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Solution Manual for Introductory and Intermediate Algebra for College Students, 6th Edition

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    ’ S S OLUTIONS M ANUAL D ANIEL S. M ILLER Niagara County Community College I NTRODUCTORY & I NTERMEDIATE A LGEBRA FOR C OLLEGE S TUDENTS S IXTH E DITION Robert Blitzer Miami Dade College

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    iii TABLE OF CONTENTS for SOLUTIONS I NTRODUCTORY AND I NTERMEDIATE A LGEBRA FOR C OLLEGE S TUDENTS 6 E Chapter 1 Variables, Real Numbers, and Mathematical Models ................................................ 1 Chapter 2 Linear Equations and Inequalities in One Variable .................................................. 59 Chapter 3 Linear Equations in Two Variables ........................................................................ 153 Chapter 4 Systems of Linear Equations .................................................................................. 221 Chapter 5 Exponents and Polynomials .................................................................................... 325 Chapter 6 Factoring Polynomials ............................................................................................ 407 Chapter 7 Rational Expressions .............................................................................................. 469 Mid-Textbook Check Point .................................................................................... 605 Chapter 8 Basics of Functions ................................................................................................. 611 Chapter 9 Inequalities and Problem Solving ........................................................................... 653 Chapter 10 Radicals, Radical Functions, and Rational Exponents ........................................... 719 Chapter 11 Quadratic Equations and Functions ........................................................................ 809 Chapter 12 Exponential and Logarithmic Functions ................................................................. 973 Chapter 13 Conic Sections and Systems of Nonlinear Equations ........................................... 1055 Chapter 14 Sequences, Series, and the Binomial Theorem ..................................................... 1159 Appendices .............................................................................................................................. 1235

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 1 1.1 Check Points 1. a. 6 2 6 2(10) 26 x + = + = b. 2( 6) 2(10 6) 32 x + = + = 2. a. 7 2 7 3 2 8 21 16 37 x y + = ⋅ + ⋅ = + = b. 6 6 3 8 10 2 2 8 2 8 3 8 5 x y y x − ⋅ − = = = − − ⋅ − − 3. a. 6 x b. 4 x + c. 3 5 x + d. 12 2 x − e. 15 x 4. a. 9 3 42 9(6) 3 42 54 3 42 51 42, false x − = − = − = = 6 is not a solution. b. 2( 3) 5 3 2(3 3) 5(3) 3 2(6) 15 3 12 12, true y y + = − + = − = − = 3 is a solution. 5. a. 5 6 x = b. 7 2 1 x − = 6. a. 4 5 4(15) 5 65 d n d = + = + = 65% of marriages end in divorce after 15 years when the wife is under 18 at the time of marriage. b. According to the line graph, 60% of marriages end in divorce after 15 years when the wife is under 18 at the time of marriage. c. The mathematical model overestimates the actual percentage shown in the graph by 5%. 1.1 Concept and Vocabulary Check 1. variable 2. expression 3. substituting; evaluating 4. equation; solution 5. formula; modeling; models 1.1 Exercise Set 1. 8 4 8 12 x + = + = 2. 10 4 10 14 x + = + = 3. 12 12 4 8 x − = − = 4. 16 16 4 12 x − = − = 5. 5 5 4 20 x = ⋅ = 6. 6 6 4 24 x = ⋅ = 7. 28 28 7 4 x = = 8. 36 36 9 4 x = = 9. 5 3 5 3 4 5 12 17 x + = + ⋅ = + = 10. 3 5 3 5 4 3 20 23 x + = + ⋅ = + = 11. 2( 5) 2(4 5) 2(9) 18 x + = + = = 12. 5( 3) 5(4 3) 5(7) 35 x + = + = = 13. 12 8 12 4 8 48 8 40 5 2 2 4 8 8 x x − ⋅ − − = = = = ⋅ 14. 5 52 5 4 52 20 52 72 6 3 3 4 12 12 x x + ⋅ + + = = = = ⋅ 15. 2 2 7 5 14 5 19 x y + = ⋅ + = + = 16. 3 3 7 5 21 5 26 x y + = ⋅ + = + = 17. 2( ) 2(7 5) 2(12) 24 x y + = + = =

