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Solution Manual for College Algebra, 10th Edition

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    ’ S S OLUTIONS M ANUAL T IM B RITT Jackson State Community College C OLLEGE A LGEBRA T ENTH E DITION Michael Sullivan Chicago State University

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    iii Table of Contents Chapter R Review R.1 Real Numbers.............................................................................................................................1 R.2 Algebra Essentials ......................................................................................................................5 R.3 Geometry Essentials.................................................................................................................11 R.4 Polynomials..............................................................................................................................16 R.5 Factoring Polynomials .............................................................................................................23 R.6 Synthetic Division ....................................................................................................................28 R.7 Rational Expressions ................................................................................................................29 R.8 n th Roots; Rational Exponents.................................................................................................39 Chapter 1 Equations and Inequalities 1.1 Linear Equations .......................................................................................................................48 1.2 Quadratic Equations .................................................................................................................64 1.3 Complex Numbers; Quadratic Equations in the Complex Number System ............................82 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations ................................88 1.5 Solving Inequalities ................................................................................................................108 1.6 Equations and Inequalities Involving Absolute Value ...........................................................119 1.7 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications 128 Chapter Review .............................................................................................................................135 Chapter Test...................................................................................................................................144 Chapter Projects .............................................................................................................................146 Chapter 2 Graphs 2.1 The Distance and Midpoint Formulas ....................................................................................147 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ..............................................159 2.3 Lines .......................................................................................................................................171 2.4 Circles .....................................................................................................................................187 2.5 Variation .................................................................................................................................199 Chapter Review .............................................................................................................................204 Chapter Test ...................................................................................................................................210 Cumulative Review .......................................................................................................................212 Chapter Project ..............................................................................................................................214 Chapter 3 Functions and Their Graphs 3.1 Functions ................................................................................................................................215 3.2 The Graph of a Function .........................................................................................................230 3.3 Properties of Functions ...........................................................................................................239 3.4 Library of Functions; Piecewise-defined Functions ...............................................................255 3.5 Graphing Techniques: Transformations .................................................................................267 3.6 Mathematical Models: Building Functions ............................................................................284 Chapter Review .............................................................................................................................290 Chapter Test ...................................................................................................................................297 Cumulative Review .......................................................................................................................300 Chapter Projects.............................................................................................................................304

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    iv Chapter 4 Linear and Quadratic Functions 4.1 Properties of Linear Functions and Linear Models ................................................................306 4.2 Building Linear Functions from Data .....................................................................................317 4.3 Quadratic Functions and Their Properties ..............................................................................322 4.4 Build Quadratic Models from Verbal Descriptions and from Data .......................................343 4.5 Inequalities Involving Quadratic Functions ...........................................................................350 Chapter Review .............................................................................................................................369 Chapter Test ...................................................................................................................................377 Cumulative Review .......................................................................................................................379 Chapter Projects .............................................................................................................................382 Chapter 5 Polynomial and Rational Functions 5.1 Polynomial Functions and Models .........................................................................................385 5.2 Properties of Rational Functions ............................................................................................407 5.3 The Graph of a Rational Function ..........................................................................................416 5.4 Polynomial and Rational Inequalities .....................................................................................469 5.5 The Real Zeros of a Polynomial Function ..............................................................................490 5.6 Complex Zeros; Fundamental Theorem of Algebra ...............................................................521 Chapter Review .............................................................................................................................530 Chapter Test ...................................................................................................................................544 Cumulative Review .......................................................................................................................549 Chapter Projects .............................................................................................................................553 Chapter 6 Exponential and Logarithmic Functions 6.1 Composite Functions ..............................................................................................................555 6.2 One-to-One Functions; Inverse Functions ..............................................................................572 6.3 Exponential Functions ............................................................................................................591 6.4 Logarithmic Functions ............................................................................................................611 6.5 Properties of Logarithms ........................................................................................................631 6.6 Logarithmic and Exponential Equations ................................................................................640 6.7 Financial Models ....................................................................................................................659 6.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models ................................................................................................................667 6.9 Building Exponential, Logarithmic, and Logistic Models from Data ....................................676 Chapter Review .............................................................................................................................681 Chapter Test ...................................................................................................................................693 Cumulative Review .......................................................................................................................697 Chapter Projects .............................................................................................................................701

