _ 1 = 2 8 a = 5 7. 1 _ 2 = 3 2 7 = 2x 8. 2 _ 1 = 7 3 5 = 3x 9. 12 _ 1 = 4 15 3x 5 10. 2 + 1 = 1 6 2x 3 11. 4 _ 3 = 10 3y y 3 12. y + 4 = -5 y 13. x - 2 = 2 x - 4 x - 4 14. y - 1 = 2 y - 3 y - 3 15. 5 = 7 4t 5t - 2 16. 3 = 5 x - 2 x + 4 17. x2 + 4 = 5 x - 1 x - 1 18. x2 - 1 = 3 x + 2 x + 2 19. 6 = a a + 1 a - 1 20. 4 = 2a a - 7 a + 3 21. 60 _ 18 = 40 t - 5 t t 22. 50 _ 16 = 30 t - 2 t t 23. 3 + x = 24 x x + 2 = x2 + 2x 24. x + 5 = 21 x + 1 + x x In Exercises 25-30, a rational function f is given. Find all values of a for which f(a) is the indicated value. 15 25. f(x) = 2x - x ; f(a) = 7 6 26. f(x) = 2x - x f(a) = 1 27. f(x)= x - 5 f(a)=3 x + 1 5 28. f(x)= x - 3, f(a) = 1 x + 2 5 29. f(x) = 12 _ 12 f(a)= 8 x 2x 30. f(x) = 6 - 6 f(a) = 5 x 2x Solve. 31. 5 _ 3 = 2x x + 2 x - 2 = 4 - x2 32. y + 3 _ y = y y + 2 y2 - 4 y-2 33. 2 + 2a - 1 = 1 a + 4 a2 + 2a - 8 = a - 2 6.4 RATIONAL EQUATIONS page 367 34. 3 + x - 2 = x x2 - 6x + 9 + 3x - 9 = 2x - 6 35. 2 _ 3x + 5 = 5 x + 3 x2 + 4x + 3 = x + l 36. 3 - 2y _ 10 = 2y + 3 y + l y2 - 1 1 - y 37. x - 1 + x + 2 = 2x + 5 x2 - 2x -3 + x2 - 9 = x2 + 4x + 3 38. 2x + 1 + x - 1 = 3x - 1 x2 - 3x - 10 + x2 - 4 = x2 - 7x + 10 39. 3 + 1 = 4 x2 - x - 12 + x2 + x - 6 = x2 + 3x - 10 40. 3 _ 1 = 2 x2 - 2x - 3 x2 - 1 = x2 - 8x + 7 41. Explain how one can easily produce rational equa-tions for which no solution exists. (Hint: Examine Example 2.) 42. Below are unlabeled graphs of f(x) = x + 2 and g(x) = (x2 - 4)/(x - 2). How could you determine which graph represents f and which graph repre-sents g? A 0 appears only in graph 1 on left graph at 4,2. graph 1 is g graph 2 is f y 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 line starts at -5 lower left corner goes to 1 and 4,2 ends at 6,4. SKILL MAINTENANCE 43. There are 70 questions on a test. The questions are either multiple-choice, true-false, or fill-in. There are twice as many true-false as fill-in and 5 fewer multiple-choice than true-false. How many of each type of ques- tion are there on the test? 44. Dinoer prices. The Danville Volunteer Fire De- partment served 250 dinners. A child's dinner cost $3 and an adult's dinner cost $8. The total amount of money collected was $1410. How many of each type of dinner was served? 45. Agriculture The perimeter of a rectangular corn field is 628 m. The length of the field is 6 m greater than the width. Find the area of the field. P = 628 m 46. Find two consecutive positive even numbers whose product is 288. 47. Determine whether each of the following systems is consistent or inconsistent. a) 2x - 3y = 4, 4x - 6y = 7 b) x + 3y = 2, 2x - 3y = 1 48. Solve: |x - 2| > 3. SYNTHESIS 49. Is the following statement true or false: "For any real numbers a, b, and c, if ac = bc, then a = b"? Explain why you answered as you did. 50. When checking a possible solution of a rational equation, is it sufficient to check that the "solution" does not make any denominator equal to 0? Why or why not? For each pair of functions f and g, find all values of a for which f(a) = g(a). x - 2 x + 2 51. f(x) = 3, g(x) = 3 x + 1 x - 3 2 2 2 - x x - 2 52. f(x) = 4 g(X) = 4 2 x + 2 2 53. = x + 3 _ x+ 4 * x + 5 _ x + 6 f(x) = x + 2 x + 3 g(x)= x + 4 x + 5 54. f(x) = 1 x 1 _ x 1 + x + 1 - x' g(x) = 1 - x 1 + x 55. f(x) = 0.793 + 18.15, g(x) = 6.034 _ 43.17 x x page 368 CHAPTER.6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 56. f(x)= 2.315 _ 12.6 g(x) = 6.71 + 0.763 x 17.4 x 57. x2 + 6x - 16 = x + 8 x =/ 2 x - 2 58. x3 + 8 = x2 - 2x + 4 x =/ -2, x =/ 2 x2 - 4 x - 2 59. Use a grapher to check your answers to Exercises 3, 27 and 33. Recall that identities are true for any possible replace- ment of the variable(s). Determine whether each of the following equations is an identity. 60. Use a grapher to check your answers to Exercises 4, 18, and 28. 61. Use a grapher with a TABLE feature to show that 2 is not in the domain of f, if f(x) = (x2 - 4)/(x - 2). (See Exercise 42.) 62. Can Exercise 61 be answered on a grapher using only graphs? Why or why not? Solving Applications 6.5 Using Rational Problems Involving Work • Problems Involving Motion Equations Now that we are able to solve rational equations, it is possible to solve problems that we could not have handled before. The five problem-solving steps remain the same. Problems Involving Work Example 1 . Sue can mow a lawn in 4 hr. Lenny can mow the same lawn in 5 hr. How long would it take both of them, working together, to mow the lawn? 1. Familiarize. We familiarize ourselves with the problem by considering two incorrect ways of translating the problem to mathematical language. a) One incorrectway to translate the problem is to add the two times: 4hr + 5hr = 9hr. Think about this. Sue can do the job alone in 4 hr. If Sue and Lenny work together, whatever time it takes them must be less than 4 hr. b) Another incorrect approach is to assume that each person mows half the lawn. Were this the case, 1 1 Sue would mow 2 the lawn in 2(4 hr), or 2 hr and 1 1 Lenny would mow' the lawn in 2(5 hr), or 2 2 hr.1 But time would be wasted since Sue would finish 2 hr before Lenny. Were Sue to help Lenny after completing her half, the entire job would take between 2 and 2 1 2 hr. This information provides a partial check on any answer we get-the answer should be between 2 and 2 1 hr. 2 6.5 SOLVING APPLICATIONS USING RATIONAL EQUATIONS page 369 Let's consider how much of the job each person completes in 1 hr, 2 hr, 3 hr, and so on. Since Sue takes 4 hr to mow the entire lawn, in 1 hr she mows 1 4 of the lawn. Since Lenny takes 5 hr to mow the entire lawn, in 1 hr he mows 1 5 of the lawn. Sue works at a rate of 1 4 lawn per hr, and Lenny works at a rate of 1 lawn per hr. 5 Working together, Sue and Lenny mow 1 1 4 + 5 of the lawn in 1 hr, so their rate-as a team-is 5 4 9 20 + 20 = 20 lawn per hr. 1 1 In 2 hr, Sue mows 4 * 2 of the lawn and Lenny mows 5 * 2 of the lawn. Working together, they mow 1 1 9 9 9 4 * 2 + 5 * 2, or 10 of the lawn in 2 hr. Note that 20 * 2 = 10. Continuing this pattern, we can form a table. Fraction of the Lawn Mowed Time By Sue By Lenny Together lhr 1 1 1 + 1 or 9 4 5 4 5 20 2hr 1 * 2 1 * 2 (1 + 1)2 or 9 * 2 or 9 4 5 4 5 20 10 3 hr 1 * 3 1 * 3 (1 + 1)3 or 9 * 3 or 27 4 5 4 5 20 20 t hr 1 * t 1 * t (1 + 1)t or 9 * t 4 5 4 5 20 This is too little. This is too much. From the table, we note that the number of hours t required for Sue and Lenny to mow exactly one lawn is between 2 hr and 3 hr. 2. Translate. From the table, we see that t must be some number for which Fraction of lawn done 1 * t + 1 * t = 1 Fraction of lawn done by Sue in t hr ~ 4 5 ~ by Lenny in t hr or t + t = 1 4 5 3. Carry out. We solve the equation: t + t = 1 4 5 t t 20 ( 4 + 5) = 20 * 1 Multiplying by the LCD 20t + 20t 4 + 5 = 20 Distributing the 20 5t + 4t = 20 Simplifying 9t = 20 20 2 t= 9 or 2 9 page 37O CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 4. Check. In 20 hr, Sue mows 1 * 20 or 5 9 4 * 9 or 9, of the lawn and Lenny mows 1 * 20 or 4 5 * 9 or 9, of the lawn. Together, they mow 5 4 1 9 + 9, or 1 lawn. The fact that our solution is between 2 and 2 2 hr (see step 1 above) is also a check. 5. State. It will take 2 2 9 hr for Sue and Lenny, working together, to mow the lawn. 6.5 SOLVING APPLICATIONS USING R.4TIONAL EQUATIONS page 371 Example3 A racer is bicycling 15 km/h faster than a person on a mountain bike. In the time it takes the racer to trave180 km, the person on the mountain bike has ; gone 50 km. Find the speed of each bicyclist. Solution 1. Familiarize. Let's guess that the person on the mountain bike is going 10 km/h. The racer would then be traveling 10 + 15, or 25 km/h. At 25 km/h, the racer will travel 80 km in 80 25 = 3.2 hr. Going 10 km/h, 50 the mountain bike will cover 50 km in 10 = 5 hr. Since 3.2 =/ 5, our guess was wrong, but we can see that if r = the rate, in kilometers per hour, of the slower bike, then the rate of the racer = r + 15. 372 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Making a drawing and constructing a table can be helpful. Distance Speed Time Mountain Bike 50 r t Racing Bike 80 r + 15 t r km/h 50 km r + 15 km/h 80 km 2. Translate. By looking at how we checked our guess, we see that in the Time column of the table, the t's can be replaced, using the formula Time = Distance/Rate, as follows. Distance Speed Time Mountain Bike 50 r 50/r Racing Bike 80 r + 15 80/(r + 15) Since we are told that the times must be the same, we can write an equation: 50 = 80 r r + 15 50 = 80 r r + 15 r(r + 15) 50 = r(r + 15) 80 Multiplying by the LCD r r + 15 50r + 750 = 80r Simplifying 750 = 30r 25 = r. 4. Check. If our answer checks, the mountain bike is going 25 km/h and the racing bike is going 25 + 15 = 40 km/h.80 Traveling 80 km at 40 km/h, the racer is riding for 40 = 2 hr. Traveling 50 km at 25 km/h, the person on the mountain bike is riding for 50 25 = 2 hr. Our answer checks since the two times are the same. 5. State. The speed of the racer is 40 km/h, and the speed of the person on the mountain bike is 25 km/h. 6.5 SOLVING APPLICATIONS USING RATIONAL EQUATIONS page 373 In the following example, although the distance is the same in both direc-tions, the key to the translation lies in an additional piece of given information. E x a m p 1 e 4 A Hudson River tugboat goes 10 mph in still water. It travels 24 mi upstream and 24 mi back in a total time of 5 hr. What is the speed of the current? Solution 1. Familiarize. Let's guess that the speed of the current is 4 mph. The tug-boat would then be moving 10 - 4 = 6 mph upstream and 10 + 4 = 14 mph downstream. The tugboat would require 24 = 4 hr to travel 24 6 24 5 mi upstream and 14 =1 7 hr to travel 24 mi downstream. Since the total time, 5 5 4 + 1 7 = 5 7 hr, is not the 5 hr mentioned in the problem, we know that our guess ~is wrong. Suppose that the current's speed = c mph. The tugboat would then travel 10 - c mph when going upstream and 10 + c mph when going downstream. A sketch and table can help display the information. Distance Speed Time tJpstream 24 10 - c tl Downstream 24 I 10 + c t 2. Translate. From examining our guess, we see that the time traveled can be represented using the formula Time = Distance/Rate: Distance Speed Time Upstream 24 10 - c t1 Downstream 24 IO + c t2 Distance Speed Time Upstream 24 10 - c 24/(10 - c) Downstream 24 IO + c 24/(10 + c) page 374 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Since the total time upstream and back is 5 hr, we use the last column of the table to form an equation: 24 + 24 = 5. 10 - c + 10 + c 3. Carry out. We solve the equation: 24 + 24 10 - c + 10 + c = 5 (10 - c) (10 + c) 24 + 24 = (10 - c) (10 + c)5 Multiplying 10 - c l0 + c by the LCD 24(10 + c) + 24(10 - c) = (100 - c2)5 480 = 500 - 5c2 Simplifying 5c2 - 20 = 0 5(c2 - 4)= 0 5(c - 2)(c + 2) = 0 c = 2 or c = -2. 4. Check. Since speed cannot be negative in this problem, -2 cannot be a solution. You should confirm that 2 checks in the original problem. 5. State. The speed of the current is 2 mph. Exercise Set 6.5 FOR EXTRA HELP AL 0""SL -111, , Digital Video Tutor CD 5 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 12 Solve. 1. The reciprocal of 3, plus the reciprocal of 6, is the reciprocal of what number? 2. The reciprocal of 5 plus the reciprocal of 7, is the reciprocal of what number? 3. The sum of a number and 6 times its reciprocal is -5. Find the number. 4. The sum of a number and 21 times its reciprocal is -10. Find the number. 5. The reciprocal of the product of two consecutive integers is 1. Find the two integers. 42 6. The reciprocal of the product of two consecutive integers is 1 . Find the two integers. 72 7. Home restoration. Cedric can refinish an apartment in 8 hr. Carolyn can refinish the floor in 6 hr. How long will it take them, working to- gether, to refinish the floor? 8. Mail order. Zoe, an experienced shipping clerk, can fill a certain order in 5 hr. Willy, a new clerk, needs 9 hr to complete the same job. Working to- gether, how long will it take them to fill the order? 9. Filling a tank. A community water tank can be filled in 18 hr by the town office well alone and in 22 hr by the high school well alone. How long will it take to fill the tank if both wells are working? 10. Filling a pool. A swimming pool can be filled in 12 hr if water enters through a pipe alone or in 30 hr if water enters through a hose alone. If water is entering through both the pipe and the hose, how long will it take to fill the pool? 5.5 , SOLVING APPLICATIONS USING RATIONAL EQUATIONS page 375 11. Hotel management. The Honeywell HQ17 air cleaner can clean the air in a 12-ft by 14-ft confer- ence room in 10 min. The HQ174 can clean the air in a room of the same size in 6 min. How long would it take the two machines together to clean the air in such a room? 12. Printing. Pronto Press can print an order of book- lets in 4.5 hr. Red Dot Printers can do the same job in 5.5 hr. How long will it take if both presses are used? 13. Cutting firewood Jake can cut and split a cord of firewood in 6 fewer hr than Skyler can. When they work together, it takes them 4 hr. How long would it take each of them to do the job alone? 14. Wood cutting. Damon can clear a lot in 5.5 hr. His partner, Tyron, can complete the same job in 7.5 hr. How long will it take them to clear the lot working together? 15. Hotel management. The Honeywell HQ17 air cleaner takes twice as long as the EV25 to clean the same volume of air. Together the two machines can clean the air in a 24-ft by 24-ft banquet room in 10 min. How long would it take each machine, working alone, to clean the air in the room? 16. Computer printers. The HP Office Jet G85 works twice as fast as the Laser Jet II. When the machines work together, a university can produce all its staff manuals in 15 hr. Find the time it would take each machine, working alone, to complete the same job. 17. Painting. Sara takes 3 hr longer to paint a floor than it takes Kate. When they work together, it takes them 2 hr. How long would each take to do the job alone? 18. Painting. Claudia can paint a neighbor's house 4 times as fast as Jan can. The year they worked to- gether it took them 8 days. How long would it take each to paint the house alone? 19. Waxing a car Rosita can wax her car in 2 hr. When she works together with Helga, they can wax the car in 45 min. How long would it take Helga, working by herself, to wax the car? 20. Newspaper deliverv. Zsuzanna can deliver papers 3 times as fast as Stan can. If they work together, it takes them 1 hr. How long would it take each to de liver the papers alone? 21. Sorting recyclables. Together, it takes John and Deb 2 hr 55 min to sort recyclables. Alone, John would require 2 more hr than Deb. How long would it take Deb to do the job alone? (Hint: Convert minutes to hours or hours to minutes.) 22. Paving Together, Larry and Mo require 4 hr 48 min to pave a driveway. Alone, Larry would re- quire 4 hr more than Mo. How long would it take Mo to do the job alone? (Hint: Convert minutes to hours.) 23. Kayaking. The speed of the current in Catamount Creek is 3 mph. Zeno can kayak 4 mi upstream in the same time it takes him to kayak 10 mi down stream. What is the speed of Zeno's kayak in still water? 24. Boating. The current in the Lazy River moves at a rate of 4 mph. Monica's dinghy motors 6 mi up- stream in the same time it takes to motor 12 mi downstream. What is the speed of the dinghy in still water? 25. Moving sidewalks. The moving sidewalk at O'Hare Airport in Chicago moves 1.8 ft/sec. Walking on the moving sidewalk, Camille travels 105 ft forward in the time it takes to trave1 51 ft in the opposite di- rection. How fast would Camille be walking on a nonmoving sidewalk? 26. Moving sidewalks. Newark Airport's moving side- walk moves at a speed of 1.7 ft/sec. Walking on the moving sidewalk, Benny can travel 120 ft forward in the same time it takes to trave1 52 ft in the oppo- site direction. How fast would Benny be walking on a nonmoving sidewalk? page 376 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 27. Train speed. The speed of the A&M freight train is 14 mph less than the speed of the A&M passenger train. The passenger train travels 400 mi in the same time that the freight train travels 330 mi. Find the speed of each train. 28. Walking. Rosanna walks 2 mph slower than Simone. In the time it takes Simone to walk 8 mi, Rosanna walks 5 mi. Find the speed of each person. Aha~ 29. Bus travel. A local bus travels 7 mph slower than the express. The express travels 45 mi in the time it takes the local to trave1 38 mi. Find the speed of each bus. 30. Train speed. The A train goes 12 mph slower than the E train. The A train travels 230 mi in the same time that the E train travels 290 mi. Find the speed of each train. 