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 2 18. 3( ) 3(7 5) 3(12) 36 x y + = + = = 19. 4 3 4 7 3 5 28 15 13 x y − = ⋅ − ⋅ = − = 20. 5 4 5 7 4 5 35 20 15 x y − = ⋅ − ⋅ = − = 21. 21 35 21 35 3 7 10 7 5 x y + = + = + = 22. 50 14 50 14 10 2 8 5 7 y x − = − = − = 23. 2 6 2 7 5 6 14 5 6 15 5 2 2 5 7 10 7 3 x y y x − + ⋅ − + − + = = = = − ⋅ − − 24. 2 24 2 5 7 24 10 7 24 27 3 2 2 7 5 14 5 9 y x x y − + ⋅ − + − + = = = = − ⋅ − − 25. 4 x + 26. 6 x + 27. 4 x − 28. 6 x − 29. 4 x + 30. 6 x + 31. 9 x − 32. 3 x − 33. 9 x − 34. 3 x − 35. 3 5 x − 36. 5 3 x − 37. 12 1 x − 38. 13 3 x − 39. 10 10 x x + 40. 20 20 x x + 41. 6 30 x + 42. 30 4 x + 43. 14 20 6 14 20 20 20, true x + = + = = The number is a solution. 44. 17 22 5 17 22 22 22, true x + = + = = The number is a solution. 45. 30 10 30 20 10 10 10, true y − = − = = The number is a solution. 46. 50 20 50 30 20 20 20, true y − = − = = The number is a solution. 47. 4 20 4(10) 20 40 20, false z = = = The number is not a solution. 48. 5 30 5(8) 30 40 30, false z = = = The number is not a solution. 49. 8 6 48 8 6 8 8, true r = = = The number is a solution. 50. 7 9 63 7 9 7 7, true r = = = The number is a solution.

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    Section 1.1 Introduction to Algebra: Variables and Mathematical Models 3 51. 4 3 23 4(6) 3 23 24 3 23 27 23, false m + = + = + = = The number is not a solution. 52. 3 4 19 3(6) 4 19 18 4 19 22 19, false m + = + = + = = The number is not a solution. 53. 5 4 2 5 5(3) 4 2(3) 5 15 4 6 5 11 11, true a a − = + − = + − = + = The number is a solution. 54. 5 3 2 6 5(3) 3 2(3) 6 15 3 6 6 12 12, true a a − = + − = + − = + = The number is a solution. 55. 6( 4) 3 6(8 4) 3(8) 6(4) 24 24 24, true p p − = − = = = The number is a solution. 56. 4( 3) 6 4(6 3) 6(6) 4(9) 36 36 36, true p p + = + = = = The number is a solution. 57. 2( 1) 3( 1) 2(7 1) 3(7 1) 2(8) 3(6) 16 18, false w w + = − + = − = = The number is not a solution. 58. 3( 2) 4( 3) 3(10 2) 4(10 3) 3(12) 4(7) 36 28, false w w + = − + = − = = The number is not a solution. 59. 4 28 x = 60. 5 35 x = 61. 14 1 2 x = 62. 1 8 4 x = 63. 20 5 x − = 64. 40 10 x − = 65. 2 6 16 x + = 66. 2 9 29 x + = 67. 3 5 7 x − = 68. 4 3 29 x − = 69. 4 5 33 x + = 70. 6 3 33 x + = 71. 4( 5) 33 x + = 72. 6( 3) 33 x + = 73. 5 24 x x = − 74. 4 25 x x = − 75. First find x . 7 2 7(5) 2 37 x y x = + = + = Evaluate the expression. 37 5 32 8 4 4 4 x y − − = = = 76. First find x . 5 2 5(4) 2 22 x y x = + = + = Evaluate the expression. 22 4 18 6 3 3 3 x y − − = = =