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    v Chapter 7 Analytic Geometry 7.2 The Parabola ...........................................................................................................................703 7.3 The Ellipse ..............................................................................................................................717 7.4 The Hyperbola ........................................................................................................................733 Chapter Review .............................................................................................................................752 Chapter Test ...................................................................................................................................756 Cumulative Review .......................................................................................................................759 Chapter Projects .............................................................................................................................761 Chapter 8 Systems of Equations and Inequalities 8.1 Systems of Linear Equations: Substitution and Elimination ..................................................763 8.2 Systems of Linear Equations: Matrices ..................................................................................784 8.3 Systems of Linear Equations: Determinants ..........................................................................808 8.4 Matrix Algebra .......................................................................................................................822 8.5 Partial Fraction Decomposition ..............................................................................................840 8.6 Systems of Nonlinear Equations.............................................................................................857 8.7 Systems of Inequalities ...........................................................................................................885 8.8 Linear Programming ...............................................................................................................899 Chapter Review .............................................................................................................................911 Chapter Test ...................................................................................................................................926 Cumulative Review .......................................................................................................................935 Chapter Projects.............................................................................................................................938 Chapter 9 Sequences; Induction; the Binomial Theorem 9.1 Sequences ...............................................................................................................................941 9.2 Arithmetic Sequences .............................................................................................................950 9.3 Geometric Sequences; Geometric Series ................................................................................957 9.4 Mathematical Induction ..........................................................................................................968 9.5 The Binomial Theorem ...........................................................................................................976 Chapter Review .............................................................................................................................981 Chapter Test ...................................................................................................................................986 Cumulative Review .......................................................................................................................989 Chapter Projects .............................................................................................................................991 Chapter 10 Counting and Probability 10.1 Counting ...............................................................................................................................993 10.2 Permutations and Combinations ...........................................................................................995 10.3 Probability ............................................................................................................................999 Chapter Review ...........................................................................................................................1005 Chapter Test .................................................................................................................................1006 Cumulative Review .....................................................................................................................1008 Chapter Projects ...........................................................................................................................1010 Appendix Graphing Utilities Section 1 The Viewing Rectangle ..............................................................................................1013 Section 2 Using a Graphing Utility to Graph Equations ............................................................1014 Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ..................1018 Section 5 Square Screens ............................................................................................................1020

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    1 Chapter R Review Section R.1 1. rational 2. 4 5 6 3 4 30 3 31 + ⋅ − = + − = 3. Distributive 4. ( ) 5 3 6 x + = 5. a 6. b 7. True 8. False; The Zero-Product Property states that if a product equals 0, then at least one of the factors must equal 0. 9. False; 6 is the Greatest Common Factor of 12 and 18. The Least Common Multiple is the smallest value that both numbers will divide evenly. The LCM for 12 and 18 is 36. 10. True 11. { } { } { } 1, 3, 4,5, 9 2, 4, 6, 7,8 1, 2,3, 4, 5, 6, 7,8, 9 A B ∪ = ∪ = 12. { } { } { } 1, 3, 4,5, 9 1, 3, 4, 6 1, 3, 4, 5, 6, 9 A C ∪ = ∪ = 13. { } { } { } 1, 3, 4,5, 9 2, 4, 6, 7,8 4 A B ∩ = ∩ = 14. { } { } { } 1, 3, 4,5, 9 1, 3, 4, 6 1, 3, 4 A C ∩ = ∩ = 15. { } { } ( ) { } { } { } { } ( ) 1, 3, 4,5, 9 2, 4, 6, 7,8 1,3, 4, 6 1, 2,3, 4,5, 6, 7,8,9 1,3, 4, 6 1, 3, 4, 6 A B C ∪ ∩ = ∪ ∩ = ∩ = 16. { } { } ( ) { } { } { } { } ( ) 1, 3, 4,5, 9 2, 4, 6, 7,8 1,3, 4, 6 4 1, 3, 4, 6 1,3, 4, 6 A B C ∩ ∪ = ∩ ∪ = ∪ = 17. { } 0, 2, 6, 7, 8 A = 18. { } 0, 2, 5, 7, 8, 9 C = 19. { } { } { } { } 1, 3, 4, 5, 9 2, 4, 6, 7, 8 4 0, 1, 2, 3, 5, 6, 7, 8, 9 A B ∩ = ∩ = = 20. { } { } { } { } 2, 4, 6, 7, 8 1, 3, 4, 6 1, 2, 3, 4, 6, 7, 8 0, 5, 9 B C ∪ = ∪ = = 21. { } { } { } 0, 2, 6, 7, 8 0, 1, 3, 5, 9 0, 1, 2, 3, 5, 6, 7, 8, 9 A B ∪ = ∪ = 22. { } { } { } 0, 1, 3, 5, 9 0, 2, 5, 7, 8, 9 0, 5, 9 B C ∩ = ∩ = 23. a. { } 2,5 b. { } 6, 2,5 − c. { } 1 6, , 1.333... 1.3, 2,5 2 − − = − d. { } π e. { } 1 6, , 1.333... 1.3, , 2,5 2 π − − = − 24. a. { } 1 b. { } 0,1 c. { } 5 , 2.060606... 2.06,1.25, 0,1 3 − = d. { } 5