32. Boating Laverne's Mercruiser travels 15 km/h in still water. She motors 140 km downstream in the same time it takes to trave1 35 km upstream. What is the speed of the river? 33. Shipping. A barge moves 7 km/h in still water. It travels 45 km upriver and 45 km downriver in a total time of 14 hr. What is the speed of the current? 34. Moped speed. Jaime's moped travels 8 km/h faster than Mara's. Jaime travels 69 km in the same time that Mara travels 45 km. Find the speed of each person's moped. 35. Aviation. A Citation II Jet travels 350 mph in still air and flies 487.5 mi into the wind and 487.5 mi with the wind in a total of 2.8 hr (Source: Eastern Air Charter). Find the wind speed. 36. Canoeing. Al paddles 55 m per minute in still water. He paddles 150 m upstream and 150 m downstream in a total time of 5.5 min. What is the speed of the current? 37. Train travel. A freight train covered 120 mi at a certain speed. Had the train been able to travel 10 mph faster, the trip would have been 2 hr shorter. How fast did the train go? 38. Boating. Julia's Boston Whaler cruised 45 mi up- stream and 45 mi back in a total of 8 hr. The speed of the river is 3 mph. Find the speed of the boat in still water. 39. Two steamrollers are paving a parking lot. Working together, will the two steamrollers take less than half as long as the slower steamroller would work- ing alone? Why or why not? 40. Two fuel lines are filling a freighter with oil. Will the faster fuel line take more or less than twice as long to fill the freighter by itself? Why? SKILL MAtNTENANCE Simplify. 41. 35a6b8 7a2b2 42. 20x9y6 4x3y2 43. 36s15t10 9s5t2 44. 6x4 - 3x2 + 9x - (8x4 + 4x2 - 2x) 45. 2(x3 + 4x3 - 5x + 7) - 5(2x3 - 4x2 + 3x - 1) 46. 9x4 + 7x3 + x2 - 8 - (-2x4 + 3x2 + 4x + 2) SYNTHESIS 47. Write a work problem for a classmate to solve. De- vise the problem so that the solution is "Liane and Michele will take 4 hr to complete the job, working together." 48. Write a work problem for a classmate to solve. De- vise the problem so that the solution is "Jen takes takes 5 hr and Pablo takes 6 hr to complete the job alone." 49. Filling a bog The Norwich cranberry bog can be filled in 9 hr and drained in 11 hr. How long will it take to fill the bog if the drainage gate is left open? 50. Filling a tub Justine's hot tub can be filled in 10 min and drained in 8 min. How long will it take to empty a full tub if the water is left on? 51. Refer to Exercise 24. How long will it take Monica to motor 3 mi downstream? 52. Refer to Exercise 23. How long will it take Zeno to kayak 5 mi downstream? 53. Escalators. Together, a 100-cm-wide escalator and a 60-cm-wide escalator can empty a 1575-person auditorium in 14 min (Source: McGraw-Hill Ency- clopedia of Science and Technology). The wider escalator moves twice as many people as the nar- rower one. How many people per hour does the 60-cm-wide escalator move? page 377 54. Aviation A Coast Guard plane has enough fuel to fly for 6 hr, and its speed in still air is 240 mph. The plane departs with a 40-mph tailwind and returns to the same airport flying into the same wind. How far can the plane travel under these conditions? 55. Boatiug. Shoreline Travel operates a 3-hr paddle- boat cruise on the Missouri River. If the speed of the boat in still water is 12 mph, how far upriver can the pilot travel against a 5-mph current before it is time to turn around? 56. Boating. The speed of a motor boat in still water is three times the speed of a river's current. A trip up the river and back takes 10 hr, and the total dis tance of the trip is 100 km. Find the speed of the current. ` 57. Travel by car. Melissa drives to work at 50 mph and arrives 1 min late. She drives to work at 60 mph and arrives 5 min early. How far does Melissa live from work? 58. At what time after 4:00 will the minute hand and the hour hand of a clock first be in the same position? 59. At what time after 10:30 will the hands of a clock first be perpendicular? Average speed is defined as total distance divided by total time. 60. Lenore drove 200 km. For the first 100 km of the trip, she drove at a speed of 40 km/h. For the sec- ond half of the trip, she traveled at a speed of 60 km/h. What was the average speed of the entire trip? (It was not 50 km/h.) 61. For the first 50 mi of a 100-mi trip, Chip drove 40 mph. What speed would he have to travel for the last half of the trip so that the average speed for the entire trip would be 45 mph? Focus: Testing a mathematical model ACTIVITY Time: 20-30 minutes Group size: 2-3 CORNER Does the Model Hold Water? 6.5 SOLVING APPLICATIONS USING RATIONAL EQUATIONS page 377 Materials: An empry 1-gal plastic jug, a kitchen or laboratory sink, a stopwatch or a watch capable of measuring seconds, an inex-pensive pen or pair of scissors or a nail or knife for poking holes in plastic. Problems like Exercises 49 and 50 can be solved algebraically and then checked at home or in a laboratory. 1. While one group member fills the empty jug with water, the other group member(s) should record how many seconds this takes. 2. After carefully poking a few holes in the bot-tom of the jug, record how many seconds it takes the full jug to empty. 3. Using the information found in parts (1) and (2) above, use algebra to predict how long it will take to fill the punctured jug. 4. Test your prediction by timing how long it takes for the pierced jug to be filled. Be sure to run the water at the same rate as in part (1). 5. How accurate was your prediction? How might your prediction have been made more accurate? page 378 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Division of Polynomials Example 1 Example 2 6.6 Divisor a Monomial • Divisor a Polynomial A rational expression indicates division. Division of polynomials, like division of real numbers, relies on our multiplication and subtraction skills. Divisor a Monomial To divide a monomial by a monomial, we can subtract exponents when bases are the same (see Section 1.6). For example, 45x10 10-4 48a2b5 = 48 3x4 = 15x = 15x6, -3ab2 -3a2-1b5-2 = -16ab3. To divide a polynomial by a monomial, we regard the division as a sum of quotients of monomials. This uses the fact that since A + B _ A + B A + B A B C C C we know that C = C + C . Divide 12x3 + 8x2 + x + 4 by 4x. Solution 12x3 + 8x2 + x + 4 = (4x) =(12x3 + 8x2 + x + 4) Writing a rational expression 4x = 12x3 + 8x2 + x + 4 Writing as a sum = 4x 4x 4x 4x of quotients Performing the 1 1 = 3x2 + 2x + 4 + x four indicated divisions Divide: (8x4y5 - 3x3y4 + 5x2y3) = x2y3. Solution 8x4y5 - 3x3y4 + 5x2y3 = 8x4y5 _ 3x3y4 + 5x2y3 x2y3 x2y3 x2y3 x2y3 Try to perform this step mentally. = 8x 2y2 - 3xy + 5 Division by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. page 379 3x - 15 Divide 3x by x 3x/x = 3. -(-3x - 15) Multiply x - 5 by 3. 0 Subtract. (x - 5)(2x + 3) = 2x2 - 7x - 15. The quotient is 2x + 3. 6.6 DIVISION OF POLYNOMIALS page 381 You may have noticed that it is helpful to have all polynomials written in descending order. t 1 Tips for Dividing Polynomials 1. Arrange polynomials in descending order. 2. If there are missing terms in the dividend, either write them with 0 coefficients or leave space for them. 3. Continue the long division process until the degree of the remainder is less than the degree of the divisor. The answer is a + 9 + a2 + 1. E x a m p 1 e 6 ' Let f(x) = 125x3 - 8 and g(x) = 5x - 2. If F(x) = ( f/g) (x), find a simplified expression for F(x). Solution Recall that ( f/g) (x) = f(x)/g(x). F(x) = 125x3 - 8 5x - 2 25x2 + lOx + 4 5x - 2 divide 125x3 - 8 Leaving space for the missing terms. 125x3 - 50x2 ~ Subtracting: 50x2 125x3 - (125x3 - 50x2) - 50x2 50x2 - 20x 20x - 8 Subtracting 20x - 8 0. a + 4 The degree of the remainder is less than the degree of the divisor, so we are . finished. Example 5 ~ Divide: (9a2+a3-5)+(a2-1). Solution We rewrite the problem in descending order: (a3 + 9a2 - 5) + (a2 - 1). a + 9 When there is a missing term, we can a2 - 1 a3 + 9a2 + Oa - 5 write it in, as in this example, or leave a3 - a space, as in Example 6 below 9a2 + a - 5 Subtracting: 9a2 -9 a3 + 9a2 -(a3-a)=9a2+a page 382 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Note that, because F(x) = f(x)/g(x), g(x) cannot be 0. Since g(x) is 0 for x = 2 5 (check this), we have F(x) = 25x2 + lOx + 4, provided x =/ 2 5. FOR EXTRA HELP Digital Video Tutor CD 5 InterAd Math Math Tutor Center MathXL MyMathLab.com yideotape 12 Divide and check. 1. 34x6 + 18x5 - 28x2 6x2 2. 30y8 - 15y6 + 40y4 5y4 3. 21a3 + 7a2 - 3a - 14 7a 4. -25x3 + 20x2 - 3x + 7 5x 5. (14y3 - 9y2 - 8y) divide (2y2) 6. (6a4 + 9a2 - 8) divide (2a) 7. (15x7 - 21x4 - 3x2) divide (- 3x2) 8. (36y6 - 18y4 - 12y2) divide (- 6Y) 9. (a2b - a3b3 - a5b5) divide (a2b) l0. (x3y2 - x3y3 - x4y2) divide (x2y2) 11. (6p2q2 - 9p2q + 12pq2) divide (- 3pq) 12. (16y4z2 - 8y6z4 + 12y8z3) divide (4y4z) Aha~13. (x2 + lOx + 21) divide (x + 7) 14. ( y2 - 8y + 16) divide ( y - 4) 15. (a2 - 8a - 16) divide (a + 4) 16. ( y2 - l0y - 25) divide ( y - 5) 17. (x2 - 9x + 21) divide (x - 5) 18. (x2 - llx + 23) divide (x - 7) 19. ( y2 - 25) divide (y + 5) 20. (a2 - 81) divide (a - 9). 21. (y3 - 4y2 + 3y - 6) divide (y - 2) 22. (x3 - 5x2 + 4x - 7) divide (x - 3) 23. (2x3 + 3x2 - x - 3) divide (x + 2) 24. (3x3 - 5x2 - 3x - 2) divide (x - 2) 25. (a3 - a + 10) divide (a - 4) 26. (x3 - x + 6) divide (x + 2) 27. (l0y3 + 6y2 - 9y + l0) divide (5y - 2) 28. (6x3 - llx2 + llx - 2) divide (2x - 3) 29. (2x4 - x3 - 5x2 + x - 6) divide (x2 + 2) 30. (3x4 + 2x3 - llx2 - 2x + 5) divide (x2 - 2) For Exercises 31-38, f(x) and g(x) are as given. Find a simplified expression for F(x) if F(x) = ( f/g) (x). (See Example 6.) 31. f(x) = 8x3 + 27, g(x) = 2x + 3 32. f(x) = 64x3 - 8, g(x) = 4x - 2 33. f(x) = 6x2 - llx - 10, g(x) = 3x + 2 34. f(x) = 8x2 - 22x - 21, g(x) = 2x - 7 35. f(x) = x4 - 24x2 - 25, g(x) = x2 - 25 36. f(x) = x4 - 3x2 - 54, g(x) = x2 - 9 37. f(x) = 8x2 - 3x4 - 2x3 + 2x5 - 5, g(x) = x2 - 1 38. f(x) = 4x - x3 - lOx2 + 3x4 - 8, g(x) = x2 - 4 39. Explain how factoring could be used to solve Example 6. 40. Explain how to construct a polynomial of degree 4 that has a remainder of 3 when divided by x + 1. SKILL MAINTENANCE Solve. 41. ab - cd = k,for c 42. xy - wz = t, for z 6:7 SYNTHETIC DIVISION page 383 43. Find three consecutive positive integers such that the product of the first and second integers is 26 less than the product of the second and third integers. 44. If f(x) = 2x3, find f(-3a). Solve. 45. |2x-3| >7 46. |3x-1| <8 SYNTHESIS 47. Explain how to construct a polynomial of degree 4 that has a remainder of 2 when divided by x + c. 48. Do addition, subtraction, and multiplication of polynomials always result in a polynomial? Does division? Why or why not? 49. (4a3b + 5a2b2 + a4 + 2ab3) = (a2 + 2b2 + 3ab) 50. (x4 - x3y + x2y2 + 2x2y - 2xy2 + 2y3) = (x2 - xy + y2) 51. (a7 + b7) divide (a + b) 52. Find k such that when x3 - kx2 + 3x + 7k is divided by x + 2, the remainder is 0. 53. When x2 - 3x + 2k is divided by x + 2, the remainder is 7. Find k. 54. Let 3x + 7 f(x) = x + 2 a) Use division to find an expression equivalent to f(x). Then graph f. b) On the same set of axes, sketch both g(x) = 1/(x + 2) and h(x) = 1/x. c) How do the graphs of f, g, and h compare? 55. Jamaladeen incorrectly states that x + 4 (x3 + 9x2 - 6) divide (x2 - 1) = x + 9 + x2 - 1. Without performing any long division, how could you show Jamaladeen that his division cannot pos-sibly be correct? 56. Check Example 3 by setting y1 = (2x2 - 7x - 15)/(x - 5) and y2 = 2x + 3. Then use either the TRACE feature (after selecting the ZOOM Z INTEGER option) or the TABLE feature (with Tb1Min = 0 and LTbl = 1) to show that yl =/ y2 for x = 5. 57. Use a grapher to check Example 5. Perform the check using yl = (9x 2 + x3 - 5)/(x2 - 1), y2 = x + 9 + (x + 4)/(x2 - 1),and y3 = y2 - yl. Synthetic Division 6.7 Streamlining Long Division • The Remainder Theorem Streamlining Long Division To divide a polynomial by a binomial of the type x - a, we can streamline the usual procedure to develop a process called synthetic division. Compare the following. In each stage, we attempt to write a bit less than in the previous stage, while retaining enough essentials to solve the problem. At the end, we will return to the usual polynomial notation. page 386 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Example 2 Use synthetic division to divide. a) (2x3 + 7x2 - 5) divide (x + 3) b) (lOx2 - 13x + 3x3 - 20) divide (4 + x) Solution a) (2x3 + 7x2 - 5) divide (x + 3) The dividend has no x-term, so we need to write 0 for its coefficient of x. Note that x + 3 = x - (-3), so we write -3 inside the _|. -3| 2 7 0 -5 _ -6 -3 9 2 1 -3 4 The answer is 2x2 + x - 3, with R4, or 2x2 + x - 3 + 4 x + 3 b) We first rewrite (lOx2 - 13x + 3x3 - 20) divide (4 + x) in descending order: (3x3 + lOx2 - 13x - 20) divide (x + 4). Next, we use synthetic division. Note that x + 4 = x - (-4). -4| 3 10 -13 -20 _ -12 8 20 3 -2 -5 | 0 The answer is 3x2 - 2x - 5. ® technology ~®~ connection In Example 1, the division by x - 2 gave a remainder of 0. The remainder theorem tells us that this means that when x = 2, the value of x3 + 6x2 - x - 30 is 0. Check this both graphically and algebraically (by substitution). Then perform a similar check for Example 2(b). The Remainder Theorem Because the remainder is 0, Example 1 shows that x - 2 is a factor of x3 + 6x2 - x - 30 and that we can write x3 + 6x2 - x - 30 as (x - 2) (x2 + 8x + 15). Using this result and the principle of zero products, we know that if f(x) = x3 + 6x2 - x - 30, then f(2) = 0 (since x - 2 is a factor of f(x)). Similarly, from Example 2(b), we know that x + 4 is a factor of g(x) = lOx2 - 13x + 3x3 - 20. This tells us that g(-4) = 0. In both examples, the remainder from the division, 0, can serve as a function value. Remarkably, this pattern extends to nonzero remainders. To see this, note that the remain-der in Example 2(a) is 4, and if f(x) = 2x3 + 7x2 - 5, then f(-3) is also 4 (you should check this). The fact that the remainder and the function value coincide is predicted by the remainder theorem, which follows. The Remainder Theorem The remainder obtained by dividing P(x) by x - r is P(r). A proof of this result is outlined in Exercise 31. Let f(x) = 8x5 - 6x3 + x - 8. Use synthetic division to find f(2). Solution The remainder theorem tells us that f(2) is the remainder when f (x) is divided by x - 2. We use synthetic division to find that remainder: _2| 8 0 -6 0 1 -8 16 32 52 104 210 8 16 26 52 105 | 202 Although the bottom line can be used to find the quotient for the division (8x5 - 6x3 + x - 8) divide (x - 2), what we are really interested in is the remain-der. It tells us that f(2) = 202. FOR EXTRA HELP Digital Video Tutor CD 5 InterAct Math Math Tutor Center MathXL MyMathLab:com Videotape 12 G.7, -c. SYNTHETIC DIVISION page 387 Exercise Set 6,7 1. (x3 - 2x2 + 2x - 7) divide (x + 1) 2. (x3 - 2x2 + 2x - 7) divide (x - 1) 3. (a2 + 8a + 11) divide (a + 3) 4. (a2 + 8a + 11) divide (a + 5) 5. (x3 - 7x2 - 13x + 3) divide (x-2) 6. (x3 - 7x2 - 13x + 3) divide (x + 2) 7. (3x3 + 7x2 - 4x + 3) divide (x + 3) 8. (3x3 + 7x2 - 4x + 3) divide (x - 3) 9. (y3 - 3y + 10) divide (y - 2) 10. (x3 - 2x2 + 8) divide (x + 2) 11. (x5 - 32) divide (x - 2) 12. (y5 - 1) divide (y - 1) 13. (3x3 + 1 - x + 7x2) divide (x + 1) 3 14. (8x3 - 1 + 7x - 6x2) divide (x - 1) 2 15. f(x) = 5x4 + 12x3 + 28x + 9; f(-3) 16. g(x) = 3x4 - 25x2 - 18; g(3) 17. P(x) = 6x4 - x3 - 7x2 + x + 2; P(-1) 18. F(x) = 3x4 + 8x3 + 2x2 - 7x - 4; F(-2) 19. f(x) = x4 - x3 - 19x2 + 49x - 30; f(4) 20. p(x) = x4 + 7x3 + llx2 - 7x - 12; p(2) 21. Why is it that we add when performing synthetic division, but subtract when performing long division? 22. Explain how synthetic division could be useful when factoring a polynomial. SKILL MAINTENANCE Solve. 23. 9 + cb = a - b,for b 24. 8 + ac = bd + ab,for a Use synthetic division to divide. Use synthetic division to find the indicated function page 388 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Find the domain of f 25. f(x) = 5 3x2 - 75 26. f(x) = 7 2x2 + 7x - 9 Graph. 3 27. y - 2 = 4 (x + 1) 28. y = _4x + 2 3 SYNTHESIS 29. Let Q(x) be a polynomial function with p(x) a fac- tor of Q(x). If p(3) = 0, does it follow that Q(3) = 0? Why or why not? If Q(3) = 0, does it follow that p(3) = 0? Why or why not? 31. To prove the remainder theorem, note that any polynomial P(x) can be rewritten as (x - r) - Q(x) + R, where Q(x) is the quotient poly- nomial that arises when P(x) is divided by x - r, and R is some constant (the remainder). a) How do we know that R must be a constant? b) Show that P(r) = R (this says that P(r) is the re- mainder when P(x) is divided by x - r). 