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 4 77. First find x . 1 4 12 1 3 1 2 4 y x x = − = − = − = Evaluate the expression. 4 3( 5) 4(2) 3(12 5) 8 3(17) 8 51 59 x y + + = + + = + = + = 78. First find x . 1 3 15 1 5 1 4 3 y x x = − = − = − = Evaluate the expression. 3 4( 6) 3(4) 4(15 6) 12 4(21) 12 84 96 x y + + = + + = + = + = 79. a. 2( 3 ) 2(4 3 1) 2(7) 14 x y + = + ⋅ = = b. 5 30 40 5(14) 30 40 70 30 40 40 40, true z − = − = − = = Yes, it is a solution. 80. a. 3(2 ) 3(2 1 5) 3(7) 21 x y + = ⋅ + = = b. 4 30 54 4(21) 30 54 84 30 54 54 54, true z − = − = − = = Yes, it is a solution. 81. a. 6 2 6 3 2 6 18 12 6 x y − = ⋅ − ⋅ = − = b. 7 45 2 7(6) 45 2(6) 42 45 12 42 33, false w w = − = − = − = No, it is not a solution. 82. a. 1 5 14 5 3 14 15 7 8 2 x y − = ⋅ − ⋅ = − = b. 4 54 5 4(8) 54 5(8) 32 54 40 32 14, false w w = − = − = − = No, it is not a solution. 83. a. 0.16 2.83 0.16(10) 2.83 4.43 T n T = + = + = According to the formula, the average price in 1990 was $4.43. The model overestimates the actual amount shown in the bar graph by $0.20. b. 0.16 2.83 0.16(30) 2.83 7.63 T n T = + = + = According to the formula, the average price in 2010 was $7.63. The bar graph shows a price in 2010 of $7.89. $7.89 $7.63 $0.26 − = The model underestimates the actual amount shown in the bar graph by $0.26. 84. a. 0.16 2.83 0.16(5) 2.83 3.63 T n T = + = + = According to the formula, the average price in 1985 was $3.63. The model overestimates the actual amount shown in the bar graph by $0.08. b. 0.16 2.83 0.16(25) 2.83 6.83 T n T = + = + = According to the formula, the average price in 2005 was $6.83. The bar graph shows a price in 2005 of $6.41. $6.83 $6.41 $0.42 − = The model overestimates the actual amount shown in the bar graph by $0.42. 85. 52 0.3 52 0.3(24) 44.8 p a p = − = − = According to the formula, 44.8% of 24-year-olds say that marriage is obsolete. The model overestimates the actual percentage shown in the bar graph by 0.8%.

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    Section 1.2 Fractions in Algebra 5 86. 52 0.3 52 0.3(66) 32.2 p a p = − = − = According to the formula, 32.2% of 66-year-olds say that marriage is obsolete. The model overestimates the actual percentage shown in the bar graph by 0.2%. 87. a. 0.8(200 ) 0.8(200 145) 44 H A H = − = − = A bowler with an average score of 145 will have a handicap of 44. b. The bowler’s final score will be 120 44, + or 164. 88. a. 0.8(200 ) 0.8(200 165) 28 H A H = − = − = A bowler with an average score of 165 will have a handicap of 28. b. The bowler’s final score will be 140 28, + or 168. 89. – 97. Answers will vary. 98. 0.8(200 ) 0.8(200 200) 0 H A H = − = − = A bowler with an average score of 200 will have no handicap. 99. makes sense 100. does not make sense; Explanations will vary. Sample explanation: 4 7 x + is not an equation. 101. makes sense 102. makes sense 103. true 104. false; Changes to make the statement true will vary. A sample change is: Equations contain the equality symbol, =. 105. true 106. false; Changes to make the statement true will vary. A sample change is: Hard work combined with a willingness to use available learning resources can greatly improve your chances of learning the material in this book. 107. Choices of variables may vary. Let h = hours worked. Let s = salary. 20 s h = 108. Choices of variables may vary. Let w = number of workers. Let t = number of televisions assembled. 10 t w = 109. 3 2 3 2 6 7 5 7 5 35 ⋅ ⋅ = = ⋅ 110. 2 7 2 5 2 5 10 3 5 3 7 3 7 21 ⋅ ÷ = ⋅ = = ⋅ 111. 9 5 9 5 4 17 17 17 17 − − = = 1.2 Check Points 1. 5 2 8 5 16 5 21 2 8 8 8 8 ⋅ + + = = = 2. 5 divided by 3 is 1 with a remainder of 2, so 5 2 1 3 3 = . 3. Begin by selecting any two numbers whose product is 36. Here is one possibility: 36 4 9 = ⋅ Because the factors 4 and 9 are not prime, factor each of these composite numbers. 36 4 9 2 2 3 3 = ⋅ = ⋅ ⋅ ⋅ Notice that 2 and 3 are both prime. The prime factorization of 36 is 2 2 3 3. ⋅ ⋅ ⋅ 4. a. 10 2 5 15 ⋅ = 3 5 ⋅ 2 3 = b. 42 2 24 = 3 ⋅ 7 2 2 2 ⋅ ⋅ ⋅ 3 ⋅ 7 4 = When reducing fractions, it may not be necessary to write prime factorizations. We can use the greatest common factor to reduce this fraction. 42 7 6 24 ⋅ = 4 6 ⋅ 7 4 =