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    Chapter R: Review 2 e. { } 5 , 2.060606... 2.06,1.25, 0,1, 5 3 − = 25. a. { } 1 b. { } 0,1 c. { } 1 1 1 0,1, , , 2 3 4 d. None e. { } 1 1 1 0,1, , , 2 3 4 26. a. None b. { } 1 − c. { } 1.3, 1.2, 1.1, 1 − − − − d. None e. { } 1.3, 1.2, 1.1, 1 − − − − 27. a. None b. None c. None d. { } 1 2, , 2 1, 2 π π + + e. { } 1 2, , 2 1, 2 π π + + 28. a. None b. None c. { } 1 10.3 2 + d. { } 2, 2 π − + e. { } 1 2, 2, 10.3 2 π − + + 29. a. 18.953 b. 18.952 30. a. 25.861 b. 25.861 31. a. 28.653 b. 28.653 32. a. 99.052 b. 99.052 33. a. 0.063 b. 0.062 34. a. 0.054 b. 0.053 35. a. 9.999 b. 9.998 36. a. 1.001 b. 1.000 37. a. 0.429 b. 0.428 38. a. 0.556 b. 0.555 39. a. 34.733 b. 34.733 40. a. 16.200 b. 16.200 41. 3 2 5 + = 42. 5 2 10 ⋅ = 43. 2 3 4 x + = ⋅ 44. 3 2 2 y + = + 45. 3 1 2 y = + 46. 2 4 6 x = ⋅ 47. 2 6 x − = 48. 2 6 y − = 49. 6 2 x = 50. 2 6 x = 51. 9 4 2 5 2 7 − + = + = 52. 6 4 3 2 3 5 − + = + = 53. 6 4 3 6 12 6 − + ⋅ = − + = 54. 8 4 2 8 8 0 − ⋅ = − = 55. 4 5 8 9 8 1 + − = − = 56. 8 3 4 5 4 1 − − = − = 57. 1 12 1 13 4 3 3 3 + + = = 58. 1 4 1 3 2 2 2 2 − − = =