32. Let f(x) = 4x3 + 16x2 - 3x - 45. Find f(-3) and 3x 75 2x +7x-9 33. Let f(x) = 6x3 - 13x2 - 79x + 140. Find f(4) and then solve the equation f(x) = 0. mad 34. Use the TRACE feature on a grapher to check your answer to Exercise 32. then solve the equation f(x) = 0. 35. Use the TRACE feature on a grapher to check your 1 answer to Exercise 33. form of P(x). Nested evaluation. One way to evaluate a polynomial function like P(x) = 3x4 - 5x3 + 4x2 - 1 is to succes- sion What adjustments must be made if synthetic divi- sively factor outx as shown: sion is to be used to divide a polynomial by a bino- mial of the form ax + b, with a > 1? P(x) = x(x(x(3x - 5) + 4) + 0) - 1. Computations are then performed using this "nested" 36. Use nested evaluation to find f(-3) in Exercise 32. Note the similarities to the calculations performed with synthetic division. 37. Use nested evaluation to find f(4) in Exercise 33. Note the similarities to the calculations performed with synthetic division. Formulas, 6.8 Applications, and Formulas • Direct variation • Inverse variation • joint and Variation Combined Variation Formulas Formulas occur frequently as mathematical models. Many formulas contain rational expressions, and to solve such formulas for a specified letter, we pro-ceed as when solving rational equations. Example Optics. The formula f = L/d tells how to calculate a camera's "f-stop." In this formula, f is the f-stop, L is the focal length (approximately the distance from the lens to the film), and d is the diameter of the lens. Solve for d. 6.8 FORMULAS, APPLICATIONS, AND VARIATION page 391 To Solve a Rational Equation for a Specified Unknown 1. If necessary, multiply both sides by the LCD to clear fractions. 2. Multiply, as needed, to remove parentheses. 3. Get all terms with the specified unknown alone on one side. 4. Factor out the specified unknown if it is in more than one term. 5. Multiply or divide on both sides to isolate the specified unknown. Variation To extend our study of formulas and functions, we now examine three real-world situations: direct variation, inverse variation, and combined variation. Direct Variation 1 i Note that for k > 0, any equation of the form y = kx indicates that as x increases, y increases as well. A hair stylist earns $18 per hour. In 1 hr, $18 is earned. In 2 hr, $36 is earned. In 3 hr, $54 is earned, and so on. This gives rise to a set of ordered pairs: (1,18), (2, 36), (3, 54), (4, 72), and so on. Note that the ratio of earnings E to time t is 18 in every case. If a situation gives rise to pairs of numbers in which the ratio is constant, we say that there is direct variation. Here earnings vary directly as the time: We have E = 18, so E = 18t or, using function notation, t E(t) = 18t The variation constant is 16. The equation of variation is y = 16x. The notation Direct Variation When a situation gives rise to a linear function of the form f(x) = kx, or y = kx, where k is a nonzero constant, we say that there is direct variation, that y varies directly as x, or that y is pro-portional to x. The number k is called the variation constant, or constant of proportionality. Example 4 F Find the variation constant and an equation of variation if y varies directly as x, and y = 32 when x = 2. Solution We know that (2,32) is a solution of y = kx. Therefore, 32 = k * 2 Substituting 32 = k, or k = 16. Solving for k 2 page 392 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNC710NS 5 cm of snow Example 5 W cm of water. Water from melting snow. The number of centimeters W of water produced from melting snow varies directly as the number of centimeters S of snow Meteorologists know that under certain conditions, 150 cm of snow will melt to 16.8 cm of water. The average annual snowfall in Alta, Utah, is 500 in. Assum-ing the above conditions, how much water will replace the 500 in. of snow? Solution 1. Familiarize. Because of the phrase "W.. varies directly as ...S," we express the amount of water as a function of the amount of snow W(S) = kS, where k is the variation constant. Knowing that 150 cm of snow becomes 16.8 cm of water, we have W(150) = 16.8. Because we are using ratios, it does not matter whether we work in inches or centimeters, pro-vided the same units are used for W and S. W cm of e E 2• Translate. We find the varx~iion constant using the data and then fmd the water ~ equation of variation: W(S) = kS W(150) = k * 150 Replacing S with 150 16.8 = k * 150 Replacing W(150) with 16.8 16.8 = 150 k Solving for k 0.112 = k. This is the variation constant. The equation of variation is W(S) = 0.1125. This is the translation. 3. Carry out. To find how much water 500 in. of snow will become, we compute W(500): W(S) = 0.112S W(500) = 0.112(500) Substituting 500 for S W = 56. 4. Check. To check, we could reexamine all our calculations. Note that our answer seems reasonable since 500/56 and 150/16.8 are equal. 5. State. Alta's 500 in. of snow will be replaced with 56 in. of water. Inverse Variation To see what we mean by inverse variation, suppose a bus is traveling 20 mi. At 20 mph, the trip will take 1 hr. At 40 mph, it will take 1 1 2 hr. At 60 mph, it will take 3 hr, and so on. This gives rise to pairs of numbers, all having the same product: 1 1 1 (20,1), (40, 2), (60, 3), (80, 4), and so on. Note that the product of each pair of numbers is 20. Whenever a situation gives rise to pairs of numbers for which the product is constant, we say that there is inverse variation. Since r * t = 20, the time t, in hours, required for the bus to travel 20 mi at r mph is given by t = 20 pr, using function notation, t(r) = 20. r r page 395 Joint variation is one form of combined variation. In general, when a vari-able varies directly and/or inversely, at the same time, with more than one other variable, there is combined variation. Fxamples 8 and 9 are both ex-amples of combined variation. Find an equation of variation if y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2. Solution The equation of variation is of the form xz Y=k * w2 105=k * 3 * 20 2 2 105 = k * 15 k = 7. xz y = 7 * w2. FOR EXTRA HELP 15.S= H ;tl 16.5= H ;H Digital Video Tutor CD 5 InterACt Math Math Tutor Center MathXL MyMathLab:com Videotape 12 6.8 FORMULAS, APPLICATIONS, AND VARIATION page 395 Exercise Set 6.8 Solve the formula for the specified letter. 1. Wl = dl d1 W2 d2 2. W1 = d1 W1 W2 d2 3. s = (V1 + V2)t v1 2 4. s = (v1 + v2)t t 2 5. 1 = 1 + 1 r1 R r1 r2 6. 1 = 1 + 1 R R r1 r2 7. I = 2V R R + 2r 8. I = 2V r R + 2r 9. R = gs g g + s 10. k = rt t r - t 11. I = nE n R + nr 12. I = ne r R + nr 13. 1 + 1 = 1 q p q f 14. 1 + 1 = 1 p p q f 15. s = H t1 m(t1 - t2) 16. s = H H m(t1 - t2) 17. E = R + r r e r 18. E = R + r R e R 19. s = a r 1 - r 20. S = a - arn a 1 - r 21. c = f a + b (a + b)c 22. d = g c + f d(c + f) 23. Taxable interest. The formula It = If 1 - T gives the taxable interest rate It equivalent to the tax-free interest rate If for a person in the (100 * T)% tax bracket. Solve for T. page 396 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 24. Interest. The formula P = A l + r is used to determine what principal P should be in- vested for one year at (100 * r)% simple interest in order to have A dollars after a year. Solve for r. 25. Electricity. Electricians regularly use the formula 1 = 1 + 1 R r1 r2 to determine the resistance R that corresponds to two resistors rl and r2 connected in parallel. Solve for r2. 26. Work rate. The formula 1 = 1 + 1 t a b gives the total time t required for two workers to complete a job, if the workers' individual times are a and b. Solve for t. 27. Average acceleration. The formula a= v2 - v1 t2 - t1 gives a vehicle's average acceleration when its velocity changes from v1 at time t1 to v2 at time t2. Solve for tl. 28. Average speed. The formula v= d2 - d1 t2 - t1 gives an object's average speed v when that object has traveled d1 miles in tl hours and d2 miles in t2 hours. Solve for t2. 29. Semester average. A = 2Tt + Qq 2T + Q gives a student's average A after T tests and Q - quizzes, where each test counts as 2 quizzes, t is the test average, and q is the quiz average. Solve for Q. 30. Astronomy. The formula L = dR D - d where D is the diameter of the sun, d is the diame- ter of the earth, R is the earth's distance from the sun, and L is some fixed distance, is used in calcu- lating when lunar eclipses occur. Solve for D. Find the variation constant and an equation of variation if y varies directly as x and the following conditions apply 31. y = 28 when x = 4 32. y = 5 when x = 12 33. y = 3.4 when x = 2 34. y = 2 when x = 5 1 35. y = 2 when x = 3 36. y = 0.9 when x = 0.5 37. Hooke's law. Hooke's law states that the distance d that a spring is stretched by a hanging object varies directly as the mass m of the object. If the distance is 20 cm when the mass is 3 kg, what is the distance when the mass is 5 kg? 38. Ohm's law. The electric current I, in amperes, in a circuit varies directly as the voltage V. When 15 volts are applied, the current is 5 amperes. What is the current when 18 volts are applied? 39. Use of aluminum cans The number N of alu- minum cans used each year varies directly as the number of people using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in Dallas, which has a population of 1,008,000? 6.8 FORMULAS, APPLICATIONS, AND VARIATION page 397 40. Weekly allowance. According to Fidelity Invest- ments Investment Vision Magazine, weekly allowance A of children varies directly as their grade level, G. In a recent year, the average allowance of a 9th-grade student was $9.66 per week. What was the average weekly allowance of a 4th-grade student? Aha~ 41. Mass of water in a human. The number of kilo- grams W of water in a human body varies directly as the mass of the body. A 96-kg person contains 64 kg of water. How many kilograms of water are in a 48-kg person? 42. Weight on Mars The weight M of an object on Mars varies directly as its weight E on Earth. A per- son who weighs 95 1b on Earth weighs 38 1b on Mars. How much would a 100-1b person weigh on Mars? 43. Relative aperture. The relative aperture, or f-stop, of a 23.5-mm lens is directly proportional to the focal length F of the lens. If a lens with a 150-mm focal length has an f-stop of 6.3, find the f-stop of a 23.5-mm lens with a focal length of 80 mm. 44. Lead pollution The average U.S. community of population 12,500 released about 385 tons of lead into the environment in a recent year.* How many tons were released nationally? Use 250,000,000 as the U.S. population. Find the variation constant and an equation of varia- tion in which y varies inversely as x, and the following conditions exist. 45. y = 3 when x = 20 46. y = 16 when x = 4 47. y = 28 when x = 4 48. y = 9 when x = 5 49. y = 27 when x = 1 3 50. y = 81 when x = 1 9 Solve 51. Ultraviolet index. At an ultraviolet, or UV, rating of 4, those people who are moderately sensitive to the sun will burn in 70 min (Source: Los Angeles Times, 3/24/98). Given that the number of minutes it takes to burn, t, varies inversely with the UV rat- ing, u, how long will it take moderately-sensitive people to burn when the UV rating is 14? 52. Current and resistance The current I in an electri- cal conductor varies inversely as the resistance R of the conductor. If the current is 1 2 ampere when the resistance is 240 ohms, what is the current when the resistance is 540 ohms? 53. Volume and pressure The volume V of a gas varies inversely as the pressure P upon it. The vol- ume of a gas is 200 cm3 under a pressure of 32 kg/cm2. What will be its volume under a pres- sure of 40 kg/cm2? 54. Pumping rate. The time t required to empty a tank varies inversely as the rate r of pumping. If a Briggs and Stratton pump can empty a tank in 45 min at the rate of 600 kL/min, how long will it take the pump to empty the tank at 1000 kL/min? 55. Work rate. The time T required to do a job varies inversely as the number of people P working. It takes 5 hr for 7 volunteers to pick up rubbish from 1 mi of roadway. How long would it take 10 volun- teers to complete the job? 56. Wavelength and frequency The wavelength W of a radio wave varies inversely as its frequency F. A wave with a frequency of 1200 kilohertz has a length of 300 meters. What is the length of a wave with a frequency of 800 kilohertz? Find an equation of variation in which: 57. y varies directly as the square of x, and y = 6 when x = 3. 58. y varies directly as the square of x, and y = 0.15 when x = 0.1. 59. y varies inversely as the square of x, and y = 6 when x = 3 60. y varies inversely as the square of x, and y = 0.15 when x = 0.1. 61. y varies jointly as x and the square of z, and y = 105 when x = 14 and z = 5. 62. y varies jointly as x and z and inversely as w, and y = 3 when x = 2, z = 3, and w = 4. 2 63. y varies jointly as w and the square of x and in- versely as z, and y = 49 when w = 3, x = 7, and z = 12. page 398 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS 64. y varies directly as x and inversely as w and the square of z, and y = 4.5 when x = 15, w = 5, and z = 2. Solve. 65. Intensiy of light. The intensity I of light from a light bulb varies inversely as the square of the dis- tance d from the bulb. Suppose I is 90 W/m2 (watts per square meter) when the distance is 5 m. What would the intensity be 7.5 m from the bulb? 66. Stopping distance of a car. The stopping distance d of a car after the brakes have been applied varies directly as the square of the speed r. If a car travel- ing 60 mph can stop in 200 ft, what stopping distance corresponds to a speed of 36 mph? 67. Volume of a gas. The volume V of a given mass of a gas varies directly as the temperature T and in- versely as the pressure P. If V = 231 cm3 when T = 42° and P = 20 kg/cm2, what is the volume when T = 30° and P = 15 kg/cm2? 68. Intensity of a signal The intensity I of a television signal varies inversely as the square of the distance d from the transmitter. If the intensity is 25 W/m2 at a distance of 2 km, what is the intensity 6.25 km from the transmitter? 69. Atmospheric drag. Wind resistance, or atmos- pheric drag, tends to slow down moving objects. Atmospheric drag Wvaries jointly as an object's surface area A and velocity v. If a car traveling at a speed of 40 mph with a surface area of 37.8 ft2 ex- periences a drag of 222 N (Newtons), how fast must a car with 51 ft2 of surface area travel in order to experience a drag force of 430 N? 70. Drag force. The drag force F on a boat varies jointly as the wetted surface area A and the square of the velocity of the boat. If a boat going 6.5 mph experiences a drag force of 86 N when the wetted surface area is 41.2 ft2, find the wetted surface area of a boat traveling 8.2 mph with a drag force of 94 N. 71. Which exercise did you find easier to work: Exercise 7 or Exercise 11? Why? 72. If y varies directly as x, does doubling x cause y to be doubled as well? Why or why not? SKILL MAINTENANCE Find the domain of f 2x - 1 73. f(x) = x2 + 1 74. f(x) = |2x - 1| 75. Graph on a plane: 6x - y < 6. 76. If f(x) = x3 - x, find f(2a). 77. Factor: t3 + 8b3. 78. Solve: 6x2 = llx + 35. SYNTHESIS 79. Suppose that the number of customer complaints is inversely proportional to the number of employ- ees hired. Will a firm reduce the number of com plaints more by expanding from 5 to 10 employees, or from 20 to 25? Explain. Consider using a graph to help justify your answer. 80. Why do you think subscripts are used in Exercises 3 and 15 but not in Exercises 17 and 18? 81. Escape velocitv A satellite's escape velocity is 6.5 mi/sec, the radius of the earth is 3960 mi, and the earth's gravitational constant is 32.2 ft/sec2. How far is the satellite from the surface of the earth? (See Example 2.) 82. The harmonic mean of two numbers a and b is a number M such that the reciprocal of M is the aver- age of the reciprocals of a and b. Find a formula for the harmonic mean. 83. Health care. Young's rule for determining the size of a particular child's medicine dosage c is c = a * d a + 12 page 399 where a is the child's age and d is the typical adult dosage (Source: Olsen, June Looby, Leon J. Ablon, and Anthony Patrick Giangrasso, Medical Dosage Calculations, 6th ed.). If a child's age is doubled, the dosage increases. Find the ratio of the larger dosage to the smaller dosage. By what percent does the dosage increase? 84. Solve for x: x2(1 - 2pq) = 2p2q3 - pq2x x -q 85. Average acceleration. The formula d4 - d3 d2 - d1 t4 - t3 t2 - t1 a = t4 - t2 can be used to approximate average acceleration, where the d's are distances and the t's are the cor-responding times. Solve for tl. 86. If y varies inversely as the cube of x and x is multi-plied by 0.5, what is the effect on y? Describe, in words, the variation given by the equation. Assume k is a constant. 