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 6 c. 13 15 ; Because 13 and 15 share no common factors (other than 1), 13 15 is already reduced to its lowest terms. d. 9 1 9 45 ⋅ = 5 9 ⋅ 1 5 = 5. a. 4 2 4 2 8 11 3 11 3 33 ⋅ ⋅ = = ⋅ b. 3 6 3 18 3 6 3 5 1 5 5 5 ⋅ = ⋅ = = c. 3 2 3 2 6 2 3 7 3 7 3 21 ⋅ ⋅ ⋅ = = = ⋅ 7 3 ⋅ 2 7 = Remember that you can divide numerators and denominators by common factors before performing multiplication. 3 2 3 7 3 ⋅ = 2 7 3 ⋅ 2 7 = d. 2 1 17 3 51 1 3 1 5 5 2 5 2 10 10 ⎛ ⎞ ⎛ ⎞ = ⋅ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 6. a. 5 3 5 8 5 4 8 4 3 4 ÷ = ⋅ = 4 ⋅ 2 10 1 3 3 3 3 ⋅ = = b. 2 2 3 2 1 2 3 3 3 1 3 3 9 ÷ = ÷ = ⋅ = c. 3 1 27 9 27 4 9 3 2 8 4 8 4 8 9 ÷ = ÷ = ⋅ = 3 4 ⋅ 4 2 ⋅ ⋅ 9 3 1 1 2 2 = = 7. a. 2 3 5 11 11 11 + = b. 5 1 4 2 6 6 6 3 − = = c. 3 1 27 9 18 9 1 3 1 2 8 8 8 8 8 4 4 − = − = = = 8. 2 2 7 14 3 3 7 21 ⋅ = = ⋅ 9. a. 1 3 1 5 3 2 5 6 11 2 5 2 5 5 2 10 10 10 ⋅ ⋅ + = + = + = ⋅ ⋅ or 1 1 10 b. 4 3 4 4 3 3 16 9 7 3 4 3 4 4 3 12 12 12 ⋅ ⋅ − = − = − = ⋅ ⋅ c. 1 11 19 23 19 2 23 3 1 6 12 6 12 6 2 12 38 23 15 12 12 12 5 1 or 1 4 4 ⋅ − = − = − ⋅ = − = = 10. 10 2 5 = ⋅ 12 2 2 3 = ⋅ ⋅ LCD 2 2 3 5 60 = ⋅ ⋅ ⋅ = 3 7 3 6 7 5 18 35 53 10 12 10 6 12 5 60 60 60 ⋅ ⋅ + = + = + = ⋅ ⋅ 11. a. 2 1 9 x x − = 2 2 2 1 1 1 7 9 7 9 2 9 1 7 9 7 9 2 1 7 7 7 1 7 1 1, true ⎛ ⎞ − = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ − = ⎜ ⎟ ⎝ ⎠ − = = = The given fraction is a solution. b. 1 1 5 3 1 3 1 3 5 20 3 20 4 3 1 20 20 20 1 1 , true 20 20 w w − = ⎛ ⎞ − = ⎜ ⎟ ⎝ ⎠ − = = The given fraction is a solution. 12. a. 2 ( 6) 3 x − b. 3 1 2 4 5 x x − =