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    Section R.1: Real Numbers 3 59. ( ) ( ) 6 3 5 2 3 2 6 15 2 1 6 17 11 − ⋅ + ⋅ − = − + ⋅         = − = − 60. ( ) ( ) [ ] [ ] 2 8 3 4 2 3 2 8 3 6 3 2 8 18 3 2 10 3 20 3 23 ⋅ − ⋅ + − = ⋅ − ⋅ −         = ⋅ − − = ⋅ − − = − − = − 61. ( ) ( ) 2 3 5 8 2 1 2 2 16 1 4 16 1 12 1 11 ⋅ − + ⋅ − = ⋅ − + − = − + − = − = 62. ( ) ( ) 1 4 3 2 2 1 12 2 2 1 12 11 − ⋅ − + = − − + = − = − 63. ( ) [ ] [ ] [ ] 10 6 2 2 8 3 2 10 6 4 5 2 10 2 5 2 10 7 2 10 14 4 − − ⋅ + − ⋅ = − − + ⋅     = − + ⋅ = − ⋅ = − = − 64. ( ) ( ) [ ] 2 5 4 6 3 4 2 20 6 1 18 6 18 6 12 − ⋅ − ⋅ − = − − ⋅ −         = − − − = − + = − 65. ( ) ( ) 1 1 5 3 2 1 2 2 − = = 66. ( ) ( ) 1 1 5 4 9 3 3 3 + = = 67. 4 8 12 6 5 3 2 + = = − 68. 2 4 2 1 5 3 2 − − = = − − 69. 3 10 3 2 5 3 5 21 5 3 7 ⋅ ⋅ ⋅ = = ⋅ ⋅ 2 5 ⋅ ⋅ 5 3 ⋅ 2 7 7 = ⋅ 70. 5 3 5 3 5 9 10 3 3 5 2 ⋅ ⋅ = = ⋅ ⋅ ⋅ 3 ⋅ 3 3 5 ⋅ ⋅ 1 6 2 = ⋅ 71. 6 10 2 3 5 2 2 3 25 27 5 5 3 9 ⋅ ⋅ ⋅ ⋅ ⋅ = = ⋅ ⋅ ⋅ 5 ⋅ 2 5 ⋅ 5 3 ⋅ ⋅ 4 45 9 = ⋅ 72. 21 100 3 7 4 25 3 25 3 25 3 ⋅ ⋅ ⋅ ⋅ = = ⋅ 7 4 25 ⋅ ⋅ ⋅ 25 3 ⋅ 28 = 73. 3 2 15 8 23 4 5 20 20 + + = = 74. 4 1 8 3 11 3 2 6 6 + + = = 75. 5 9 25 54 79 6 5 30 30 + + = = 76. 8 15 16 135 151 9 2 18 18 + + = = 77. 5 1 10 3 13 18 12 36 36 + + = = 78. 2 8 6 40 46 15 9 45 45 + + = = 79. 1 7 3 35 32 16 30 18 90 90 45 − − = = − = − 80. 3 2 9 4 5 14 21 42 42 − − = = 81. 3 2 9 8 1 20 15 60 60 − − = = 82. 6 3 12 15 3 35 14 70 70 − − = = − 83. 5 18 5 27 5 9 3 5 9 18 11 9 2 11 11 27     ⋅ ⋅ ⋅   = ⋅ = = ⋅ ⋅       3 9 ⋅ 15 22 2 11 = ⋅ ⋅ 84. 5 21 5 35 5 7 5 5 7 21 2 7 3 2 2 35     ⋅ ⋅ ⋅   = ⋅ = = ⋅ ⋅       5 7 ⋅ 25 6 3 2 = ⋅ ⋅