87. Q = kp2 q3 88. W = km1M1 d2 89. Tension of a musical string The tension T on a string in a musical instrument varies jointly as the string's mass per unit length m, the square of its length l, and the square of its fundamental fre- quency f. A 2-m-long string of mass 5 gm/m with a fundamental frequency of 80 has a tension of 100 N. How long should the same string be if its tension is going to be changed to 72 N? 90. Volume aod cost. A peanut butter jar in the shape of a right circular cylinder is 4 in. high and 3 in. in diameter and sells for $1.20. If we assume that cost is proportional to volume, how much should a jar 6 in. high and 6 in. in diameter cost? 91. Golf distance finder. A device used in golf to esti- mate the distance d to a hole measures the size s that the 7-ft pin appears to be in a viewfinder. The viewfinder uses the principle, diagrammed here, that s gets bigger when d gets smaller. If s = 0.56 in. when d = 50 yd, find an equation of variation that expresses d as a function of s. What is d when s = 0.40 in.? 12 in wide 7 feet height 6.8 FORMULAS, APPLICATIONS, AND VARIATION page 399 HOW IT WORKS: Just sight the flagstick through the viewfinder... at flag between top dashed line and the solid line below... read the distance, 50 - 220 yards. Nothing to focus. . Gives you exact distance that your ball lies from the flagstick, Choose proper club on every approach shot. Figure new pin placement instantly. Train your naked eye for formal and tournament play. Eliminate the need to remember every stake, tree, and bush on the course. page 400 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS CORNER How Many Is a Million? Focus: Direct variation and estimation Time. 15 minutes Group size: 2 or 3 and entire class The National Park Service's estimates of crowd sizes for static (stationary) mass demonstrations vary directly as the area covered by the crowd. Park Service officials have found that at basic "shoulder-to-shoulder" demonstrations, 1 acre of land (about 45,000 ftZ) holds about 9000 people. Using aerial photographs, officials impose a grid to estimate the total area covered by the demonstrators. Once this has been accomplished, estimates of crowd size can be prepared. ACTIVITY 1. In the grid imposed on the photograph below, each square represents 10,000 ft2. Esti- mate the size of the crowd photographed. Then compare your group's estimate with those of other groups. What might explain discrepancies between estimates? List ways in which your group's estimate could be made more accurate. 2. Park Service officials use an "acceptable mar-gin of error" of no more than 20%. Using all estimates from part (1) above and allowing for error, find a range of values within which you feel certain that the actual crowd size lies. 3. The Million Man March of 1995 was not a static demonstration because of a periodic turnover of people in attendance (many people stayed for only part of the day's festiv-ities). How might you change your methodol-ogy to compensate for this complication? SUMMARY AND REVIEW: CHAPTER 6 page 401 Summary and Review 6 Rational expression, p. 334 Rational equation, p. 362 Inverse variation, p. 392 Rational function, p. 334 Clear fractions, p. 362 ~oint variation, p. 394 Simplified, p. 337 Motion problem, p. 371 Combined variation, p. 395 Least common multiple, LCM, Synthetic division, p. 385 p. 345 Direct variation, p. 391 Least common denominator, Variation constant, p. 391 LCD, p. 346 Complex rational expression, Constant of proportionality, p. 391 p. 352 Important Properties and formulas Addition: A + B = A + B C C C Subtraction: A _ B = A - B C C C Multiplication: A * C = AC B D BD Division: A divide C = A * D B D B C To find the least common multiple, LCM, use each factor the greatest number of times that it occurs in any one prime factorization. a = the time needed for A to complete the work alone, Simplifying Complex b = the time needed for B to complete the Rational Expressions work alone, and I: By using multiplication by 1 t = the time needed for A and B to com-plete the work together, I. Find the LCD of all rational expres- sions within the complex rational then expression. t + t = 1 a b 2. Multiply the complex rational ex- pression by 1, writing 1 as the LCD 1 * t + 1 * t = 1 a b divided by itself. and 1 1 1 a + b = t . 3. Distribute and simplify so that the numerator and the denominator of the complex rational expression are polynomials. 4. Factor and, if possible, simplify. Modeling Work Problems If _A _B _ A + B II. By using division C + C C 1. Add or subtract, as necessary to get one rational expression in the numerator. 2. Add or subtract, as necessary to get one rational expression in the denominator. 3. Perform the indicated division (in-vert the divisor and multiply). 4. Simplify, if possible, by re,moving any factors equal to 1. page 402 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Motion Formula d = rt or r = d/t or t = d/r The Remainder Theorem The remainder obtained by dividing P(x) by x - r is P(r) Motion Formula Variation y varies directly as x if there is some d = rt or r = d/t or t = d/r nonzero constant k such that y = kx. y varies inversely as x if there is some nonzero constant k such that y = k/x. Review Exercises 1. If f(t)= t2 - 3t + 2 t2 - 9 find the following function values. a) f(0) b) f(-1) c) f(2) Find the LCD. 2. 7 Y 6x3' 16x2 3. x + 8 x x2 + x - 20 x2 + 3x - 10 Perform the indicated operations and, if possible, simplify. 4. x2 _ 9 x - 3 x - 3 5. 4x - 2 _ 3x + 2 x2 - 5x + 4 x2 - 5x + 4 6. 3a2b3 15c9d4 5c3d2 * 9a7b 7. 5 7 6m2n3p + 9mn4p2 8. y2 - 64, y + 5 2y + 10 * y + 8 9. x3 - 8 * x2 + 10x + 25 x2 - 25 * x2 + 2x + 4 10. 9a2 - 1 divide 3a + 1 a2 - 9 a + 3 11. x3 - 64 divide x2 + 5x + 6 x2 - 16 x2 - 3x - 18 12. x _ 2 x2 + 5x + 6 x2 + 3x + 2 13. -4xy + x + y x2 - y2 x - y 14. 2x 2 + 2y2 x - y y - x 15. 3 _ y + y2 + 3 y + 4 y - 1 + y2 + 3y - 4 Simplify. 16. 5 _ 5 x 7 _ 7 x 17. 2 2 a + b 4 4 a3 + b3 y2 + 4y - 77 18. y2 - 10y + 25 y2 - 5y - 14 y2 - 25 19. 5 _ 3 x2 - 9 x + 3 4 + 2 x2 + 6x + 9 + x - 3 Solve. 20. 6 + 4 = 5 x x y varies jointly as x and z if there is some nonzero constant k such that y = kxz. 21. 5 = 3 3x + 2 2x 22. 4x + 4 + 9 = 4 x + l x x2 + x REVIEW EXERCISES: CHAPTER 6 page 403 23. x + 6 x = x + 2 x2 + x - 6 + x2 + 4x + 3 x2 - x - 2 24. If f(x)= 2 + 2 x - 1 x + 2 find all a for which f(a)=1 Solve 25. Kim can set up for a banquet in 12 hr. Kelly can set up for the same banquet in 9 hr. How long would it take them, working together, to set up for the banquet? 26. A research company uses personal computers to process data while the owner is not using the com- puter. A Pentium III 850 megahertz processor can process a megabyte of data in 15 sec less time than a Celeron 700 megahertz processor. Working to- gether, the computers can process a megabyte of data in 18 sec. How long does it take each com- puter to process one megabyte of data? 27. The Gold River's current is 6 mph. A boat travels 50 mi downstream in the same time that it takes to travel 30 mi upstream. What is the speed of the boat in still water? 28. A car and a motorcycle leave a rest area at the same time, with the car traveling 8 mph faster than the motorcycle. The car then travels 105 mi in the time it takes the motorcycle to travel 93 mi. Find the speed of each vehicle. Divide. 29. (20r2S3 + 15r2S2 - 10r3s3) divide (5r2S) 30. (y3 + 125) divide (y + 5) 31. (4x3 + 3x2 - 5x - 2) divide (x2 + l) 32. Divide using synthetic division: (x3 + 3x2 + 2x - 6) divide (x - 3). 33. If f(x) = 4x3 - 6x2 - 9, use synthetic division to find f(5). Solve. 34. R = gs for s g + s 35. S= H m(tl - t2),for m 36. 1 = 2 _ 3 , for c ac ab bc 37. T = A v(t2 - tl) , for tl 38. The amount of waste generated by a restaurant varies directly as the number of customers served. A typical McDonalds that serves 2000 customers per day generates 238 lb of waste daily (Source: En- vironmental Defense Fund Study, November 1990). How many pounds of waste would be generated daily by a McDonalds that serves 1700 customers a day? 39. A warning dye is used by people in lifeboats to aid search planes. The volume V of the dye used varies directly as the square of the diameter d of the cir cular patch of water formed by the dye. If 4 L of dye is required for a 10-m wide circle, how much dye is needed for a 40-m wide circle? 40. Find an equation of variation in which y varies in-versely as x, and y = 3 when x = 1 4. SYNTHESIS 41. Discuss at least three different uses of the LCD studied in this chapter. 42. Explain the difference between a rational expres-sion and a rational equation. Solve. 43. 5 _ 5 = 65 x - 13 x x2 - 13x 44. x + 2 x2 - 25 x - 5 3 _ 4 = 1 x - 5 x2 - lOx + 25 45. A Pentium 4 1.5-gigahertz processor can process a megabyte of data in 20 sec. How long would it take a Pentium 4 working together with the Pentium III and Celeron processors (see Exercise 26) to process a megabyte of data? page 404 CHAPTER 6 RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS Chapter Test 6 Simplify. 1. t - 1 * 3t + 9 t + 3 4t2 - 4 2. x3 + 27 divide x2 + 8x + 15 x2 - 16 x2 + x - 20 3. Find the LCD: 3x x + 1 x2 + 8x - 33 x2 - 12x + 27 Perform the indicated operation and simplify when possible. 4. 25x x3 x + 5 + x + 5 5. 3a2 _ 3b2 - 6ab a - b b - a 6. 4ab a2 + b2 a2 - b2 + a + b 7. 6 _ 4 x3 - 64 x2 - 16 8. 4 _ y + y2 + 4 y + 3 y - 2 y2 + y - 6 Simplify 9. 2 3 a + b 5 1 ab + a2 10. x2 - 5x - 36 x2 - 36 x2 + x - 12 x2 - 12x + 36 11. 4 _ 2 x + 3 x2 - 3x + 2 3 1 x - 2 + x2 + 2x - 3 Solve 12. 4 = 6 2x - 5 = 5x + 3 13. t + 11 + 1 = 4 t2 - t - 12 + t - 4 = t + 3 For Exercises 14 and 15, let f(x) = x + 3 x - 1. 14. Find f(2) and f(-3). 15. Find all a for which f(a) = 7. 16. Kyla can lay vinyl in a kitchen in 3.5 hr. Brock can lay the same vinyl in 4.5 hr. How long will it take them, working together, to lay the vinyl? Divide. 17. (16ab3c - lOab2C2 + 12a2b2c) divide (4a2b) 18. (y2 - 20y + 64) divide (y - 6) 19. (6x4 + 3x2 + 5x + 4) divide (x2 + 2) 20. Divide using synthetic division: (x3 + 5x2 + 4x - 7) divide (x - 4). 21. If f(x) = 3x4 - 5x3 + 2x - 7, use synthetic divi- sion to find f(4). h(bl + b2) 22. Solve A = 2 for bl. 23. The product of the reciprocals of two consecutive integers is 1 30. Find the integers. 24. Georgia bicycles 12 mph with no wind. Against the wind, she bikes 8 mi in the same time that it takes to bike 14 mi with the wind. What is the speed of the wind? 25. The number of workers n needed to clean a sta- dium after a game varies inversely as the amount of time t allowed for the cleanup. If it takes 25 workers to clean the stadium when there are 6 hr allowed for the job, how many workers are needed if the stadium must be cleaned in 5 hr? 26. The surface area of a balloon varies directly as the square of its radius. The area is 325 in 2 when the radius is 5 in. What is the area when the radius is 7 in.? SYNTHESIS 27. Let f(x)= 1 + 5 x + 3 x - 2 Find all a for which f(a)= f(a + 5). 28. Solve: 6 _ 6 = 90 x - 15 x x2 - 15x 29. Find the x- and y-intercepts for the function given by 5 _ 3 x + 4 x - 2 f(x)= 2 + 1 x - 3 x + 4 30. One summer, Hans mowed 4 lawns for every 3 lawns mowed by his brother Franz. Together, they mowed 98 lawns. How many lawns did each mow? CUMULATIVE REVIEW: CHAPTERS 1-6 page 405 Cumulative Review 1-6 1. Evaluate 2x - y2 x + y for x = 3 and y = -4. 2. Convert to scientific notation: 5,760,000,000. answer is 576. 3. Determine the slope and the y-intercept for the line given by 7x - 4y = 12. 4. Find an equation for the line that passes through the points (-1, 7) and (2, -3). 5. Solve the system 5x - 2y = -23, 3x + 4y = 7. 6. Solve the system -3x - 4y + z = -5 x - 3y - z = 6 2x + 3y + 5z = -8 7. Briar Creek Elementary School sold 45 pizzas for a fundraiser. Small pizzas sold for $7.00 each and large pizzas for $10.00 each. The total amount of funds received from the sale was $402. How many of each size pizza were sold? 8. The sum of three numbers is 20. The first number is 3 1ess than twice the third number. The second number minus the third number is -7. What are the numbers? 9. Trex Company makes decking material from waste wood fibers and reclaimed polyethylene. Its sales rose from $3.5 million in 1993 to $74.3 million in 1999 (Source: Business Week, May 29, 2000). Calcu- late the rate at which sales were rising. 10. In 1989, the average length of a visit to a physician in an HMO was 15.4 min; and in 1998, it was 17.9 min (Sources: Rutgers University Study and National Center for Health Statistics). Let V repre- sent the average length of a visit t years after 1989. a) Find a linear function V(t) that fits the data. b) Use the function of part (a) to predict the average length of a visit in 2005. 11. If f(x) = x - 2 x - 5 find (a) f(3) and (b) the domain of f. 12. 8x = 1 + 16x2 13. 625 = 49y2 14. 20 > 2 - 6x 15. 1 1 1 _ 1 3x - 5 _> 5x 3 16. -8 < x + 2 < 15 17. 3x - 2 < -6 or x + 3 > 9 18. |x| > 6.4 19. |4x - 1| <_ 14 20. 2 _ 7 = 3 n n 21. 6 = 2 x - 5 2x 22. 3x _ 6 = 24 x - 2 x + 2 x2 - 4 23. 3x2 + 5x - 22 = -48 x + 2 x - 2 x2 - 4 24. Let f(x) = |3x - 5|. Find all values of x for which f(x) = 2. ______ 25. Write the domain of f using interval notation if f(x)= V x - 7. Solve. 26. 5m - 3n = 4m + 12, for n 27. P = 3a,for a a + b Graph on a plane. 28. 4x _> 5y + 20 29. y = 1 3x - 2 Perform the indicated operations and simplify. 30. (2x2 - 3x + 1) + (6x - 3x3 + 7x2 - 4) 31. (5x3y2) (-3xy2) 32. (3a + b - 2c) - (-4b + 3c - 2a) page 406 CUMULATIVE REVIEW: CHAPTERS 1-6 33. (5x2 - 2x + 1) (3x2 + x - 2) 34. (2x2 - y)2 2 after x is upper small size numbers. 35. (2x2 - y) (2x2 + y) 36. (-5m3n2 - 3mn3) + (-4m2n2 + 4m3n2) - (2mn3 - 3m2n2) 37. y2 - 36 * y + 4 2y + 8 * y + 6 38. x4 - 1 divide x2 + 1 x2 - x - 2 x - 2 39. 5ab + a + b a2 - b2 + a - b 40. 2 3 _ m2 - 1 m + 1 + m - 5 m2 - 4m - 5 41. y _ 2 3y 42. Simplify: 1 _ 1 x y. x + y 43. Divide: (9x3 + 5x2 + 2) divide (x + 2). FACTOR 44. 4x3 + 18x2 45. x2 + 8x - 84 46. 16y2 - 81 47. 64x3 + 8 48. t2 - 16t + 64 49. x6 - x2 50. 0.027b3 - 0.008c3 51. 20x2 + 7x - 3 52. 3x2 - 17x - 28 53. x5 - x3y + x2y - y2 54. If f(x) = x2 - 4 and g(x) = x2 - 7x + 10, find the domain of f/g. 55. A digital data circuit can transmit a particular set of data in 4 sec. An analog phone circuit can transmit the same data in 20 sec. How long would it take, working together, for both circuits to transmit the data? 56. The floor area of a rental trailer is rectangular. The length is 3 ft more than the width. A rug of area 54 ft2 exactly fills the floor of the trailer. Find the perimeter of the trailer. 57. The sum of the squares of three consecutive even integers is equal to 8 more than three times the square of the second number. Find the integers. 58. Logging. The volume of wood V in a tree trunk varies jointly as the height h and the square of the girth g (girth is distance around). If the volume is 35 ft3 when the height is 20 ft and the girth is 5 ft, what is the height when the volume is 85.75 ft3 and the girth is 7 ft? SYNTHESIS 59. Multiply: (x - 4)3. 60. Find all roots for f(x) = x4 - 34x2 + 225. Solve. 61.4 <_ |3 - x| <_6 62. 18 + 10 = 28x x - 9 x + 5 = x2 - 4x - 45 63. 16x3 = x page 409 Example 2 Principal Square Root The principal square root of a nonnegative number is its nonnegative square root. The symbol ___ V is called a radical sign and is used to indicate the principal square root of the number over which it appears. Simplify each of the following. a) __ V25 __ b) V25 64 __ c)-V64 ______ d) V0.0049 Solution __ __ a) V25 = 5 V indicates the principal square root.=/-5. __ b) V25 = 5 2 = 25 64 = 8 Since 8 = 64 c) -V64 = -8 ______ d) V0.0049 = 0.07 (0.07)(0.07)= 0.0049 The square root function given by _ f(x) = Vx has the interval [0,8issideways8. _ f(x) = Vx x Vx (x,f(x)) 0 0 (0,0) 1 1 (1,1) 4 2 (4,2) 9 3 (9,3) indicating line start to end 1,1 4,2 9,3. f(1)= 3 * 1 - 2 = 1 7.1 RADICAL EXPRESSIONS AND FUNCTIONS page 411 Simplfy each expression. Assume no radicals were formed by raising negative quantities to even powers. __ ___ ____________ a) Vy2 b)Va10 c) V9x2 - 6x + 1 2 10 2 are small upper numbers. Solution __ a) Vy2 = y =/ y __ b) Va10 = a5 (a5)2 = a10 ______________ ______ c) V9x2 - 6x + 1 = V(3x - 1)2 = 3x - 1 page 412 CHAPTER 7 EXPONENTS AND RADICALS Cube Roots We often need to know what number was cubed in order to produce a certain value. When such a number is found, we say that we have found a cube root. For example, 2 is the cube root of 8 because 2 3 = 2 * 2 * 2 = 8; -4 is the cube root of -64 because (-4)3 = (-4) (-4) (-4) = -64. Cube Root The number c is the cube root of a if c3 = a. In symbols, we write 3_ Va to denote the cube root of a. The cube-root function, given by 3_ f(x) = VX has R as its domain. We can draw its graph by selecting convenient values for x and calculating the corresponding outputs. Once these ordered pairs have been graphed, a smooth curve can be drawn. 