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    Section 1.2 Fractions in Algebra 7 13. 5 ( 32) 9 5 5 (77 32) (45) 25 9 9 C F C = − = − = = 77 F ° is equivalent to 25 C. ° 1.2 Concept and Vocabulary Check 1. numerator; denominator 2. mixed; improper 3. 5; 3; 2; 5 4. natural 5. prime 6. factors; product 7. a b 8. a c b d ⋅ ⋅ 9. reciprocals 10. d c 11. a c b + 12. least common denominator 1.2 Exercise Set 1. 3 2 8 3 16 3 19 2 8 8 8 8 ⋅ + + = = = 2. 7 2 9 7 18 7 25 2 9 9 9 9 ⋅ + + = = = 3. 3 7 5 3 35 3 38 7 5 5 5 5 ⋅ + + = = = 4. 2 6 5 2 30 2 32 6 5 5 5 5 ⋅ + + = = = 5. 7 8 16 7 128 7 135 8 16 16 16 16 ⋅ + + = = = 6. 5 9 16 5 144 5 149 9 16 16 16 16 ⋅ + + = = = 7. 23 divided by 5 is 4 with a remainder of 3, so 23 3 4 5 5 = . 8. 47 divided by 8 is 5 with a remainder of 7, so 47 7 5 8 8 = . 9. 76 divided by 9 is 8 with a remainder of 4, so 76 4 8 9 9 = . 10. 59 divided by 9 is 6 with a remainder of 5, so 59 5 6 9 9 = . 11. 711 divided by 20 is 35 with a remainder of 11, so 711 11 35 20 20 = . 12. 788 divided by 25 is 31 with a remainder of 13, so 788 13 31 25 25 = . 13. composite; 22 2 11 = ⋅ 14. composite; 15 3 5 = ⋅ 15. composite; 20 4 5 2 2 5 = ⋅ = ⋅ ⋅ 16. composite; 75 3 25 3 5 5 = ⋅ = ⋅ ⋅ 17. 37 has no factors other than 1 and 37, so 37 is prime. 18. 23 has no factors other than 1 and 23, so 23 is prime. 19. composite; 36 4 9 2 2 3 3 = ⋅ = ⋅ ⋅ ⋅ 20. composite; 100 4 25 2 2 5 5 = ⋅ = ⋅ ⋅ ⋅ 21. composite; 140 10 14 2 5 2 7 2 2 5 7 = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ 22. composite; 110 10 11 2 5 11 = ⋅ = ⋅ ⋅ 23. 79 has no factors other than 1 and 79, so 79 is prime.

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 8 24. 83 has no factors other than 1 and 83, so 83 is prime. 25. composite; 81 9 9 3 3 3 3 = ⋅ = ⋅ ⋅ ⋅ 26. composite; 64 8 8 2 4 2 4 2 2 2 2 2 2 = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 27. composite; 240 10 24 2 5 2 12 2 5 2 3 4 2 5 2 3 2 2 2 2 2 2 3 5 = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 28. composite; 360 10 36 2 5 6 6 2 5 2 3 2 3 2 2 2 3 3 5 = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 29. 10 2 5 5 16 2 8 8 ⋅ = = ⋅ 30. 8 2 4 4 14 2 7 7 ⋅ = = ⋅ 31. 15 3 5 5 18 3 6 6 ⋅ = = ⋅ 32. 18 9 45 = 2 9 ⋅ 2 5 5 = ⋅ 33. 35 5 7 7 50 5 10 10 ⋅ = = ⋅ 34. 45 5 9 9 50 5 10 10 ⋅ = = ⋅ 35. 32 16 2 2 80 16 5 5 ⋅ = = ⋅ 36. 75 5 80 = 15 5 ⋅ 15 16 16 = ⋅ 37. 44 2 22 22 50 2 25 25 ⋅ = = ⋅ 38. 38 2 19 19 50 2 25 25 ⋅ = = ⋅ 39. 120 2 60 60 86 2 43 43 ⋅ = = ⋅ 40. 116 2 58 58 86 2 43 43 ⋅ = = ⋅ 41. 2 1 2 1 2 5 3 5 3 15 ⋅ ⋅ = = ⋅ 42. 3 1 3 1 3 7 4 7 4 28 ⋅ ⋅ = = ⋅ 43. 3 7 3 7 21 8 11 8 11 88 ⋅ ⋅ = = ⋅ 44. 5 3 5 3 15 8 11 8 11 88 ⋅ ⋅ = = ⋅ 45. 4 9 4 9 4 36 1 9 or 5 7 1 7 1 7 7 7 ⋅ ⋅ = ⋅ = = ⋅ 46. 3 8 3 8 3 24 3 8 or 3 7 1 7 1 7 7 7 ⋅ ⋅ = ⋅ = = ⋅ 47. 1 5 1 5 5 5 1 1 10 6 10 6 60 5 12 12 ⋅ ⋅ ⋅ = = = = ⋅ ⋅ 48. 1 2 1 2 2 2 8 3 8 3 24 ⋅ ⋅ = = = ⋅ 1 2 ⋅ 1 12 12 = ⋅ 49. 5 6 5 6 30 2 15 15 1 or 1 4 7 4 7 28 2 14 14 14 ⋅ ⋅ ⋅ = = = = ⋅ ⋅ 50. 7 6 7 6 42 2 4 11 4 11 44 ⋅ ⋅ = = = ⋅ 21 2 ⋅ 21 22 22 = ⋅ 51. 3 3 15 8 120 20 3 1 4 5 4 5 20 ⎛ ⎞ ⎛ ⎞ = ⋅ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 6 20 ⋅ 6 1 = ⋅ 52. 4 1 14 5 70 10 2 1 5 4 5 4 20 ⎛ ⎞ ⎛ ⎞ = ⋅ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 7 10 ⋅ 2 7 1 or 3 2 2 ⋅ = 53. 5 4 5 3 5 3 15 4 3 4 4 4 4 16 ⋅ ÷ = ⋅ = = ⋅ 54. 7 2 7 3 7 3 21 5 or 1 8 3 8 2 8 2 16 16 ⋅ ÷ = ⋅ = = ⋅