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    Chapter R: Review 4 85. 1 3 7 3 7 3 7 10 1 2 5 10 10 10 10 10 + ⋅ + = + = = = 86. 2 4 1 2 2 2 2 2 2 2 2 3 5 6 3 5 3 2 3 5 3 2 3 15 2 5 2 10 2 10 2 12 3 5 15 15 15 15 15 4 3 4 3 4 5 3 5 3 5 ⋅ ⋅ + ⋅ = + = + = + ⋅ ⋅ ⋅ ⋅ + = ⋅ + = + = = ⋅ ⋅ = = = ⋅ ⋅ 87. 3 3 2 3 3 6 3 6 2 3 2 4 8 1 4 8 4 8 4 2 8 12 3 12 3 15 8 8 8 8 ⋅ + = ⋅ + = + = ⋅ + + = + = = 88. 5 1 3 5 1 3 5 1 3 5 1 3 6 2 1 6 2 3 2 2 3 2 2 5 1 5 1 4 2 2 2 2 2 ⋅ ⋅ ⋅ − = ⋅ − = − = − ⋅ ⋅ − = − = = = 89. ( ) 6 4 6 24 x x + = + 90. ( ) 4 2 1 8 4 x x − = − 91. ( ) 2 4 4 x x x x − = − 92. ( ) 2 4 3 4 12 x x x x + = + 93. 3 1 3 1 2 3 2 2 2 2 4 2 4 2 2 2 2 2 3 2 3 1 2 2 2 2 x x x x x ⋅   − = ⋅ − ⋅ = −   ⋅   ⋅ = − = − ⋅ 94. 2 1 2 1 3 2 3 3 3 3 3 6 3 6 3 3 2 3 2 3 1 2 3 3 2 2 x x x x x ⋅   + = ⋅ + ⋅ = +   ⋅   ⋅ = + = + ⋅ 95. ( )( ) 2 2 2 4 4 2 8 6 8 x x x x x x x + + = + + + = + + 96. ( )( ) 2 2 5 1 5 5 6 5 x x x x x x x + + = + + + = + + 97. ( )( ) 2 2 2 1 2 2 2 x x x x x x x − + = + − − = − − 98. ( )( ) 2 2 4 1 4 4 3 4 x x x x x x x − + = + − − = − − 99. ( )( ) 2 2 8 2 2 8 16 10 16 x x x x x x x − − = − − + = − + 100. ( )( ) 2 2 4 2 2 4 8 6 8 x x x x x x x − − = − − + = − + 101. ( ) ( ) 2 3 2 3 2 3 5 5 x x x x x x x + = ⋅ + ⋅ = + ⋅ = ⋅ = 102. 2 3 4 2 12 14 + ⋅ = + = since multiplication comes before addition in the order of operations for real numbers. ( ) 2 3 4 5 4 20 + ⋅ = ⋅ = since operations inside parentheses come before multiplication in the order of operations for real numbers. 103. ( ) ( ) 2 3 4 2 12 24 ⋅ = = ( ) ( ) ( )( ) 2 3 2 4 6 8 48 ⋅ ⋅ ⋅ = = 104. 4 3 7 1 2 5 7 + = = + , but 4 3 4 5 3 2 20 6 26 13 2.6 2 5 10 10 10 5 ⋅ + ⋅ + + = = = = = 105. Subtraction is not commutative; for example: 2 3 1 1 3 2 − = − ≠ = − . 106. Subtraction is not associative; for example: ( ) ( ) 5 2 1 2 4 5 2 1 − − = ≠ = − − . 107. Division is not commutative; for example: 2 3 3 2 ≠ . 108. Division is not associative; for example: ( ) 12 2 2 6 2 3 ÷ ÷ = ÷ = , but ( ) 12 2 2 12 1 12 ÷ ÷ = ÷ = . 109. The Symmetric Property implies that if 2 = x, then x = 2.

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    Section R.2: Algebra Essentials 5 110. From the principle of substitution, if 5 x = , then ( )( ) ( )( ) 2 2 2 5 5 25 25 5 30 x x x x x x x =  =  + = +  + = 111. There are no real numbers that are both rational and irrational, since an irrational number, by definition, is a number that cannot be expressed as the ratio of two integers; that is, not a rational number Every real number is either a rational number or an irrational number, since the decimal form of a real number either involves an infinitely repeating pattern of digits or an infinite, non- repeating string of digits. 112. The sum of an irrational number and a rational number must be irrational. Otherwise, the irrational number would then be the difference of two rational numbers, and therefore would have to be rational. 113. Answers will vary. 114. Since 1 day = 24 hours, we compute 12997 541.5416 24 = . Now we only need to consider the decimal part of the answer in terms of a 24 hour day. That is, ( )( ) 0.5416 24 13 ≈ hours. So it must be 13 hours later than 12 noon, which makes the time 1 AM CST. 115. Answers will vary. Section R.2 1. variable 2. origin 3. strict 4. base; exponent (or power) 5. 3 1.2345678 10 × 6. d 7. b 8. True. 9. True 10. False; the absolute value of a real number is nonnegative. 0 0 = which is not a positive number. 11. False; a number in scientific notation is expressed as the product of a number , x , 1 10 x ≤ < or 10 1 x − < ≤ − , and a power of 10. 12. False; to multiply two expressions with the same base, retain the base and add the exponents. 13. −2.5 −1 5 2 1 0 0.25 3 4 14. 1 3 3 2 −2 2 −1.5 0 2 3 15. 1 0 2 > 16. 5 6 < 17. 1 2 − > − 18. 5 3 2 − < − 19. 3.14 π > 20. 2 1.41 > 21. 1 0.5 2 = 22. 1 0.33 3 > 23. 2 0.67 3 < 24. 1 0.25 4 =