3_ f (x) = Vx 3_ x Vx (x,f(x)) 0 0 (0,0) 1 1 (1,1) 8 2 (8,2) -1 -1 (-1,-1) -8 -2 (-8,-2) In the real-number system, every number has exactly one cube root. The cube root of a positive number is positive, and the cube root of a negative number is negative. Absolute-value signs are not used when finding cube roots. Example 7 For each function, find the indicated function value. 3_ 3_____ a) f(y) = Vy; f(125) b) g(x) = Vx - 3; g(-24) Solution 3____ a) f(125) = V125 = 5 Since 5 * 5 * 5 = 125 3________ b) g(-24) = V -24 - 3 3___ = V-27 = -3 Since (-3) (-3) (-3) = -27 7.1 RADICAL EXPRESSIONS AND FUNCTIONS page 415 Digita) Video Tutor CD 5 InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 13 For each number, find the square roots. 1. 16 2. 49 3. 144 4. 9 5. 81 6. 400 7. 900 8. 225 Simplify 49 9.-V36 10. 361 -V 9 ___ 11. V441 ___ 12. V196 __ 13. -V16 81 __ 14. -V81 144 ____ 15. V0.09 ____ 16. V0.36 ______ 17. -V0.0049 ______ 18. V0.0144 Identify the radicand and the index for each expression. ______ 19. 5Vp2 + 4 ______ 20. -7Vy2 - 8 3_____ 21. x2y3V x y + 4 ______ 3 a 22. a2b3 V a2 - b For each function, find the specified function value, if it exists. _______ 23. f(t) = V5t - l0; f(6), f(2), f(1), f(-1) _______ 24. g(x) = Vx2 - 25; g(-6),g(3),g(6),g(13) ______ 25. t(x) = -V2x + 1; t(4), t(0), t(-1), t(-1) 2 ________ 26. p(z) = V2z2 - 20; p(4), p(3), p(-5), p(o) ______ 27. f(t) = Vt2 + 1; f(o), f(-1), f(-l0) ______ 28. g(x) = -V(x + 1)2; g(-3), g(4), g(-5) ______ 29. g(x) = Vx3 + 9; g(-2),g(-3),g(3) _______ 30. f(t) = Vt3 - 10; f(2),f(3),f(4) Simplify. Remember to use absolute-value notation when necessary. ____ 31. V 36x2 ____ 32. V 25t2 _____ 33. V(-6b)2 ______ 34. V(-7c)2 ________ 35. V(7 - t)2 ________ 36. V(a + 1)2 ______________ 37. V y2 + 16y + 64 ______________ 38. V x2 - 4x + 4 ______________ 39. V9x2 - 30x + 25 ______________ 40. V4x2 + 28x + 49 4___ 41. - V625 42. 4___ V256 43. - 5__ V35 44. 5___ V -1 45. 5___ V 1 -32 46. 5___ V 32 -243 47. 8__ Vy8 48. 6__ v x6 49. 4_____ V(7b)4 ____ 50. 4 V(5a)4 _______ 51. l2 (-10)12 V 10_____ 52. V(-6)l0 53. 1976 ___________ V(2a + b)1976 54. 414__________ V(a + b)414 ___ 55. Vxl0 ___ 56. Va22 ___ 57. Va14 ___ 58. Vxl6 Simplify. Simplify. Assume that no radicands were formed by rais-ing negative quantities to even powers. ____ 59. V25t2 ____ 60. V16x2 ____ 61. V(7c)2 ____ 62. V(6b)2 ________ 63. V(5 + b)2 ________ 64. V(a + 1)2 ______________ 65. V9x2 + 36x + 36 _____________ 66. V4x2 + 8x + 4 ______________ 67. V25t2 - 20t + 4 ____________ 68. V9t2 - 12t + 4 69. - 3__ V64 70. 3__ V27 answer can be 9 x 3 = 27. 71. 4_____ V81x4 72. 4_____ V16x4 _________ 73. - 5 -100,000 V _____ 74. 3 -216 V 75. - 3_____ V-64x3 76. - 3______ V-125y3 ___ 77. Va14 ___ 78. Va22 _________ 79. V(x + 3)l0 ________ 80. V(x - 2)8 For each function, find the specified function value, if it exists. _____ 81. f(x) = 3 x + 1; f(7),f(26),f(-9),f(-65) V ______ 82. g(x) = - 3 2x - 1; g(o), g(-62), g(-13), g(63) V _____ 83. g(t) = 4 t- 3; g(l9),g(-13),g(1),g(84) V ______ 84. f(t) = 4 t + 1; f(0), f(15), f(-82), f(80) V Determine the domain of each function described. _____ 85. f(x) = Vx - 5 _____ 86. g(x) = Vx + 8 page 416 CHAPTER 7 EXPONENTS AND RADICALS 4____ 87. g(t) = Vt + 3 88. f (x) = 4_____ Vx - 7 89. g(x) = 4_____ V 5 - x 90. g(t) = 3_____ V2t - 5 5______ 91. f(t) = V2t + 9 6______ 92. f(t) =-V2t + 5 _______ 93. h(z) = - 6 -5z + 3 V 4_______ 94. d(x) = - V 7x - 5 8___ Aha' 95. f(t) = 7 + Vt8 6___ 96. g(t) = 9 + V t6 97. Explain how to write the negative square root of a number using radical notation. 98. Does the square root of a number's absolute value always exist? Why or why not? SKILL MAINTENANCE Simplify. Do not use negative exponents in your answer. 99. (a3b2C5)3 100. (5a7b6)(2a3b) 101. (2a-2b3c-4)-3 102. (5x-3y-lz2)-2 103. 8x-2y5 4x-6z-2 104. l0a-6b-7 2a-2c-3 SYNTHESIS 105. Under what conditions does the nth root of x3 exist? Explain your reasoning. The 3 is upper small size number. 106. Under what conditions does the nth root of x2 exist? Explain your reasoning. 107. Spaces in a parking lot. A parking lot has atten- dants to park the cars. The number N of stalls needed for waiting cars before attendants can get to them is given by the formula N = 2.5VA-, where A is the number of arrivals in peak hours. Find the number of spaces needed for the given number of arrivals in peak hours: (a) 25; (b) 36; (c) 49; (d) 64. Determine the domain of each function described. Then draw the graph of each function. _____ 108. f(x) = Vx + 5 _____ 109. g(x) = Vx + 5 _ 110. g(x) = Vx - 2 _____ 111. f(x) =Vx - 2 112. Find the domain of f if V______ f(x)= x + 3 4 2 - x V 113. Find the domain of g if 4 V_____ 5 - x g(x) =6_____ Vx + 4 114. Use a grapher to check your answers to Exercises 31, 39, and 49. On some graphers, a MATH key is needed to enter higher roots. 115. Use a grapher to check your answers to Exercises 112 and 113. (See Exercise 114.) Rational Numbers 7•2 as Exponents Rational Exponents • Negative Rational Exponents • Laws of Exponents • Simplifying Radical Expressions In Section 1.1, we considered the natural numbers as exponents. Our discus-sion of exponents was expanded to include all integers in Section 1.6. In this section, we expand the study still further-to include all rational numbers. This will give meaning to expressions like a1/3, 7-l/2, and (3x)4/5. Such notation will help us simplify certain radical expressions. Rational Exponents Consider a1/2 . a1/2. If we still want to add exponents when multiplying, it must follow that a1/2 * a1/2 = a1/2 + l/2, or a1. This suggests that a1/2 is a square root of a. Similarly, a1/3 * a1/3 * a1/3 = a1/3 + 1/3 + 1/3 or a1, so a1/3 should mean 3_ Va. page 417 7.2 • RATIONAL NUMBERS AS EXPONENTS page 479 1 c) 64-z/3 = 642/3 64-Z/3 is the reciprocal of 64Z/3. 1 Since 64z/3 = ( 3 64)z = 4z = 16, the answer simplifies to 16 . d) 4x-2/3y1/5 = 4 • 2/3 ' yl/5 = 4 2/3 x x e) In Section 1.6, we found that (a/b)-n = (b/a)n. This property holds for any  negative exponent: _3r -5/z _ _7s 5/z Writing the reciprocal of the base and (7s) - (3r) ' changing the sign of the exponent Laws of Exponents The same laws hold for rational exponents as for integer exponents. Laws of Exponents For any real numbers a and b and any rational exponents m and n for which am, an, and bm are defined: 1. am ~ an = am+n In multiplying, add exponents if the bases are the same. a"2 In dividing, subtract exponents if the bases a~ = am-n are the same. (Assume a ~ 0.)  3. (am)" = am'n To raise a power to a power, multiply the exponents. 4. (ab)"a = a"2b"' To raise a product to a power, raise each factor to the power and multiply. E x a m p 1 e 6 Use the laws of exponents to simplify. y 3i/s , 33/5 _, b) ai/4/ai/z C) (7.22/3)3/4 d) (a-1/3b2/5)1/2  SO I U fIOYI a) 31/5 , 33/5 = 31/5+3/5 = 34/5 Adding exponents b) al/4 _ al/4-1/z = al/4-2/4 Subtractingexponentsafterfindinga ai/z - common denominator = a-1/4, oral/4 a-1/4 is the reciprocal of al/4 c) (7.22/3)3/4 = 7.22/3~3/4 = 7.2s/iz Multiplying exponents = 7.21/z Using arithmetic to simplify theexponent d) (a-1/3bz/5)i/z = a-i/s'i/z , bz/su/z Raising a product to a power and multiplying exponents bi/s = a-i/sbi/5 or ai/s page 419 Example Exponents a) 3 1/5 * 3 3/5 Solution is a) 3 1/5 * 3 3/5 = 3 1/5+3/5 = 3 4/5 page 420 CHAPTER 7 EXPONENTS AND RADICALS Simplifying Radical Expressions Many radical expressions can be simplified using rational exponents. To Simplify Radical Expressions 1. Convert radical expressions to exponential expressions. 2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation as needed. Example 7 ; Use rational exponents to simplify. Do not use fractional exponents in the final answer. + A technology r'nd connection B One way to check Example 7(a) is to let yl = (5x)3/6 and y2 = V-5x. Then see if the graphs of yl and yz coincide. An alternative is to let y3 = yz - Yl and see if y3 = 0. Check Example 7(a) using one of the checks just described. 1. Why are rational exponents especially useful when work-ing on a grapher? division or divide the following. 6_____ a) V(5x)3 5_____ b) Vt20 3________ c) V(ab2c)12 ___ d) V3x_ V Solution _____ a) 6V (5x)3 = (5x)3/6 Converting to exponential notation (5x) 1 / 2 Simplifying the exponent __ = V5x Returning to radical notation ____ b) 5Vt2o = t20/5 Converting to exponential notation = t4 Simplifying the exponent c) ( 3 ab2c)12 = (ab2c)l2/3 Converting to exponential notation _____ _ (ab2c)4 Simplifying the exponent = a4b8c4 Using the laws of exponents xl/3 Converting the radicand to exponential notation _ (X 113)1/2 Try to go directly to this step. = xl/s Using the laws of exponents ' + -_ ~ Returning to radical notation Exercise Set 7.2 Note: Assume for all exercises that even roots are of nonnegative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. FOR EXTRA HELP Digital Video Tutor CD S InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape 13 1. x l/4 2. y 1/5 3. 16 1/2 4. 8 1/3 5. 81 1/4 6. 64 1/6 7. 9 1/2 8. 25 1/2 9. (xyz)1/3 10. (ab) 1/4 11. (a2b2)1/5 12. (x3y3)1/4 13. a 2/3 14. b 3/2 15. 16 3/4 16. 4 7/2 17. 49 3/2 18. 27 4/3 7.2 RATIONAL NUMBERS AS EXPONENTS page 421 19. 9 5/2 20. 81 3/2 21. (81x)3/4 22. (125a)2/3 23. (25x4)3/2 24. (9y6)3/2 Write an equivalent expression using exponential notation. 3__ 25. V20 26. 3__ V19 __ 27. V17 _ 28. V6 __ 29. Vx3 __ 30. Va5 31. 5__ Vm2 32. 5__ Vn4 33. 4__ Vcd 34. 5__ Vxy 35. 5____ Vxy2z 36. 7_____ Vx3y2z2 ___ 37. (V3mn)3 ___ 38. (3V7xy)4 39. 7______ V(8x2y)5 40. 6_______ V(2a5b)7 41. 2x_ 3Z2 V 42. 3a_ 5 c2 V Write an equivalent expression with positive exponents and, if possible, simplify. 43. x -1/3 44. y-1/4 45. (2rs) -3/4 46. (5xy) -5/6 47. 1 (8)-2/3 48. 1 (16) -3/4 49. 1 a-5/7 50. 1 a-3/5 51. 2a3/4b-1/2c2/3 52. 5x-2/3y4/5z 53. 2-1/3x4y-2/7 -3/5 54. 3-5/2a3 b-7/3 55. (7x)-3/5 8yz 56. (2ab)-5/6 3c 57. 7x 3_ Vz 58. 6a 4_ Vb 59. 5a 3c-1/2 60. 2z 5x-1/3 Use the 61. 5 3/4 * 5 1/8 62. 11 2/3 . 11 1/2 63. 3 5/8 3-1/8 64. 8 7/11 8-2/11 65. 4.1-1/6 4.1-2/3 66. 2.3-3/10 2.3 -1/5 67. (10 3/5)2/5 68. (5 5/4)3/7 69. a 2/3 * a5/4 70. x3/4 * x 2/3 Aha" 71. (64 3/4)4/3 72. (27 -2/3)3/2 73. (m2/3n-1/4)1/2 74. (x-1/3y2/5)1/4 Use the divide 75. 6__ Va2 76. 6__ Vt4 77. 3___ Vx15 78. 4___ Va12 79. 6___ Vxl8 80. 5___ Va10 3____ 81. (V ab)15 7___ 82. (Vxy)14 83. 8____ V(3x)2 84. 4____ V(7a)2 85. (10__)5 V3a 86. (8__)6 V2x 87. 4____ V V x 88. 3 _ 6__ V Vm ___ 89. V(ab)6 90. 4____ V(xy)l2 ____ 91. (3Vx2y5)12 ____ 92. (5Va2b4)15 _____ 93. 3 4 xy VV 94. 5____ VV2a 95. If f(x) = (x + 5) 1/2(X + 7)-1/2, find the domain Use rational exponents to simplify. Do not use fractional z/s 4 s/z s s/z exponents in the final answer. off. Explain how you found your answer. __ 96. Explain why 3/x6 = x2 for any value of x, whereas 2__ Vx6 = x3 only when x _> 0. SKILL MAINTENANCE Simplify. 97. 3x(x3 - 2x2) + 4X2 (2X2 + 5x) 98. 5t3(2t2 - 4t) - 3t4(t2 - 6t) 99. (3a - 4b) (5a + 3b) 100. (7x - y)2 101. Real estate taxes. For homes under $100,000, the real-estate transfer tax in Vermont is 0.5% of the selling price. Find the selling price of a home that had a transfer tax of $467.50. 102. What numbers are their own squares? Use the laws of exponents to simplify. Do not use negative exponents in any answers. SYNTHESIS 103. Let f(x) = 5x-1/3. Under what condition will we have f(x) > 0? Why? 104. If g(x) = x3/n, in what way does the domain of g depend on whether n is odd or even? Use rational exponents to simplify. ______ 105. 5 x2yVXy V ______ 106. Vx 3 x2 V __________ 107. 4 3 8x3y6 V V _____________ 108. 12p2 + 2pq + q2 V page 422 CHAPTER 7 EXPONENTS AND RADICAIS Music. The function f(x) = k2x/l2 can be used to deter- mine the frequency, in cycles per second, of a musical note that is x half-steps above a note with frequency k.* 109. The frequency of middle C on a piano is 262 cycles per second. Find the frequency of the C that is one octave (12 half-steps) higher. 110. The frequency of concert A for a trumpet is 440 cycles per second. Find the frequency of the A that is two octaves (24 half-steps) above concert A (few trumpeters can reach this note.) 111. Show that the G that is 7 half-steps (a "perfect fifth") above middle C (see Exercise 109) has a fre- quency that is about 1.5 times that of middle C. 112. Show that the C sharp that is 4 half-steps "major third") above concert A (see Exercise 110) has a frequency that is about 25% greater than that of concert A. 113. Road pavement messages. In a psychological study, it was determined that the proper length L of the letters of a word printed on pavement is given by 0.000169d2 27 L = h where d is the distance of a car from the lettering and h is the height of the eye above the surface of the road. All units are in meters. This formula says that if a person is h meters above the surface of the road and is to be able to recognize a message d meters away, that message will be the most recognizable if the length of the letters is L. Find L to the nearest tenth of a meter, given d and h. *This application was inspired by information provided by Dr. Homer B. Tilton of Pima Community College East. a) h = 1 m, d = 60 m b) h = 0.9906 m, d = 75 m c) h = 2.4 m, d = 80 m d) h = 1.1 m, d = 100 m 114. Dating fossils The function r(t} = 10-12 2-r/5700 expresses the ratio of carbon isotopes to carbon atoms in a fossil that is t years old. What ratio of carbon isotopes to carbon atoms would be pres- ent in a 1900-year-old bone? 115. Physics. The equation m = mo(1 - v2c-2)-l/2, developed by Albert Einstein, is used to deter- mine the mass m of an object that is moving v meters per second and has mass mo before the motion begins. The constant c is the speed of light, approximately 3 x 10 8 m/sec. Suppose that a particle with mass 8 mg is accelerated to a speed of 3 5 x 10 8 m/sec. Without using a calcula- tor, find the new mass of the particle. 116. Use a grapher in the SIMULTANEOUS mode with the LABEL OFF format to graph yl = x1/2 y2 = 3x2/5, y3 = x4/7 and y4 = 1 5x3/4. Then, looking only at coordinates, match each graph with its equation. 7.3 MULTIPLYING RADICAL EXPRESSIONS page 423 CORNER Are Equivalent Fractions Equivalent Exponents? Focus: Functions and rational exponents one group member should graph f, a second W Time. 10-20 minutes Group size: 3 Materials: Graph paper In arithmetic, we have seen that 1 1 * 1 3, 6 2, and 2 * 6 all represent the same number. Interestingly, f(x) = x1/3 g(x) = (X1/6)2, and h(x) = (x2)1/6 represent three different functions. ACTIVITY 1. Selecting a variety of values for x and using the definition of positive rational exponents, group member should graph g, and a third group member should graph h. Be sure to check whether negative x-values are in the domain of the function. 2. Compare the three graphs and check each other's work. How and why do the graphs differ? 3. Decide as a group which graph, if any, would best represent the graph of k(x) = x2/16. Then be prepared to explain your reasoning to the entire class. (Hint: Study the definition of Clm/n on p. 417 carefully.) Multiplying Radical 7• IYeSSIOnS Multiplying Radical Expressions • Simplifying by Factoring • - , Multiplying and Simplifying Multiplying Radical Expressions _ __ ______ ___ Note that V4 V25 = 2 * 5 = 10. Also V4 * 25 = V100 = 10. Likewise, __ _ ______ ___ 3 27 3 8 = 3 . 2 = 6 and 3 27 * 8 = 3 216 = 6. V V V V These examples suggest the following. _ _ The Product Rule for Radicals For any real numbers n a and n b. V V n_ n_ = n______ Va * Vb V a * b (To multiply, when the indices match, multiply the radicands.) page 424 CHAPTER 7 EXPONENTS AND RADICALS Fractional exponents can be used to derive this rule: n_ * n_ 1/n * 1/n = 1/n = n______ Va Vb = a b (a * b) V a * b Example 1 ! Multiply. _ _ _____ _____ a) V3 * V5 b) Vx + 3 Vx - 3 _ _ _ _ c) 3 4 * 3 5 d) 4 y * 4 7 V V V 5 V x technology connection A To check Example 1(b), let y1= x+3 x-3 and y2 = x2 - 9 and compare. 