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    Section 1.2 Fractions in Algebra 9 55. 18 18 1 2 5 5 2 18 1 18 2 9 9 4 or 1 5 2 10 2 5 5 5 ÷ = ⋅ ⋅ ⋅ = = = = ⋅ ⋅ 56. 12 12 1 3 7 7 3 12 1 12 3 4 4 7 3 21 3 7 7 ÷ = ⋅ ⋅ ⋅ / = = = = ⋅ ⋅ / 57. 18 2 5 10 2 5 5 2 5 1 18 18 2 9 9 ⋅ ÷ = ⋅ = = = ⋅ 58. 12 3 7 21 3 7 7 3 3 or 1 7 1 12 12 3 4 4 4 ⋅ / ÷ = ⋅ = = = ⋅ / 59. 3 1 3 4 3 4 12 3 4 4 4 1 4 1 4 ⋅ ÷ = ⋅ = = = ⋅ 60. 3 1 3 7 3 7 21 3 7 7 7 1 7 1 7 ⋅ ÷ = ⋅ = = = ⋅ 61. 7 5 7 3 7 3 21 3 7 7 6 3 6 5 6 5 30 3 10 10 ⋅ ⋅ ÷ = ⋅ = = = = ⋅ ⋅ 62. 7 3 7 8 7 8 56 4 14 4 8 4 3 4 3 12 4 3 14 2 or 4 3 3 ⋅ / ⋅ ÷ = ⋅ = = = ⋅ / ⋅ = 63. 1 1 1 7 7 7 1 1 14 7 14 1 14 7 2 2 ⋅ ÷ = ⋅ = = = ⋅ 64. 1 1 1 4 4 4 1 1 8 4 8 1 8 4 2 2 / ⋅ ÷ = ⋅ = = = / ⋅ 65. 3 1 33 11 6 1 5 10 5 10 33 10 11 5 11 ÷ = ÷ = ⋅ = 3 5 ⋅ 5 ⋅ 2 11 ⋅ 6 6 1 = = 66. 3 5 7 21 1 2 4 8 4 8 7 8 56 28 4 21 84 ÷ = ÷ = ⋅ = = 2 28 ⋅ 2 3 3 = ⋅ 67. 2 4 2 4 6 11 11 11 11 + + = = 68. 5 2 5 2 7 13 13 13 13 + + = = 69. 7 1 8 4 2 2 12 12 12 4 3 3 ⋅ + = = = ⋅ 70. 5 1 6 2 16 16 16 + = = 3 2 ⋅ 3 8 8 = ⋅ 71. 5 5 10 2 5 5 1 or 1 8 8 8 2 4 4 4 ⋅ + = = = ⋅ 72. 3 3 6 2 3 3 8 8 8 2 4 4 ⋅ + = = = ⋅ 73. 7 5 2 2 12 12 12 − = = 1 2 ⋅ 1 6 6 = ⋅ 74. 13 5 8 2 18 18 18 − = = 4 2 ⋅ 4 9 9 = ⋅ 75. 16 2 14 7 7 7 7 − = = 2 7 ⋅ 2 1 = ⋅ 76. 17 2 15 3 5 5 5 − = = 77. 1 1 1 5 1 2 2 5 2 5 5 2 5 2 5 2 7 10 10 10 10 + = ⋅ + ⋅ + = + = = 78. 1 1 1 5 1 3 3 5 3 5 5 3 5 3 5 3 8 15 15 15 15 + = ⋅ + ⋅ + = + = = 79. 3 3 3 5 3 4 20 4 5 20 15 3 20 20 18 2 20 + = ⋅ + = + = = 9 2 ⋅ 9 10 10 = ⋅ 80. 2 2 2 3 2 5 15 5 3 15 6 2 8 15 15 15 + = ⋅ + = + =