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    Chapter R: Review 6 25. 0 x > 26. 0 z < 27. 2 x < 28. 5 y > − 29. 1 x ≤ 30. 2 x ≥ 31. Graph on the number line: 2 x ≥ − 0 −2 32. Graph on the number line: 4 x < 4 0 33. Graph on the number line: 1 x > − −1 0 34. Graph on the number line: 7 x ≤ 0 7 35. ( , ) (0,1) 1 0 1 1 d C D d = = − = = 36. ( , ) (0, 3) 3 0 3 3 d C A d = − = − − = − = 37. ( , ) (1,3) 3 1 2 2 d D E d = = − = = 38. ( , ) (0,3) 3 0 3 3 d C E d = = − = = 39. ( , ) ( 3,3) 3 ( 3) 6 6 d A E d = − = − − = = 40. ( , ) (1, 1) 1 1 2 2 d D B d = − = − − = − = 41. 2 2 2 3 2 6 4 x y + = − + ⋅ = − + = 42. 3 3( 2) 3 6 3 3 x y + = − + = − + = − 43. 5 2 5( 2)(3) 2 30 2 28 xy + = − + = − + = − 44. 2 2( 2) ( 2)(3) 4 6 2 x xy − + = − − + − = − = − 45. 2( 2) 4 2 4 2 3 5 5 x x y − − = = = − − − − 46. 2 3 1 1 2 3 5 5 x y x y − + + = = = − − − − − 47. 3( 2) 2(3) 6 6 3 2 0 0 2 2 3 5 5 x y y − + − + + = = = = + + 48. 2( 2) 3 4 3 2 3 7 3 3 3 x y − − − − − = = = − 49. 3 ( 2) 1 1 x y + = + − = = 50. 3 ( 2) 5 5 x y − = − − = = 51. 3 2 3 2 5 x y + = + − = + = 52. 3 2 3 2 1 x y − = − − = − = 53. 3 3 1 3 3 x x = = = 54. 2 2 1 2 2 y y − = = = − − − 55. 4 5 4(3) 5( 2) 12 10 22 22 x y − = − − = + = = 56. 3 2 3(3) 2( 2) 9 4 5 5 x y + = + − = − = = 57. 4 5 4(3) 5( 2) 12 10 12 10 2 2 x y − = − − = − − = − = = 58. 3 2 3 3 2 2 3 3 2 2 9 4 13 x y + = + − = ⋅ + ⋅ = + = 59. 2 1 x x − Part (c) must be excluded. The value 0 x = must be excluded from the domain because it causes division by 0.

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    Section R.2: Algebra Essentials 7 60. 2 1 x x + Part (c) must be excluded. The value 0 x = must be excluded from the domain because it causes division by 0. 61. 2 ( 3)( 3) 9 x x x x x = − + − Part (a) , 3 x = , must be excluded because it causes the denominator to be 0. 62. 2 9 x x + None of the given values are excluded. The domain is all real numbers. 63. 2 2 1 x x + None of the given values are excluded. The domain is all real numbers. 64. 3 3 2 ( 1)( 1) 1 x x x x x = − + − Parts (b) and (d) must be excluded. The values 1, and 1 x x = = − must be excluded from the domain because they cause division by 0. 65. 2 2 3 5 10 5 10 ( 1)( 1) x x x x x x x x x + − + − = − + − Parts (b), (c), and (d) must be excluded. The values 0, 1, and 1 x x x = = = − must be excluded from the domain because they cause division by 0. 66. 2 2 3 2 9 1 9 1 ( 1) x x x x x x x x − − + − − + = + + Part (c) must be excluded. The value 0 x = must be excluded from the domain because it causes division by 0. 67. 4 5 x − 5 x = must be exluded because it makes the denominator equal 0. { } Domain 5 x x = ≠ 68. 6 4 x − + 4 x = − must be excluded sine it makes the denominator equal 0. { } Domain 4 x x = ≠ − 69. 4 x x + 4 x = − must be excluded sine it makes the denominator equal 0. { } Domain 4 x x = ≠ − 70. 2 6 x x − − 6 x = must be excluded sine it makes the denominator equal 0. { } Domain 6 x x = ≠ 71. 5 5 5 ( 32) (32 32) (0) 0 C 9 9 9 C F = − = − = = ° 72. 5 5 5 ( 32) (212 32) (180) 100 C 9 9 9 C F = − = − = = ° 73. 5 5 5 ( 32) (77 32) (45) 25 C 9 9 9 C F = − = − = = ° 74. 5 5 ( 32) ( 4 32) 9 9 5 ( 36) 9 20 C C F = − = − − = − = − ° 75. 2 ( 4) ( 4)( 4) 16 − = − − = 76. 2 2 4 (4) 16 − = − = − 77. 2 2 1 1 4 16 4 − = = 78. 2 2 1 1 4 16 4 − − = − = − 79. 6 4 6 4 2 2 1 1 3 3 3 3 9 3 − − + − ⋅ = = = = 80. 2 3 2 3 1 4 4 4 4 4 − − + ⋅ = = = 81. ( ) ( )( ) 1 2 1 2 2 3 3 3 9 − − − − = = =