1. What should the graph of Y= x2-9 x+3V - 3 look like? Solution a) When no index is written, roots are understood to be square roots with an unwritten index of two. We apply the product rule: _ _ _____ V3 * V5 = V3 * 5 __ =V15. _____ _____ ______________ b) Vx + 3 Vx - 3 = V(x + 3)(x - 3) The product of two square roots ______ =Vx2 - 9 is the square root of the product. Caution! ______ __ _ Vx2 - 9 =/ Vx2 - V9. _ _ c) Both 3 4 and 3 5 have indices of three, so to multiply we can use V V the prod-uct rule: 3_ 3_ 3_____ 3__ V4 * V5 = V4 * 5 = V20. d) 4y 4 7 y 7 4 7y V5 * V x = V 5 * x = V 5x 7Y In Section 7.4, we discuss other ways 5x to write answers like this. Important: The product rule for radicals applies only when radicals have the same index. Simplifying by Factoring An integer p is a perfect square if there exists a rational number q for which q2 = p. We say that p is a perfect cube if q3 = p for some rational number q. In general, p is a perfect nth power if qn = p for some rational number q. The product rule allows us to simplify " ab when a or b is a perfect nth power. Using the Product Rule to Simplify _-~l -a - N~~b. 1/3 x 0.09 converted to exact value of 0.03. to a decimal (0.333) page 427 Example 5 Multiply and simplify. __ _ 3__ 3_ 4_____ 4______ a) V15 V6 b) 3V25 * 2V5 c) V8x3y5 V4x2y3 Solution __ _ ______ a) v15 v6 = v15 * 6 Multiplying radicands. __ ______ =V90 = V9 * 10 __ =3V10 b) 3__ 3_ 3______ 3V25 * 2V5 = 3 * 2 * V25 * 5 3___ =6 * V125 =6 * 5 or 30 c) 4_____ 4_____ 4______ V8x3y5 V4x2y3 = V32x5y8 = 4____________ V16x4y8 * 2x 4__ 4__ 4__ 4__ =V16 Vx4 Vy8 V2x 2 4__ = 2xy V2x page 427 Multiply. __ _ 1. V10 V7 _ _ 2. V5 V7 3_ 3_ 3. V2 V5 3_ 3_ 4. V7 V2 4_ 4_ 5. V8 V9 4_ 4_ 6. V6 V3 __ __ 7. V5a V6b __ ___ 8. V2x V13y 9. 5___ 5__ V9t2 V2t 10. 5___ 5___ V8y3 Vl0y _____ _____ 11. Vx - a Vx + a _____ _____ 12. Vy - b Vy + b 13. 3___ 3___ V0.5x V0.2x 14. 3____ 3___ V0.7y V0.3y ______ __________ 15. 4 x - 1 4 x2 + x + 1 V V 16. 5_____ 5_____ Vx - 2 V(x- 2)2 17. _ _ /x /7 V 6 V y 18. _ __ /7 /s V t V 11 19. ______ _____ 7/ x - 3 7/ 5 V 4 V x + 2 20. ______ _____ 6/ a 6/ 3 V b - 2 V b + 2 Simplify by factoring. __ 21. V50 __ 22. V27 __ 23. V28 __ 24. V45 _ 25. V8 __ 26. V18 ___ 27. V198 ___ 28. V325 _____ 29. V36a4b _____ 30. V175y8 3_____ 31. V8x3y2 3_____ 32. v27ab6 3_____ 33. V-16x6 3_____ 34. V-32a6 Find a simplified form of f(x). Assune that x can be any real number. 3_____ 35. f(x) = V125x5 3____ 36. f(x) = V16x6 __________ 37. f(x) = V49(x - 3)2 __________ 38. f(x) = V81(x - 1)2 _____________ 39. f(x) = V5x2 - lOx + 5 ____________ 40. f(x) = V2x2 + 8x + 8 Simplify. Assume that no radicands were formed by mis- ing negative numbers to even powers. 41. ____ Va3b4 42. ____ Vx6y9 43. 3_______ Vx5y6z10 44. 3_______ Va6b7c13 45. 5________ V-32a7b11 46. 4_______ V16x5y11 7.3 MULTIPLYING RADICAL EXPRESSIONS page 427 a) 15 A/6- = 15-6 Multiplying radicands = 90 = 9-10 9 is a perfect square. = 3V-1-0 b) 3-3 25 - 21V5_ = 3 - 2 - 3 /_2_5- 5 Using a commutative law; multiplying radicands = 6 3 125 125 is a perfect cube. = 6 5, or 30 c) 4 8x3y5 4 4x2y3 = 4 32x5y$ Multiplying radicands = 4 16x4y$ - 2x Identifying perfect fourth-power factors = 4 16 ~ ~//? ~1_2x Factoring into radicals = 2xy2,~1_2X Finding the fourth roots; assume x ? 0. Digital Video Tutor CD 5 InterAct Math Math Tutor Center MathXL MyMathlab.com Videotape 13 3 ing negative numbers to even powers. page 428 CHAPTER 7 EXPONENTS AND RADICALS ______ 47. 5 a6b8c9 V _______ 48. 5 x13y8z17 V _____ 49. 4 810x9 V ______ 50. 3 -80a14 V Multiply and simplify. __ _ 51. V15 V5 _ _ 52. V6 V3 __ __ 53. V10 V14 __ __ 54. V15 V21 3 3 55. V2 V4 56. 3 3 V9 V3 ____ ____ Aha~ 57. V18a3 V18a3 ____ ____ 58. V75x7 V75x7 ___ __ 59. 3 5a2 3 2a V V __ ___ 60. 3 7x 3 3x2 V V ___ ____ 61. V3x5 V15x2 ___ ____ 62. v5a7 V15a3 63. 3 S2t4 3 S4t6 V V 64. 3 x2y4 3 x2y6 V V _______ ________ 65. 3 (x + 5)2 3 (x + 5)4 V V 66. 3 (a - b)5 3 (a - b)7 67. 4 12a3b7 4 4a2b5 68. 4 9x7y2 4 9x2y9 __________ __________ 69. 5 x3(y + z)4 5 x3(y + z)6 70. 5 a3(b - c)4 5 a7(b - c)4 71. Why do we need to know how to multiply radical expressions before learning how to simplify radical expressions? __ 72. Why is it incorrect to say that, in general, Vx2 = x? SKILL MAINTENANCE Perform the indicated operation and, if possible, simplify. 73. 3x + 5y 16y 64x 74. 2 6 3 4 + a b a4b 75. 4 _ 7 x2 - 9 2x - 6 76. 8 _ 3 x2 - 25 2x - 10 Simplify. 77. 9a4b7 3a2b5 78. 12a2b7 4ab2 SYNTHESIS 79. Explain why it is true that n__ n__ n__ Vab = v a * Vb _________ 80. Is the equation V(2x + 3)8 = (2x + 3)4 always, sometimes, or never true? Why? 81. Speed of a skidding car Police can estimate the speed at which a car was traveling by measuring its skid marks. The function __ r(L) = 2V5L can be used, where L is the length of a skid mark, in feet, and r(L) is the speed, in miles per hour. Find the exact speed and an estimate (to the nearest tenth mile per hour) for the speed of a car that left skid marks (a) 20 ft long; (b) 70 ft long; (c) 90 ft long. 82. Wind chill temperature When the temperature is T degrees Celsius and the wind speed is v meters per second, the wind chill temperature, Tw, is the temperature (with no wind) that it feels like. Here is a formula for finding wind chill temperature: _ TW = 33 - (10.45 + 10Vv - v) (33 - T) 22 Estimate the wind chill temperature (to the nearest tenth of a degree) for the given actual temperatures and wind speeds. a) T = 7°C, v = 8 m/sec b) T = 0°C, v = 12 m/sec c) T= -5 C, v = 14 m/sec d) T = -23°C, v = 15 m/sec Simplify. Assume that all variables are nonnegative. ___ 83. (Vr3t)7 _____ 84. (3 25x4)4 V ____ 85. (3 a2b4)5 V ____ 86. (Va3b5)7 Draw and compare the graphs of each group of equations. ___________ 87. f(x) = Vx2 - 2x + 1, g(x) = x - 1, h(x) = |x - 1| ___________ 88. f(x) = Vx2 + 2x + 1, g(x) = x + 1 h(x) = |x + 1| 7.4 DIVIDING RADICAL EXPRESSIONS page 429 page 429 ___________ 89. If f(t) = Vt2 - 3t - 4, what is the domain of f? ___________ 90. What is the domain of g, if g(x) = Vx2 - 6x + 8? Solve 3_____ 3____ 91. V5xk+1 V25xk = 5x7, for k 5______ 5_____ 92. V4a3k+2 V8a6-k = 2a4, for k 93. Use a grapher to check your answers to Exer-cises 15, 35, and 59. 94. Rony is puzzled. When he uses a grapher to graph _ _ y = Vx * Vx he gets the following screen. Explain why Rony did not get the complete line y = x. 10 10 10 10 Dividing Radical 7•4 Expressions Dividing and Simplifying • Rationalizing Denominators and Numerators StudyTip It is always best to study for a final exam over a period of at least two weeks. If you have only one or two days of study time, however, begin by studying the formulas, problems, properties, and procedures in each chapter Summary and Review. Then do the exercises in the Cumulative Reviews. Make sure to attend a review session if one is offered. Dividing and Simplifying Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots. For example, 3__ = 3__ V27 - 3 and V27 = 3 8 2 3 8 2 V This example suggests the following. The Quotient Rule for Radicals For any real numbers n_ and n_ Va Vb b =/0 n_ n_ a = Va Vb n b V Remember that an nth root is simplified when its radicand has no factors that are perfect nth powers. Recall too that we assume that no radicands repre-sent negative quantities raised to an even power. page 430 CHAPTER 7 EXPONENTS AND RADICALS Example 1 Simplify by taking the roots of the numerator and the denominator. __ __ a) 3 27 b) V25 V125 y2 Solution __ __ a) 3 27 = 3 27 = 3 V125 3 125=5 V b) __ __ 25 = 25 = 5 Vy2 =Vy2 = y 7.4 DIVIDING RADICAL EXPRESSIONS page 433 c) To change the radicand 2xzy3 into a perfect fifth power, we need four more factors of 2, three more factors of x, and two more factors of y. Thus we mul tiply by 1, using 5 24x3yz/ 5 24x3yz, or 5 16x3yz/ 5 16x3yz: 5 3x - 3x . 5 16x3y2 Multiplying by 1 2xzy3 52xzy3 16x3yz _ 3x-5 16x3yz - This radicand is now a 5 _32X 5y5 perfect fifth power. 3x-5 16x3y2 3 5 16x3y2 = 2x1' _ - 11 2y Always simplify if possible. Sometimes in calculus it is necessary to rationalize a numerator. To do so, we multiply by 1 to make the radicand in the numerator a perfect power. E x a m p 1 e 6 ~ Rationalize each numerator: (a) v 5 . 7 _ 49 35 _ ~ 49 35 7 7 7 Multiplying by 1 under the radical. We also could have multiplied by N/T/V~7 outside the radical. 35 s 4a2 s 4az - IV 2--a Multiplying by 1 IV 8-a3 < - This radicand is now a perfect cube. 3 lOba 2a 3 l0ab In Section 7.5, we will discuss rationalizing denominators and numerators in which two terms appear. The numerator is now a perfect square. Using the quotient rule for radicals page 434 CHAPTER 7 EXPONENTS AND RADICALS Section 7.4, page 434, Problems 8, 22, 50 Exercise Set 7,4 FOR EXTRA HELP Digital Video Tutor CD 6 InterAct Math Videotape 13 Simplify by taking the roots of the numerator and the denominator. Assume all variables represent positive numbers. 8. ____ 36a5 V b6 5 is small upper number. 22. 3_ V40 3_ V5 50. _____ 21x2y V75xy5 2 after x is small upper number. 7.5 EXPRESSIONS CONTAINING SEVERAL RADICAL TERMS page 436 Example 1 _ _ a)6v7 + 4v7 3_ 3_ 3_ b) V2 - 7xV2 + 5V2 Solution _ _ _ a) 6V7 + 4V7 = (6 + 4) V7 _ = 10V7 3_ 3_ 3_ 3_ b) V2 - 7xV2 + 5V2 =(1 - 7x + 5) V2 3_ = (6 - 7x)V2 page 439 Example 8 Divide and if possible simplify 4________ V(x + y)3 _____ Vx + y Solution 4________ V(x + y)3 = (x + y)3/4 Converting to exponential notation. _____ __________ vx + y = (x + y)1/2 =(x + y) 3/4 - 1/2 Since the bases are identical we can subtract exponents. 3 _ 1 = 3 _ 2 = 1 4 2 4 4 4 = (x + y) 1/4 4_____ Converting back to radical notation. =Vx + y page 440 Chapter 7 EXPONENTS AND RADICALS Section 7.5, page 440, Problems 16, 30, 56, 70, 92 If factors are raised to powers that share a common denominator, we can write the final result as a single radical expression. Example 9 Divide and, if possible, simplify: 16. 3__ 3_ V27 - 5 V8 30. _ __ _ V2(3v10 - 2v2) 56. _ __ 5V3 - V11 2V3 - 5V2 70. 5___ __ Va3b Vab _ _ 92. f(x)= x - V2 g(x) = x + V6 7.5 EXPRESSIONS CONTAINING SEVERAL RADICAL TERMS page 441 33. ~( 3 az + 3 24az ) 34. ~( 3 3xz - 3 81xz ) 73. 3 xyzz x3yzz 74. a4b3c4 3 abzc 35. (5+~)(5-~) 36. (2-~)~2+~) 75. 3xz 76. 3az 37. (3 - 2~) (3 + 2~) 38. (4 + 3~) (4 - 3~) 77. 5 a4b 78. 4 xzy3 39. (3 + ~)z 40. (7 + ~)z ;.~ a ~ 41. (2~ - 4~) (3~ + 6~) 79. s x3Y4 80. ab3 s azbs 42. (4~ + 3~) (3~ - 4~) s (2 + 5x)z (3x - 1)3 43. (2~ - ~) (~ + 2~) 81. 4 2 + 5x 82. s (3x - 1)s 44. (3~ + ~) (2~ - 3~) 83. 4 (5 + 3x)3 84. 3 (2x + 1)z 45. (~ + ~)z 3 (5 + 3x)z 5 (2x + 1)z 46. (~ - ~)z , 85. 3 xzY(~ - 5 xY3~ 86. 4 azb ( 3 azb - s azbz ) 3. Rationalize each denominator. ` 2 3 87. (m + 3 nz)(2m + ~) ~, 47. 3+~ 48. 4-~ 88. (r- 4s3)(3r-~) 2 + ~ 1 + ~ In Exercises 89-92, f(x) and g(x) are as given. Find `} 49. 6 - ~ 50. 3 + ~ ( f . g) (x). Assume all variables represent nonnegative : ~ real numbers. 51. ~~~ 52. ~~~ 89. f(x) _ ~~ g(x) _ ~ - 4 xii Ah ~ 53. ~ - ~ 54. ~ + ~ 90. f(x) = 4 x~ + 4 3xz~ g(x) _ ~ ~+~ 91. f(x)=x+~~ S'(x)=x-~ 55. 3~ - ~ 56. 5~ 11 92. f(x) = x - ~~ g(x) = x + ~ 4~ + ~ 2~ - 5~ Let f(x) = xz. Find each of the following. , 57. 5~ - 3~ 58. 7~ + 4~ 93. f(5 - ~~ 94. f(7 + ~) 3~-2~ 4~-3~ 95. f(~+~~ 96. f(~-~~ Rationalize each numerator. ~ 97. Why do we need to know how to multiply radical ~ + 2 ~ + 1 expressions before learning how to add them? 59. 5 60. 4 gg, In what way(s) is combining like radical terms the same as combining like terms that are 61. ~ - 2 62. 10 + 4 monomials? ~+7 ~-3 63. ~ - ~ 64. ~ + ~ SKILL MAINTENANCE ~ + ~ ~ - ~ Solve. 12x _ 3xz _ 384 9g~ x-4 x+4 xz-16 65. ~ ' a3 66. 3 xz 6 x5 100. ? + 1 = 67. 5 bz b3 68. 4 a3 3 az 3 t 5 Perform the indicated operation and simplify. Assume all variables represent nonnegative real numbers. `~, s 101. The width of a rectangle is one-fourth the length. 69. xy3 xzy 70. a3b ~ The area is rivice the perimeter. Find the dimen- 71. 4 9ab3 3~4b 72. 2x3y3 3 4xyz sions of the rectangle. 442 CHAPTER 7 EXPONENTS AND RADICALS 102. The sum of a number and its square is 20. Find Simplify. the number. 115. 1' 36a5bc4 - Z 3 64a4bc6 + 6 144a3bc6 103. 5x2 - 6x + 1 = 0 116. 7x (x -+y) 3 - 5xy x -+y - 2y (x -+y) 3 104. 7t z - 8t + 1 = 0 117. 27a5(b + 1) 3 81a(b + 1)4 SYNTHESIS 118. 8x( y + z)5 3 4xz( y + z)Z 105. Ramon incorrectly writes = - 1 5 x2 . x3 = x2/5 , x3/2 = 5 x3. 119. What mistake do you suspect he is making? VI-w- + 1 ~~ 106. After examining the expression 4 25xy3 5x4y Dyan (correctly) concludes that x and y are both nonnegative. Explain how she could reach this conclusion. 1 1 1 120.4+V3- +73+V3- -4 For Exercises 107-110, fill in the blanks by selecting from the following words: cal expressions. 121. x-5 122. y-7 123. x-a radicands, indices, bases, denominators. Words can be used more than once. Multiply. 107. To add radical expressions, the and 124. 9 + 3V-5 9 - 3V-5 the must be the same. 125. (x+2- x-2)2 108. To multiply radical expressions, the must be the same. For Exercises 126-129, assume that all radicands are 109. To add rational expressions, the must positive and that no denominator is 0. be the same. Rationalize each denominator. 110. To find a product by adding exponents, the must be the same. 126. a a+ b 127. b+A/b-a+b-b l+b+Vb Find a simplified form for f(x). Assume x ? 0. Rationalize each numerator. 111. f(x) = 20x2 + 4x3 - 3x 45 -+9x + 5x2 -+x3 112. f(x) = x3 _ x2 + 9x3 --9x2 - 4x3 - 4x2 128. 18 y + 18 - ~ 129. x + 6 - 5 x+6+5 113. f(x) = 4 x5 - x4 + 3 4 x9 --x8 MA;' 130. Use a grapher to check your answers to 114. f(x) = 4 -16x4+ 16x5 - 24x-8+ x9 Exercises 19, 33, and 75. Express each of the following as the product of two radi-  CQUtlOi7! A common error in solving equations like  is to obtain 1 + (x - 3) as the square of the right side. This is wrong be-cause (A + B)z ~ Az + Bz. For example, (1+2)z~lz+2z 3z~1+4 9~5. 7.6 SOLVING RADICAL EQUATIONS 447 E x A m p 1 e 6 ~ Let f (x) = x + 5 - x - 7. Find all x-values for which f (x) = 2. Solution We must have f(x) = 2, or x + 5 - x - 7 = 2. Substituting for f (x) To solve, we isolate one radical term and square both sides: x + 5 = 2 + x - 7 Adding x - 7 to both sides. This isolates one of the radical terms. V___ ( x + 5)z = (2 + x - 7)z Using the principle of powers (squaring both sides) 5 = 4 x - 7 - 3 Adding -x to both sides and combining like terms 8 = 4 x - 7 . Isolating the remaining radical term Check: f(11) = 11 + 5 - 11 - 7 = 16-~ =4-2=2. We will have f (x) = 2 when x = 11. Squaring both sides page 448 CHAPTER 7 EXPONENTS AND RADICALS Section 7.6, page 448, Problems 8, 34, 48, 64 Exercise Set 7,6 FOR EXTRA HELP Digital Video Tutor CD 6 InterAct Math Math Tutor Center MathXL MyMaihLab.com Videotape 14 Solve. _____ 8. Vx - 7 + 3 = 10 _____ 34. x = Vx - 1 + 3 _____ 48. If f(t) = 7 + V2t -5 and g(t) = 3(t + 1)1/2 find a such that f(a)= g(a). where g is the force of gravity r is the planet or star's radius and h is the height of the satellite above the planet or star's surface. 