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 10 81. 3 5 3 3 5 2 8 12 8 3 12 2 9 10 19 24 24 24 + = ⋅ + ⋅ = + = 82. 3 2 3 3 2 2 10 15 10 3 15 2 9 4 13 30 30 30 + = ⋅ + ⋅ = + = 83. 11 2 11 2 2 11 4 7 18 9 18 9 2 18 18 18 − = − ⋅ = − = 84. 17 4 17 4 2 17 8 9 18 9 18 9 2 18 18 18 9 − = − ⋅ = − = = 1 9 ⋅ 1 2 2 = ⋅ 85. 4 3 4 4 3 3 3 4 3 4 4 3 16 9 7 12 12 12 − = ⋅ − ⋅ = − = 86. 3 2 3 3 2 2 2 3 2 3 3 2 9 4 5 6 6 6 − = ⋅ − ⋅ = − = 87. 7 3 7 8 3 5 10 16 10 8 16 5 56 15 41 80 80 80 − = ⋅ − ⋅ = − = 88. 7 5 7 4 5 5 30 24 30 4 24 5 28 25 3 120 120 120 3 − = ⋅ − ⋅ = − = = 1 3 ⋅ 1 40 40 = ⋅ 89. 3 1 15 7 3 2 4 3 4 3 15 3 7 4 4 3 3 4 45 28 17 5 or 1 12 12 12 12 − = − = ⋅ − ⋅ = − = 90. 2 1 11 5 3 2 3 2 3 2 11 2 5 3 3 2 2 3 22 15 7 1 or 1 6 6 6 6 − = − = ⋅ − ⋅ = − = 91. 7 28 2 7 8 28 2 7 8 28 2 1 7 2 x = ⋅ = ⋅ = 4 2 ⋅ ⋅ 28 1 28 28, true = = The given number is a solution. 92. 5 30 3 5 18 30 3 5 18 30 3 1 5 3 x = ⋅ = ⋅ = 6 3 ⋅ ⋅ 30 1 30 30, true = = The given number is a solution. 93. 2 3 3 4 w − = 5 2 3 1 12 3 4 17 2 3 12 3 4 17 8 3 12 12 4 9 3 12 4 3 3 , true 4 4 − = − = − = = = The given number is a solution.

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    Section 1.2 Fractions in Algebra 11 94. 3 7 4 4 w − = 1 3 7 2 2 4 4 5 3 7 2 4 4 10 3 7 4 4 4 7 7 , true 4 4 − = − = − = = The given number is a solution. 95. 1 1 20 3 2 1 1 20 12 12 3 2 1 12 1 12 20 3 1 2 1 20 4 12 16 12, false z z − = − ⋅ = ⋅ − ⋅ = ⋅ − = = The given number is not a solution. 96. 1 1 12 4 2 1 1 12 20 20 4 2 1 20 1 20 12 4 1 2 1 12 5 10 7 10, false z z − = − ⋅ = ⋅ − ⋅ = ⋅ − = = The given number is not a solution. 97. 2 1 3 9 3 7 y y + = 2 27 1 27 3 9 35 3 35 7 2 9 ⋅ + ⋅ = 9 ⋅ 3 1 35 3 ⋅ + 3 ⋅ 9 3 35 7 6 9 3 35 35 7 15 3 35 7 3 5 ⋅ = + = = ⋅ 7 5 ⋅ 3 7 3 3 , true 7 7 = = The given number is a solution. 98. 2 5 2 3 6 y y + = 2 1 5 1 1 1 2 3 3 6 3 2 4 5 4 2 3 3 6 3 8 20 2 9 18 8 10 2 9 9 18 2 9 2 2, true ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = + = = = The given number is a solution. 99. 1 1 ( 2) ( 4) 3 5 1 1 (26 2) (26 4) 3 5 1 1 (24) (30) 3 5 8 6, false x x − = + − = + = = The given number is not a solution. 100. 1 3 ( 2) 3 (3 4) 2 8 1 3 (4 2) 3 (3 4 4) 2 8 1 3 (2) 3 (12 4) 2 8 3 1 3 (8) 8 4 3, false x x − + = − − + = ⋅ − + = − + = = The given number is not a solution.