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    Chapter R: Review 8 82. ( ) ( )( ) 3 1 3 1 3 2 2 2 8 − − − − = = = 83. 2 25 5 5 = = 84. 2 36 6 6 = = 85. ( ) 2 4 4 4 − = − = 86. ( ) 2 3 3 3 − = − = 87. ( ) ( ) 2 2 3 2 3 6 8 8 64 x x x = = 88. ( ) 1 2 2 2 1 1 4 4 4 x x x − − = = − − 89. ( ) ( ) ( ) 4 2 2 2 2 1 2 1 4 2 2 x x y x y x y y − − − = ⋅ = = 90. ( ) ( ) 3 3 3 1 1 3 3 3 3 y x y x y x y x − − − = ⋅ = = 91. 2 3 2 1 3 4 1 1 4 x y x x y x y y xy − − − = = = 92. 2 2 1 1 2 3 1 2 3 1 x y x y x y x y x y − − − − − − = = = 93. 3 4 2 4 2 2 2 3 3 4 1 2 3 2 1 3 1 1 3 ( 2) ( ) 8 3 9 8 9 8 9 8 9 x y z x y z x y z x y z x y z x y z x z y − − − − − − = − = − = = − 94. 2 1 2 1 1 3 4 4 2 4 1 1 1 6 2 1 6 2 4 ( ) 4 2 8 4 8 1 2 1 2 x y z x y z x y x y x y z x y z x y z − − − − − − − − − − − − − = = = = 95. 2 2 2 1 2 2 2 1 2 2 2 3 3 4 4 16 4 3 4 3 9 y x x x x x y y y y − − − −       = = = =             96. ( ) ( ) 3 3 3 2 2 2 2 2 2 3 3 2 6 3 6 3 2 5 5 6 6 6 5 6 216 125 5 y x x y x y x x y y − − − −       = =             = = 97. ( ) ( ) 1 2 2 2 2 4 1 x xy y − = = = − − 98. ( ) ( ) 1 3 1 3 3 3 2 2 y x y x − − − − − = = = 99. ( ) ( ) 2 2 2 2 2 1 4 1 5 x y + = + − = + = 100. ( ) ( ) 2 2 2 2 2 1 4 1 4 x y = − = ⋅ = 101. ( ) ( ) ( ) ( ) 2 2 2 2 1 2 4 xy = ⋅ − = − = 102. ( ) ( ) ( ) ( ) 2 2 2 2 1 1 1 x y + = + − = = 103. 2 2 2 x x = = = 104. ( ) 2 2 x x = = 105. ( ) ( ) 2 2 2 2 2 1 4 1 5 x y + = + − = + = 106. 2 2 2 1 2 1 3 x y x y + = + = + − = + = 107. 1 1 2 2 y x − = = 108. ( ) 2 1 1 x y = − =