64. Solve for h 65. Solve for r. _ Sighting to the horizon. The function D(h) = 1.2Vh can be used to approximate the distance D in miles that a person can see to the horizon from a height h in feet. h D 7.6 SOLVING RADICAL EQUATIONS page 449 A formula for the escape velocity v of a satellite is page 450 CHAPTER 7 EXPONENTS AND RADICALS Focus: Radical equations and problem solving Time. 15-25 minutes ~r Group size: 2-3 Materials: Calculators or square-root tables Q m J CORNER TailgaterAlert The faster a car is traveling, the more distance it needs to stop. Thus it is important for drivers to allow sufficient space between their vehicle and the vehicle in front of them. Police recommend that for each 10 mph of speed, a driver allow 1 car length. Thus a driver going 30 mph should have at least 3 car lengths between his or her ve-hicle and the one in front. In Exercise Set 7.3, the function r(L) = 2 5L was used to find the speed, in miles per hour, that a car was traveling when it left skid marks L feet long. Column 1 gives a car's speed s, column 2 lists the minimum amount of space between cars traveling s miles per hour, as recommended by police. Column 3 is the speed that a vehi-cle could travel were it forced to stop in the distance listed in column 2, using the above function. Column 1 s (in miles per hour) 20 30 40 50 60 70 Column 2 L(s) (in feet) Column 3 r(L) (in miles per hour) ACTIVITY 3. Determine whether there are any speeds at 1. Each group member should estimate the which the "1 car length per 10 mph" guideline length of a car in which he or she frequently might not suffice. On what reasoning do you travels. (Each should use a different length, if base your answer? Compare tables to deter- possible.) mine how car length affects the results. What 2. Using a calculator as needed, each group recommendations would your group make to member should complete the table below a new driver? Wal Geometric Applications 7,7 Using the Pythagorean Theorem • Two Special Triangles Using the Pythagorean Theorem There are many kinds of problems that involve powers and roots. Many also involve right triangles and the Pythagorean theorem, which we studied in Sec-tion 5.8 and restate here. 7.7 GEOMETRIC APPLICATIONS 451 The Pythagorean Theorem* In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a2+b2=c2. Hypotenuse a Leg 90° b Leg In using the Pythagorean theorem, we often make use of the following principle. The Principle of Square Roots For any nonnegative real number n, _ _ If x2 = n, then x = Vn or x = -Vn. Example 1 ~ Baseball. A baseball diamond is actually a square 90 ft on a side. Suppose € a catcher fields a ball along the third-base line 10 ft from home plate. How far would the catcher's throw to first base be? Give an exact answer and an ; approximation to three decimal places. Solution We first make a drawing and let d = the distance, in feet, to first base. Note that a right triangle is formed in which the length of the leg from home to first base is 90 ft. The length of the leg from home to where the catcher fields the ball is 10 ft. *The converse of the Pythagorean theorem also holds. That is, if a, b, and c are the lengths of the sides of a triangle and a2 + b2 = cz, then the triangle is a right triangle. page 452 CHAPTER 7 EXPONENTS AND RADICALS We substitute these values into the Pythagorean theorem to find d: d2 = 90 2 + 10 2 2 is upper small numbers. d2 = 8100 + 100 d2 = 8200. ____ We now use the principle of square roots: If d2 = 8200, then d =V8200 or d = -V200. In this case, since d is a length, it follows that d is the positive square root of 8200: ____ d = V8200 ft This is an exact answer. d - 90.6 ft. Using a calculator for an approximation Example 2 Guy wires. The base of a 40-ft-long guy wire is located 15 ft from the tele-phone pole that it is anchoring. How high up the pole does the guy wire reach? Give an exact answer and an approximation to three decimal places. Solution We make a drawing and let h =the height on the pole that the guy wire reaches. A right triangle is formed in which the length of one leg is 15 ft and the length of the hypotenuse is 40 ft. Using the Pythagorean theorem, we have h2 + 15 2 = 40 2 2 are small size upper numbers. h2 + 225 = 1600 h2 = 1375 h =V1375. ____ Exact answer: h = V1375 ft Approximation: h = 37.081 ft Using a calculator Two Special Triangles When both legs of a right triangle are the same size, we call the triangle an 45° isosceles right triangle, as shown at left. If one leg of an isosceles right triangle has length a, we can find a formula for the length of the hypotenuse as follows: c2 = a2 + b2 Because the triangle is isosceles, both legs 45° are the same size: a = b. c2 = a2 + b2. c2 = 2a2 Combining like terms Next, we use the principle of square roots. Because a, b, and c are lengths, there is no need to consider negative s quare roots or absolute values. ___ c = V2a2 Using the principle of square roots ______ _ c = Va2 * 2 = aV2 7.7 GEOMETRICAPPLICATIONS page 455 Lengths Within Isosceles and 30°-60°-90° Right Triangles The length of the hypotenuse in an isosceles right triangle is the length of a leg times ~. 45° a 45° a The length of the longer leg in a 30°-60°-90° right triangle is the length of the shorter leg times ~. The hypotenuse is twice as long as the shorter leg. 30° Exercise Set 7,7 In a right triangle, find the length of the side notgiven. Give an exact answer and, where appropriate, an ap-proximation to three decimal places. Section 7.7, page 455, Problems 2, 18, 38, 44 2. a = 8,b = 10 page 456 CHAPTER 7 EXPONENTS AND RADICALS 18. Speaker placement. A stereo receiver is in a corner of a 12-ft by 14-ft room. Speaker wire will run under a rug, diagonally, to a speaker in the far corner. If 4 ft of slack is required on each end, how long a piece of wire should be purchased? 7.7 GEOMETRIC APPLICATIONS page 457 38. Triangle ABC has sides of lengths 25 ft, 25 ft, and 30 ft. Triangle PQR has sides of lengths 25 ft, 25 ft, and 40 ft. Which triangle has the greater area and by how much? 44. Find all points on the x-axis of a Cartesian coordi-nate system that are 5 units from the point (0, 4). page 459 Imaginary Numbers A imaginary number is a number can be written in the form a + bi where a and b are real numbers and b =/ 0. 5 + 4i Here a = 5,b = 4. _ V5 - tti Here a = v5,b = -tt. 17i Here a = 0, b = 17. page 460 Example 2 add or subtract a) (8 + 6i) + (3 + 2i) Solution a) (8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i) =11 + (6 + 2)i = 11 + 8i page 462 Example 4 Find the conjugate. a) -3 + 7i Solution a) -3 + 7i Example 5 Multiply (5 + 7i)(5 - 7i) Solution (5 + 7i)(5 - 7i) = 5 2 - (7i)2 = 25 - 49i2 = 25 - 49(-1) i2 = -1 =25 + 49 = 74 Using (A + B)(A - B) = a2 - B2 2 are is small upper size numbers. The Complex Numbers Imaginary and Complex Numbers • Addition and Subtraction • Multiplication • Conjugates and Division • Powers of i Imaginary and Complex Numbers Negative numbers do not have square roots in the real-number system. How-ever, a larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system, and it makes use of a number that is a square root of -1. We call this new number i. 7.8 THE COMPLEX NUMBERS page 461 Multiplication For complex numbers, the property V-a Vb _ Vab does not hold in general, but it does hold when a = -1 and b is nonnegative. To multiply square roots of negative real numbers, we first express them in terms of i. For example, = i V-2 i N/5-= i2 V-10 _ -1 10 = - 10 is correct! Caution! With complex numbers, simply multiplying radicands is incorrect: W V '-5 0 10. With this in mind, we can now multiply complex numbers. Example 3 ; Multiply and simplify. When possible, write answers in the form a + bi. c) -3i - 8i e) (1 + 2i)(1 + 3i) Solution a) -16- -25=W 16W 25 = i - 4 - i 5 =i2-20 _ -1 - 20 i2 = -1 _ -20 c) -3i - 8i = -24 i2 _ -24 (-1) i2 = -1 - 24 d) -4i(3 - 5i) _ -4i - 3 + (-4i) (-5i) Using the distributive law _ -12i + 20i2 =-12i-20 iz=-1 _ -20 - 12i Writing in the form a + bi e) (i + 2i) (1 + 3i) = 1 + 3i + 2i + 6i2 Multiplying every term of one number by every term of the other (FOIL) =1+3i+2i-6 i2=-1 _ -5 + 5i Combining like terms page 462 CHAPTER 7 EXPONENTS AND RADICALS Conjugates and Division Conjugates of complex numbers are defined as follows. Conjugate of a Complex Number The conjugate of a complex number a + bi is a - bi, and the conjugate of a - bi is a + bi. a) -3 + 7i b) 14-5i c) 4i Solution a) -3 + 7i The conjugate is -3 - 7i. b) 14 - 5i The conjugate is 14 + 5i. c) 4i The conjugate is -4i. Note that 4i = 0 + 4i. The product of a complex number and its conjugate is a real number. Example 5 Multiply: (5 + 7i) (5 - 7i). Solution (5 + 7i) (5 - 7i) = 52 - (7i)2 Using (A + B) (A - B) = AZ - BZ =25-49iZ = 25 - 49(-1) i2 = -1 =25+49=74 Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators in Section 7.5. Example 6 Divide and simplify to the form a + bi. a) -5 + 9i 1 - 2i ` b) 7+3i 5i page 463 Powers of i ___ i or V -1 i2 = -1 i3 = i2 * i =(-i)i = -i, i4 = (i2)2 =(-1)2 = 1 i5 = i4 * i = (i2) * i = (-1)2 * i = i i6 = (i2)3 = (-1)3 = -1. 2 - 6 numbers are are small upper size. page 465 Example 7 Simplify (a) i18 Solution a) i18 =(i2)9 =(-1)9 = -1 -1 to a odd power is -1. Section 7.8, page 464, Problems 4, 34, 52, 64, 106 ___ 4. V-19 ___ __ 34. V-36 V-9 page 466 CHAPTER 7 EXPONENTS AND RADICALS 52. (-2 + 3i)(-2 + 5i) 64. 4i___ 5 - 3i __ 106. 5 - V5i V5i page 462 CHAPTER 7 EXPONENTS AND RADICALS Conjugates and Division Conjugates of complex numbers are defined as follows. Conjugate of a Complex Number The conjugate of a complex number a + bi is a - bi, and the conjugate of a - bi is a + bi. a) -3 + 7i b) 14-5i c) 4i Solution a) -3 + 7i The conjugate is -3 - 7i. b) 14 - 5i The conjugate is 14 + 5i. c) 4i The conjugate is -4i. Note that 4i = 0 + 4i. The product of a complex number and its conjugate is a real number. Example 5 Multiply: (5 + 7i) (5 - 7i). Solution (5 + 7i) (5 - 7i) = 52 - (7i)2 Using (A + B) (A - B) = AZ - BZ =25-49iZ = 25 - 49(-1) i2 = -1 =25+49=74 Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators in Section 7.5. Example 6 Divide and simplify to the form a + bi. a) -5 + 9i 1 - 2i b) 7+3i 51 page 462 CHAPTER 7 EXPONENTS AND RADICALS Conjugates and Division Conjugates of complex numbers are defined as follows. Conjugate of a Complex Number The conjugate of a complex number a + bi is a - bi, and the conjugate of a - bi is a + bi. a) -3 + 7i b) 14-5i c) 4i Solution a) -3 + 7i The conjugate is -3 - 7i. b) 14 - 5i The conjugate is 14 + 5i. c) 4i The conjugate is -4i. Note that 4i = 0 + 4i. The product of a complex number and its conjugate is a real number. E x a m p I e 5 Multiply: (5 + 7i) (5 - 7i). Solution (5 + 7i) (5 - 7i) = 52 - (7i)2 Using (A + B) (A - B) = A2 - B2 =25-49iZ = 25 - 49(-1) iz = -1 =25 + 49 = 74 Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators in Section 7.5. E x a m p / e 6 Divide and simplify to the form a + bi. a) -5 + 9i 1 - 2i `i b) 7 + 3i 5i 7.8 THE COMPLEX NUMBERS page 463 SO I U t10Y1 a) To divide and simplify (-5 + 9i)/(1 - 2i), we multiply by 1, using the con- jugate of the denominator to form 1: -5 + 9i _ -5 + 9i 1 + 2i Multiplying by 1 using the conjugate of 1 - 2i 1 - 2i . I + 2i the denominator in the symbol for 1 _ (-5 + 9i) (1 + 2i) Multiplying numerators; (1 - 2i) (1 + 2i) multiplying denominators b) When the denominator is a pure imaginary number, it is easiest if we mul- tiply by i/i: 7 + 3i _ 7 + 3i i Multiplying by 1 using i/i. We can 5i 5i . i also use the conjugate of 5i to write -5i/(-5i). Powers of i Answers to problems involving complex numbers are generally written in the form a + bi. In the following discussion, we show why there is no need to use powers of i (other than 1) when writing answers. Recall that -1 raised to an even power is 1, and -1 raised to an odd power is -1. Simplifying powers of i can then be done by using the fact that iz = -1 and expressing the given power of i in terms of iz. Consider the following: _ 7i + 3i2 5,12 _ -5 - 10i + 9i + 18i2 lz - 4i2 Using FOIL _-5-i-18 _-23-i 5 23 1 . =-5 - 5t Writing in the form a + bi Multiplying - 7i + 3(-1) iz - -1 5(-1) 7i - 3 _ -5 -3 7 3 7 _ =5 + 5 i, or 5 - 5 i. i, or v- 1, i2 = -1, i3 = iz . i = (-1)i = -i, i4 = (iz)a = (-1)a = 1, i5 = i4 . i = (iZ)Z . i = (-1)z . i = i, ~ The pattern is now repeating. i6 = (lZ)3 = (-1)3 = -1. 464 CHAPTER 7 EXPONENTS AND RADICALS Note that the powers of i cycle themselves through the values i, -1, -i, and 1 and that even powers of i are -1 or 1 whereas odd powers of i are i or -i. Example Simplify: (a) i18; (b) iz4; (c) iz9; (d) i75. Solution a) iia = (iz)s Using the power rule -1 to an odd power is -1 Using the power rule _ (-1)lz = 1 -1 to an even power is 1 _ (-1)9 = -1 b) i24 = (i2)i2 c) i29 = izsil Using the product rule. This is a key step when i is raised to an odd power. _ (i2)t4i _ (-1)14i =1~i=i Using the power rule Using the product rule Using the power rule 3-2i 5-2i page 480 Chapter 7 7.7 GEOMETRIC APPLICATIONS page Example 13 Free falling objects. The formula s = 16t2 is used to appro>dmate the dis-~ tance s, in feet, that an object falls freely from rest in t seconds. The RCA Building in New York City is 850 ft tall. How long will it take an object to fall from the top? Solution 1. Familiarize. We make a drawing to help visualize the problem. 2. Translate. We substitute into the formula: s = 16t2 850 = 16t2. 3. Carry out. We solve for t: 850 = 16t2 sso = t2 ls 53.125 = t2 53.125 = t Using the principle of square roots; rejecting the negative square root since t cannot be negative in this problem 7.3 ~ t. Using a calculator to approximate the square root and rounding to the nearest tenth 4. Check. Since 16(7.3)2 = 852.64 ~ 850, our answer checks. 5. State. It takes about 7.3 sec for an object to fall freely from the top of the RCA Building. s = 16r2 8.I QUADRATIC EQUATIONS page 481 As we saw in Section 5.8, a grapher can be used to find y = x2 - 8x - 7 approximate solutions of any quadratic equation that has 25 real-number solutions. To check Example 10(a), we graph y = xZ - 8x - 7 and use the ZERO or ROOT option of the CALC menu. When asked for a Left and Right Bound, we enter cursor positions to the left of and to the right of the root. A Guess between the bounds is entered and a value for the root then appears. -25 Zero Yscl = 5 x = -.7958315, y = 0 Section 8.1, page 481, Problems 12, 16, 50, 70 2 is upper small size numbers. 12. (x + 1)2 = -9 16. x2 - 10x + 25 = 64 page 482 CHAPTER 8 QUADRATIC FUNCTIONS AND EQUATIONS 50. x2 + 23 = 10x 70. Reaching 745 ft above the water, the towers of Cali- fornia's Golden Gate Bridge are the world's tallest bridge towers (Source: The Guinness Book of Records). How long would it take an object to fall freely from the top? page 482 CHAPTER 8 QUADRATIC FUNCTIONS AND EQUATIONS 43. x + 8x + 7 = 0 44. x + lOx + 9 = 0 45. x2 - lOx + 21 = 0 page 484 CHAPTER 8 QUADRATIC FUNCTIONS AND EQUATIONS 2 2 x2 + a x + 4 2 = - a + 4 2 Adding 4aZ to complete the square b lz - __4ac _bz x + 2a J 4az + 4az C blz-bz-4ac x + 2a/l 4az Factoring on the left side; finding a common denominator on the right side b bz - 4ac x+2a=± 2a -b ± bz - 4ac x= . 2a Using the principle of square roots and the quotient rule for radicals; since a > 0, 4a2 = 2a Adding - 2a to both sides It is important that you remember the quadratic formula and know how to use it. The Quadratic Formula The solutions of ~2 + bx + c = 0, a ~ 0, are given by x= -b ± bz - 4ac 2a E x p m p 1 e 1 Solve 5xz + 8x = -3 using the quadratic formula. Solution We first find standard form and determine a, b, and c: 5xz + 8x + 3 = 0; Adding 3 to both sides to get 0 on one side a=5~ b=8~ c=3. Next, we use the quadratic formula: x= -b ± b2 - 4ac 2a -g± gz-4.5.g x = 2 , 5 Substituting _ -8 ± 64 - 60 ~- Be sure to write the fraction x - 10 bar all the way across. -8±~ -8±2 x= _ lo la -s+a -s-a x = 10 or x = 10 page 484 CHAPTER 8 QUADRATIC FUNCTIONS AND EQUATIONS Solve by completing the square. Show your work. 54. 3xZ + 5x - 2 = 0 55.4xZ+8x+3=0 56.9x2+18x+8=0 57. 6xZ - x = 15 x2 + a x + 4 z = - a + 4 2 Adding 4aZ to complete the square b lz - __4ac _bz x + 2a J 4az + 4az C blz-bz-4ac x + 2a/l 4az Factoring on the left side; finding a common denominator on the right side b bz - 4ac x+2a=± 2a -b ± bz - 4ac x= . 2a Using the principle of square roots and the quotient rule for radicals; since a > 0, 4a2 = 2a Adding - 2a to both sides It is important that you remember the quadratic formula and know how to use it. The Quadratic Formula The solutions of ~2 + bx + c = 0, a ~ 0, are given by -b ± bz - 4ac 2a E x p m p 1 e 1 Solve 5xz + 8x = -3 using the quadratic formula. Solution We first find standard form and determine a, b, and c: 5xz + 8x + 3 = 0; Adding 3 to both sides to get 0 on one side a=5~ b=8~ c=3. Next, we use the quadratic formula: x= -b ± b2 - 4ac 2a -g± gz-4.5.g x = 2 , 5 Substituting _ -8 ± 64 - 60 ~- Be sure to write the fraction x - 10 bar all the way across. -8±~ -8±2 x= _ lo la -s+a -s-a x = 10 or x = 10 Section 8.2, page 488, Problems 16, 26, 38, 46 page 488 16. 15t2 + 7t = 2 26. x2 + 7 = 3x 2 after x is small size upper number. 38. x2 + 6x + 4 = 0 46. Donuts South Street Bakers charges $1.10 for a cream-filled donut and 85 cents for a glazed donut. On a recent Sunday a total of 90 glazed and cream- filled donuts were sold for $88.00. How many of each typee were sold? page 493 Example 4 Falling distance A object tossed downward with a initial speed velocity of v will travel a distance of s meters where s = 4.9t2 + v t and t is measured in sconds. Solve for t. Solution Since t is squared in one term and raised to the first power in the other term the equation is quadratic in t. 4.9t2 + v t = s 4.9t2 + v^ t - s = 0 a = 4.9, b = v^ c= -s _______________ t= -v^ + Vv2 - 4(4.9)(-s) 2(4.9) Using the quadratic formula. Since the negative square root would yield a negative value for t we use only the positive root. t= -v^ + Vv2 + 19.6s 9.8 page 494 Chapter 8 Section 8.3, page 494, Problems 4, 8, 34 4. Car trips Petra's Plymouth travels 200 mi averag- ing a certain speed. If the car had gone 10 mph faster the trip would have taken 1 hr less. Find Petra's average speed. 8. Car speed On a sales trip Gail drives the 600 mi to Richmond averaging a certain speed. The return trip is made at a average speed that is 10 mph slower. Total time for the round trip is 22 hr. Find Gail's average speed on each part of the trip. 34. Falling distance a) A object is dropped 75 m from a airplane. How long does it take the object to reach the ground? b) A object is thrown downward with a initial velocity of 30 m/sec from a plane 75 m above the ground. How long does it take the object to reach the ground? c)How far will a object fall in 2 sec if thrown downward at a initial velocity of 30 m/sec? page 501 Section 8.4, page 501, Problems 14, 28, 62 2 upper small size numbers. 14. 10x2 - x - 2 = 0 quadratic equation. 