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    Chapter 1 Variables, Real Numbers, and Mathematical Models 12 101. 2 7 ( 6) ( 2) 3 9 y y ÷ + = ÷ − 1 2 1 7 4 6 4 2 3 3 3 9 13 6 2 13 2 7 3 1 3 3 1 9 13 1 2 13 1 7 3 6 3 3 2 9 13 2 13 7 18 3 6 9 13 12 39 14 18 18 18 18 25 25 , true 18 18 ⎛ ⎞ ⎛ ⎞ ÷ + = ÷ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ÷ + = ÷ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⋅ + = ⋅ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = − + = − = The given number is a solution. 102. 1 5 ( 6) ( 2) 3 9 y y ÷ + = ÷ − 2 1 2 5 2 6 2 2 3 3 3 9 8 6 1 8 2 5 3 1 3 3 1 9 8 1 1 8 1 5 3 6 3 3 2 9 8 1 8 5 18 3 6 9 8 6 24 10 18 18 18 18 14 14 , true 18 18 ⎛ ⎞ ⎛ ⎞ ÷ + = ÷ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ÷ + = ÷ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⋅ + = ⋅ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = − + = − = The given number is a solution. 103. 1 5 x 104. 1 6 x 105. 1 4 x x − 106. 1 3 x x − 107. 1 1 4 2 x x − = 108. 1 1 3 2 x x − = 109. 1 1 12 7 8 x x + = 110. 1 1 15 9 10 x x + = 111. 2 ( 6) 3 x + 112. 3 ( 9) 4 x + 113. 2 6 3 3 x x + = − 114. 3 9 2 4 x x + = − 115. 3 3 3 4 5 4 5 20 a a a ⋅ ⋅ = = ⋅ 116. 2 2 7 2 7 14 3 7 3 3 3 a a a a ⋅ ÷ = ⋅ = = ⋅ 117. 11 9 11 9 20 x x x x + + = = 118. 10 6 10 6 4 y y y y − − = = 119. 1 1 5 3 2 5 2 3 8 6 6 8 1 5 6 8 1 8 8 2 6 5 30 ⎛ ⎞ ⎛ ⎞ − ÷ = − ÷ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ÷ = ⋅ = = 4 2 ⋅ 4 15 15 = ⋅ 120. 1 1 1 1 2 1 3 2 2 4 2 3 4 4 6 6 3 5 4 6 3 6 4 5 18 2 20 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + ÷ + = + ÷ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = ÷ = ⋅ = = 9 2 ⋅ 9 10 10 = ⋅

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    Section 1.2 Fractions in Algebra 13 121. ( ) 1 1 1 2 5 2 5 x x ⎛ ⎞ + = − ⎜ ⎟ ⎝ ⎠ 1 5 1 5 1 2 5 8 2 8 5 1 5 2 1 5 1 5 8 1 2 8 5 1 5 16 1 25 8 5 8 8 2 40 40 1 21 1 17 5 8 2 40 21 17 40 80 42 17 , false 80 80 ⎛ ⎞ ⎛ ⎞ + = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = = The given number is not a solution. 122. ( ) ( ) 12 3 2 4 3 x x x − − = − + 1 1 1 12 3 3 2 4 3 3 3 2 2 2 12 3 7 2 4 7 7 3 1 1 2 1 1 2 2 1 12 3 7 4 2 2 1 1 2 2 ⎛ ⎞ ⎛ ⎞ − − = ⋅ − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ − − = ⋅ − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⋅ ⎛ ⎞ − − = ⎜ ⎟ ⎝ ⎠ 7 1 2 ⋅ 7 6 2 2 12 3 3 14 13 1 1 2 1 2 24 9 28 13 2 2 2 2 15 15 , true 2 2 ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ − = − ⎜ ⎟ ⎝ ⎠ − = − = The given number is a solution. 123. 5 ( 32) 9 5 5 (68 32) (36) 20 9 9 C F C = − = − = = 68 F ° is equivalent to 20 C. ° 124. 5 ( 32) 9 5 5 (41 32) (9) 5 9 9 C F C = − = − = = 41 F ° is equivalent to 5 C. ° 125. a. ( ) ( ) ( ) 7 220 10 7 220 20 10 7 200 10 140 H a H = − = − = = The lower limit of the heart rate for a 20-year- old with this exercise goal is 140 beats per minute. b. ( ) ( ) ( ) 4 220 5 4 220 20 5 4 200 5 160 H a H = − = − = = The upper limit of the heart rate for a 20-year- old with this exercise goal is 160 beats per minute. 126. a. ( ) ( ) ( ) 1 220 2 1 220 30 2 1 190 2 95 H a H = − = − = = The lower limit of the heart rate for a 30-year- old with this exercise goal is 95 beats per minute. b. ( ) ( ) ( ) 3 220 5 3 220 30 5 3 190 5 114 H a H = − = − = = The upper limit of the heart rate for a 30-year- old with this exercise goal is 114 beats per minute.

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