28. -1,-3 62. x2 - kx + 2 = 0 one solution is 1 + i. page 507 Chapter 8 Section 8.5, page 507, Problems 8, 18, 32, 54 _ 8. x - 2Vx - 6 = 0 18. w2/3 - 2w1/3 - 8 = 0 32. f(x) = x1/2 - x1/4 - 6 54. x6 + 7x3 - 8 = 0 6 3 after x is small upper size numbers. page 509 Chapter 8 Quadratic functions and their graphs x f(x) = x2 (x,f(x)) -3 9 (-3,9) -2 4 (-2,4) -1 1 (-1,1) 0 0 (0,0) 1 1 (1,1) 2 4 (2,4) 3 9 (3,9) x h(x) =2x2 -3 18 -2 8 -1 2 0 0 1 2 2 8 3 18 page 515 Chapter 8 Section 8.6, page 515, Problems 34, 43 For each of the following graph the function and find the vertex the axis of symmetry and the maximum value or the minimum value. 34. f(x) = (x - 1)2 + 2 35. g(x) = (x + 4)2 + 1 Without graphing find the vertex the axis of symmetry and the maximum value or the minimum value. 43. f(x) =8(x - 9)2 + 7 2 after ) is small size numbers. page 521 a graph can look like y 6 5 1 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -3 Find the x- and y-intercepts of the graph of f(x)= x2 - 2x - 2. Solution The y-intercept is simply (0,f(0)) or (0,-2). To find the x-intercepts we solve the equation 0 = x2 - 2x - 2 x = 1 + V3 1 - V3,0 and 1 + V3, 0. If graphing would get -0.7,0 and 2.7,0. page 522 Chapter 8 Section 8.7, page 522, Problems 2, 24 For each quadratic function a find the vertex and the axis of symmetry and b graph the function. 2. f(x) = x2 + 2x - 5 24. h(x) = 1x2 - 3x + 2 2 page 526 Chapter 8 Quadratic Functions and Equations Example 3 Hydrology. a river depths D in feet of a river at distances x in feet from one bank. x = distance from left bank in feet D(x)= depth of river in feet. Distance x Depth D from left bank of the river in feet. 0 0 15 10.2 25 17 50 20 90 7.2 100 0 page 528 Chapter 8 Section 8.8, page 528, Problems 2, 8, 46, 48 2. Minimizing cost Aki's Bicycle Designs has deter- mined that when x hundred bicycles are built the average cost per bicycle is given by c(x) = 0.1x2 - 0.7x + 2.425 2 after x is small size upper number. where C(x) is in hundreds of dollars. What is the minimum average cost per bicycle and how many bicycles should be built to achieve that minimum? 8. Patio design A stone mason has enough stones to enclose a rectangular patio with 60 ft of perimeter assuming that the attached house forms one side of the rectangle. What is the maximum area that the mason can enclose? What should the dimen- sions of the patio be in order to yield this area? 46. Cover charges When the owner of Sweet Sounds charges a $10 cover charge a average of 80 people will attend a show. For each 25 cents increase in admis- sion price the average number attending decreases by 1. What should the owner charge in order to make the most money? 48. Bridge design The cables supporting a straight line suspension bridge are nearly parabolic in shape. Suppose that a suspension bridge is being designed with concrete supports 160 ft apart and with vertical cables 30 ft above road level at the midpoint of the bridge and 80 ft above road level at a point 50 ft from the midpoint of the bridge. How long are the longest vertical cables? 160 ft <___> page 539 Chapter 8 Section 8.9, page 539, Problems 12, 52 12. x2 + 6x > -8 2 after x is small size upper number. 52. Height of a thrown object The function S(t) = -16t2 + 32t + 1920 2 is small size upper number after t. gives the height S in feet of a object thrown from a cliff that is 1920 ft high. Here t is the time in sec- onds that the object is in the air. a) For what times does the height exceed 1920 ft? b) For what times is the height less than 640 ft? page 558 Chapter 8 Section 9.1, page 558, Problems 2, 20, 56 Find (f g)(1),(g f)(1),(f g)(x) and (g f)(x) a small 0 between f g g f f g g f 2. f(x) = 2x + 1; g(x)= x2 - 5 Determine whether each function is one to one. 20. f(x) = 1 - x2 2 aftere x is small size upper corner number. Graph each function and its inverse using the same set of axes. 56. f(x) = _1x + 1 2 page 567 Chapter 9 Section 9.2, page 567, Problems 28, 38 Graph 28. x = (4)y 3 Solve. 38. Salvage value A photocopier is purchased for $5200. Its value each year is about 80% of the value of the preceding year. Its value in dollars after t years is given by the exponential function v(t) = 5200(0.8)t. a) Find the value of the machine after 0 yr. 1 yr. 2 yr. 5yr. and 10 yr. b) Graph the function. page 572 Chapter 9 Example 3 Graph y = f(x) = log5 x Solution If y = log5 x then 5y = x. 5 after log is small number. 0 1 2 after -1 -2 are all small upper size numbers. For y = 0,x = 5 0 = 1. For y = 1,x = 5 1 = 5. For y = 2,x = 5 2 = 25. For y = -1,x = 5 -1 = 1 5 For y = -2,x = 5 -2 = 1 25. log5 1 = 0 log5 5 = 1 log5 25 = 2 log 1 = -1 2 Example 7 Solve (a) log10 1000 = x (b) log4 1 = t Solution a) Convert log10 1000 = x to exponential form and solve. 10x = 1000 10x = 10 3 x = 3. solution is 3. b) convert log 4 1 = t to exponential form and solve. 4t = 1 4t = 4 0 t = 0. Solution is 0. Example 5 y = log 3 5 Solution is 3y = 5. page 575 Section 9.3, page 575, Problems 2, 26, 46, 62, 86 numbers is small size upper number after log. 2. log10 1000 solution answer is 3. 26. log 27 9 solution answer is 9 27. 46. 10 0.3010 = 2 solution answer is 62. log10 3 = 0.4771 86. log8 x = 2 3 page 580 Example 9 a) 1 logax - 7 loga y + loga z 2 Solution _ = log a Zvx y7 page 577 Example 1 Express as a sum of logarithms log2 (4 * 16) Solution log2 (4 * 16) = log2 4 + log2 16. log2 (4 * 16) = log2 64 = 6 2 6 = 24 2 6 = 64 6 is small upper number. 6 after 2 is small upper number.2 after log. log2 4 + log2 16 = 2 + 4 = 6. 2 2 = 4 and 2 4 = 16. 2 3 after 2 2 small. page 582 Chapter 9 Section 9.4, page 582, Problems 4, 30, 46, 56 4. log5 (25 * 125) 30. logh x2 y5 w4 z7 46. loga (2x + 10) - loga (x2 - 25) 56. logh 45 page 586 Example 3 use a calculator to find in 4568. Solution We enter 4568 then press In. Find that In 4568 = 8.4268 page 587 Example 6 find log4 31 Solution log4 31 = loge 31 substituting into logbM=logaM loge 4 loga b =In 31 = 3.433987204 In4 = 1.386294361 use In twice. = 2.4771. use a calculator. 4 2.4771=31. page 589 Section 9.5, page 589, Problems 24, 36 24. In 1900 0.07 36. log5 42 page 595 Section 9.6, page 595, Problems 2, 30, 50 2. 2x = 8 x is size upper letter. 30. log3 x = 4 3 is small size upper number. 50. log6 (x + 3) + log6 (x + 2) = log6 20 6 after log is small size upper numbers. Section 9.7, page 597, Problems 2, 8, 14, 16, 18, 22 only examples appeared on page 597. page 623 A Vertex is (11,5) 2 2 we find the vertex by first computing its y-coordinate -b/(2a) then substituting to find the x-coordinate. y = _b = _ 10 = 5 2a 2(-2) 2 x = -2y2 + 10y - 7 = -2(5)2 + 10(5) - 7 2 2 =11 2 page 624 The lengths of the legs are |x2 - x1| and |y2 - y1|. We find d the length of the hypotenuse by using the Pythagorean theorem d2 = |x2 - x1|2 + |y2 - y1|2. The Distance Formula The distance d between any two points x1,y1 and x2,y2 is given by ________________________ d = V(x2 - x1)2 + (y2 - y1)2. page 626 Chapter 10 Equation of a circle The equation of a circle centered at (h,k) with radius r is given by (x - h)2 + (y - k)2 = r2. 2 numbers are all small size numbers. Example 6 Find a equation of the circle having center (4,5) and radius 6. Solution Using the standard form we obtain (x - 4)2 + (y - 5)2 = 6 2 Using (x - h)2 + (y - k)2 = r2. or (x - 4)2 + (y - 5)2 = 36. Example 4 Find the distance between (5,-1) and (-4,6). Find a exact answer and a approximation to three decimal places. Solution We substitute into the distance formula ______________________ d = V(-4 -5)2 + [6 - (-1)]2 ___________ = v(-9)2 + 7 2 ___ = V130 = 11.402 The distance formula is needed to develop the formula for a circle. page 628 Chapter 10 Section 10.1, page 628, Problems 14, 22, 28, 44, 54, 72 Graph label each vertex. 14. y = x2 + 2x + 1 Find distances between each pair of points. find approximation to three decimal places. where appropriate. 22.(1, 10) and (7, 2) 28. (5.9, 2) and (3.7, -7.7) 44. (4.1, 6.9) and (5.2, -6.9) _ 54. Center (-2, 7) radius 2V5 72. x2 + y2 + 6x - 4y - 15 = 0 2 after x y is small upper size numbers. page 633 Chapter 10 Using a and b to Graph a Ellipse For the ellipse x2 + y2 = 1 a2 b2 the x-intercepts are -a,0 and a,0. the y-intercepts are 0, -b and 0,b. page 635 Example 3 Graph the ellipse (x - 1)2 (y + 5)2 4 + 9 = 1 Solution (x - 1)2 + (y + 5)2 = (x - 1)2 + (y + 5)2 4 9 2 2 3 2 2 after 3 2 are small size 2. (x - 1)2 + (y + 5)2 = (x - 1)2 + (y - (-5))2 2 2 3 2 2 2 3 2 second 2s are small size upper numbers. (3,-5) (-1,-5), (1,-2) and (1,-8) page 636 Chapter 10 Section 10.2, page 636, Problems 2, 22, 38 Graph each of the following equations. 2. x2 + y2 = 1 4 1 22. (x + 5)2 + (y - 2)2 = 1 4 36 38. (-7,0),(7,0),(0,-5), and (0,5) page 642 Chapter 10 Hyperbolas Nonstandard form The equations for hyperbolas are the standard ones there are other hyperbolas. Equation of a Hyperbola in Nonstandard Form. Hyperbolas having the x- and y- axes as asymptotes have equations as follows. xy = c where c is a nonzero constant. Example 3 Graph xy = -8. Solution We first solve for y _8 y = x Dividing both sides by x. Note that x =/ 0. x y 2 -4 -2 4 4 -2 -4 2 1 -8 -1 8 8 -1 -8 1 xy = -8. y 9 8 7 6 5 4 3 2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 x -1 -2 -9 page 644 Parabola equation y = ax2 + bx + c a > o y = ax2 + bx + c a < 0 = a(x - h)2 + k = a(x - h)2 +k graph 1 graph 2 Circle Center at the origin Center at (h,k) x2 + y2 = r2 (x - h)2 + (y - k)2 = r2 page 645 Hyperbola Center at the origin x2 _ y2 y2 _ x2 a2 b2 = 1 b2 a2 = 1 xy = c c > 0 xy = c c < 0 Center at (h,k)* (x - h)2 _ (y - k)2 = 1 (y - k)2 _ (x - h)2 = 1 a2 b2 b2 a2 (h,k) (h - a,k) (h + a,k) Ellipse Center at the origin Center at (h,k) x2 _ y2 = 1 (x - h)2 + (y - k)2 = 1 a b2 a2 b2 Graph 1 shows Graph 2 shows y (o,b) y (h,k + b) (-a, 0) (a, 0) xaxis (h - a,k) (h,k) (h + a, k) (0,-b) (h,k - b) Example 4 a) 5x2 = 20 - 5y2 Solution 5x2 + 5y2 = 20 adding 5y2. 5(x2 + y2) = 20 Factoring out 5. x2 + y2 = 4 Dividing both sides by 5. x2 + y2 = 2 2 This is a equation for a circle. 2 after 2 x y are small size numbers. page 647 Chapter 10 Section 10.3, page 647, Problems 4, 22, 26, 32, 54 Graph each hyperbola. Label all vertices sketch all asymptotes. 4. y2 _ x2 = 1 16 9 Classify each of the following as the equation of a circle, a ellipse, a parabola, or a hyperbola. 22. 1 + 3y = 2y2 - x 2 after y small upper size number. 26. 2y + 13 + x2 = 8x - y2 32. x2 = 16 + y2 2 after x and y small size upper number. hyperbola complete the square write standard form. Find the center, vertices, the asymptotes. Then graph the hyperbola. 54. 25(x - 4)2 - 4(y - 5)2 = 100 page 655 Chapter 10 Section 10.4, page 655, Problems 2, 42 Solve. Graphs can be used to confirm all real solutions. 2. x2 + y2 = 100, y - x = 2 2 after x and y are small size upper numbers. 42. Geometry A rectangle has a area of 2 yd2 and a perimeter of 6 yd. Find its dimensions. 2 after yd is small size upper number. page 663 Chapter 11 The notation a n is the same as a(n). a1 = 2 1 = 2 a2 = 2 2 = 4 a3 = 23 = 8 a6 = 226 = 64 The 1 2 after 3 2 are small size numbers. or 2,4,8 2n Example 1 Solution a1 = (-1)1 = _1 1 + 1 = 2 a2 = (-1)2 = 1 2 + 1 = 3 a3 = (-1)3 = -1 3 + 1 = 4 57+ 1 = 58 Example 2 Sequence predict the general term. a) 1, 4, 9, 16, 25,... c)2, 4, 8,... Soulution a) 1, 4, 9, 16, 25,... These are squares of consecutive positive integers term is n2. c) 2, 4, 8,... The pattern as powers of 2 16 would be the next term. 2n the general term. Could be written 2,4,8,16,32,64,128,... or written 2,4,8,14,22,32,44,58,... page 666 Chapter 11 Section 11.1, page 666, Problems 8, 24, 46, 54 the nth term of a sequence is given. find the first 4 terms the 10th term, a10 and the 15th term a15. 8. an = (_1)n-1 2 Look for a pattern then predict the general term or nth term an of of each sequence. 24. 2,4,6,8.... Write out and evaluate each sum. 46. 4 k - 2 E k + 3 k=1 Rewrite each sum using sigma notation. 54. 3 + 6 + 9 + 12 + 15 page 670 Example 3 For the sequence in example 2 which term is 300? Find n if a n = 300. Solution We substitute into the formula for the nth term of a arithmetic sequence and solve for n. a n = a1 + (n - 1)d 300 = 6 + (n - 1) * 3 300 = 6 + 3n - 3 297 = 3n 99 = n. The term 300 is the 99th term of the sequence. page 672 Example 5 Find the sum of the first 100 positive even numbers. Solution The sum is 2 + 4 + 6 + ... + 198 + 200. a1 = 2, n = 100, and a n = 200. s n = n (a1 + a n) 2 s100 = 100(2 + 200) 2 =50(202) = 10,100. page 675 Chapter 11 Section 11.2, page 675, Problems 4, 20, 32, 42 4. -9, -6, -3,0,... 20. Find a20 when a1 = 14 and d = -3. 32. Find the sum of the first 400 natural numbers. 42. Accumulated savings If 10 cents is saved on October 1, another 20 cents on October 2, another 30 cents on October 3 and so how much is saved during October? (October has 31 days.) page 678 Chapter 11 Example 1 For each geometric sequence find the common ratio. a) 3, 6, 12, 24, 48,... b) 3, -6, 12, -24, 48, -96,... c) $5200, $3900 $2925 $2193.75.... Solution Common Sequence Ratio 6 12 a) 3, 6, 12, 24, 48,... 2 3 = 2 6 = 2 and so on. -6 12 b) 3, -6, 12, -24, 48, -96,... -2 3 = -2 -6 = -2 and so on. $3900 $29.25 c) $5200 $3900, $2925, $2193.75,... 0.75 $5200 = 0.75 $39.00 = 0.75 To develop a formula for the general or nth term of a geometric sequence let a1 be the first term and let r be the common ratio. We write out the first few terms as follows a1 a2 = a1r a3 = a2r = (a1r)r = a1r2 a4 = a3r = (a1r2)r = a1r3. 2 3 after r are small size upper numbers. Generalizing we obtain the following To find an for a Geometric Sequence n after a is small numberlower. The nth term of a geometric sequence with common ratio r is given by a n = a1rn-1, for any integer n_> 1. page 679 Find the common ratio we can divide any term other than the first by the term preceding it. Since the second term is 20 and the first is 4. 20 r = 4 * or 5 The formula a n = a1rn-1 gives us a7 = 4 * 5 7-1 = 4 * 5 6 = 4 * 15,625 = 62,500. page 684 Chapter 11 Section 11.3, page 684, Problems 2, 12, 26, 56 Find the common ratio for each geometric sequence. 2. 2, 16, 18, 54,... Find the indicated term for each geometric sequence. 12. 2,8,32,...;the 9th term use the formula for Sn to find the indicated sum. 26. s6 for the geometric series 16 - 8 + 4 - ... 56. Shrinking population A population of 5000 fruit flies is dying off at a rate of 4% per minute. How many flies will be alive after 15 min? page 690 Chapter 11 Example 11 Expand (u - v)5. Solution Using the binomial theorem we have a = u, b= -v and n = 5. We use the 6th row of Pascal's triangle 1 5 10 10 5 1. (u - v)5 = [u +(-v)]5 as a sum u - v =1(u)5 + 5(u)4(-v)1 + 10(u)3(-v)2 + 10(u)2(-v)3 + 5(u)1(-v)4 + 1(-v)5 = u5 - 5u4v + 10u3v2 - 10u2v3 + 5uv4 - v5. 5 4 3 2 numbers after letters are all upper small size numbers. page 692 Chapter 11 Example 3 Simplify 8! 5! 3! Solution 8! = 8 * 7 * 6 * 5! = 8 * 7 Removing a factor equal to 1. 6 * 5 = 1 5! 3! 5! * 3 * 2 * 1 5! * 3 * 2 =56. Example 4 Simplify (a) (7) 2 Solution (a) (7) = 7! 2 (7 - 2)!2! = 7! = 7 * 6 * 5! = 7 * 6 5!2! = 5! * 2 * 1 = 2 =7 * 3 = 21 page 694 Chapter 11 Section 11.4, page 694, Problems 4, 14, 30, 36, 56 Simplify. 4. 11! 14. (30) 3 Expand. Use both of the methods shown in this section. 30. (x3 - 2y)5 Find the indicated term for binomial expression. 36. 6th, (x + y)7 56. Baseball In reference to Exercise 54 the probabil- ity that Jeter will get at most 3 hits is found by adding the last 4 terms of the binomial expansion of (0.325 + 0.675)5. Find these terms and use a cal- culator to estimate the probability. page 555 Example 8 Consider g(x) = x3 + 2. a) Determine whether the function is one to one. Solution a) The graph of g(x)= x3 + 2 is shown below. It passes the horizontal line test and has a inverse. y 12 g(x) = x3 + 2 10 8 6 4 -12 -10 -8 -6 -4 2 4 6 8 10 12 x -4 -6 -8 -10 -12 line starts at -12 ends at 12 on top. Example 4 E (-1)k(2k) k=4 Solution 6 (-1)k(2k) = (-1)4(2 * 4) + (-1)5(2 * 5) + (-1)6(2 * 6) E k=4 = 8 - 10 + 12 = 10 From: robrain To: Subject: hi Date: 01/07/04 05:47pm Attachments: ROB1.JPG - 8.0KB (Displayed) Hi I received your note and I have attached a picture of me for you. I live in Kirkland near Seattle in U.S.A. I like music, and once in awhile travel. from Robert. robrain@blarg.net From: robrain To: Subject: hi Date: 1 10 2004 11:45am Hi I originally was from Manhattan, New York. I currently live in Kirkland near Seattle in Washington on the west coast. I like to learn more about more new things it becomes interesting sometimes. I never been married before. I live by myself for now. I have some college education. from, Robert Alberg. robrain@blarg.net
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