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    Section R.2: Algebra Essentials 9 109. If 2, x = 3 2 3 2 2 3 5 4 2 2 3 2 5 2 4 16 12 10 4 10 x x x − + − = ⋅ − ⋅ + ⋅ − = − + − = If 1, x = 3 2 3 2 2 3 5 4 2 1 3 1 5 1 4 2 3 5 4 0 x x x − + − = ⋅ − ⋅ + ⋅ − = − + − = 110. If 1, x = 3 2 3 2 4 3 2 4 1 3 1 1 2 4 3 1 2 8 x x x + − + = ⋅ + ⋅ − + = + − + = If 2, x = 3 2 3 2 4 3 2 4 2 3 2 2 2 32 12 2 2 44 x x x + − + = ⋅ + ⋅ − + = + − + = 111. 4 4 4 4 (666) 666 3 81 222 (222)   = = =     112. ( ) 3 3 3 3 3 3 3 3 1 (0.1) (20) 2 10 10 1 2 10 10 2 8   = ⋅ ⋅     = ⋅ ⋅ = = 113. 6 (8.2) 304, 006.671 ≈ 114. 5 (3.7) 693.440 ≈ 115. 3 (6.1) 0.004 − ≈ 116. 5 (2.2) 0.019 − ≈ 117. 6 ( 2.8) 481.890 − ≈ 118. 6 (2.8) 481.890 − ≈ − 119. 4 ( 8.11) 0.000 − − ≈ 120. 4 (8.11) 0.000 − − ≈ − 121. 2 454.2 4.542 10 = × 122. 1 32.14 3.214 10 = × 123. 2 0.013 1.3 10 − = × 124. 3 0.00421 4.21 10 − = × 125. 4 32,155 3.2155 10 = × 126. 4 21, 210 2.121 10 = × 127. 4 0.000423 4.23 10 − = × 128. 2 0.0514 5.14 10 − = × 129. 4 6.15 10 61,500 × = 130. 3 9.7 10 9700 × = 131. 3 1.214 10 0.001214 − × = 132. 4 9.88 10 0.000988 − × = 133. 8 1.1 10 110, 000, 000 × = 134. 2 4.112 10 411.2 × = 135. 2 8.1 10 0.081 − × = 136. 1 6.453 10 0.6453 − × = 137. A lw = 138. ( ) 2 P l w = + 139. C d π = 140. 1 2 A bh = 141. 2 3 4 A x = 142. 3 P x = 143. 3 4 3 V r π = 144. 2 4 S r π =

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    Chapter R: Review 10 145. 3 V x = 146. 2 6 S x = 147. a. If 1000, x = 4000 2 4000 2(1000) 4000 2000 $6000 C x = + = + = + = The cost of producing 1000 watches is $6000. b. If 2000, x = 4000 2 4000 2(2000) 4000 4000 $8000 C x = + = + = + = The cost of producing 2000 watches is $8000. 148. 210 80 120 25 60 32 5 $98 + − + − − − = His balance at the end of the month was $98. 149. We want the difference between x and 4 to be at least 6 units. Since we don’t care whether the value for x is larger or smaller than 4, we take the absolute value of the difference. We want the inequality to be non-strict since we are dealing with an ‘at least’ situation. Thus, we have 4 6 x − ≥ 150. We want the difference between x and 2 to be more than 5 units. Since we don’t care whether the value for x is larger or smaller than 2, we take the absolute value of the difference. We want the inequality to be strict since we are dealing with a ‘more than’ situation. Thus, we have 2 5 x − > 151. a. 110 108 110 2 2 5 x − = − = − = ≤ 108 volts is acceptable. b. 110 104 110 6 6 5 x − = − = − = > 104 volts is not acceptable. 152. a. 220 214 220 6 6 8 x − = − = − = ≤ 214 volts is acceptable. b. 220 209 220 11 11 8 x − = − = − = > 209 volts is not acceptable. 153. a. 3 2.999 3 0.001 0.001 0.01 x − = − = − = ≤ A radius of 2.999 centimeters is acceptable. b. 3 2.89 3 0.11 0.11 0.01 x − = − = − = ≤ / A radius of 2.89 centimeters is not acceptable. 154. a. 98.6 97 98.6 1.6 1.6 1.5 x − = − = − = ≥ 97˚F is unhealthy. b. 98.6 100 98.6 1.4 1.4 1.5 x − = − = = < 100˚F is not unhealthy. 155. The distance from Earth to the Moon is about 8 4 10 400, 000, 000 × = meters. 156. The height of Mt. Everest is about 3 8848 8.848 10 = × meters. 157. The wavelength of visible light is about 7 5 10 0.0000005 − × = meters. 158. The diameter of an atom is about 10 1 10 0.0000000001 − × = meters. 159. The smallest commercial copper wire has a diameter of about 4 0.0005 5 10 − = × inches. 160. The smallest motor ever made is less than 2 0.05 5 10 − = × centimeters wide.

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    Section R.3: Geometry Essentials 11 161. ( )( ) ( )( ) 5 1 1 2 2 1.86 10 6 10 2.4 10 3.65 10 186, 000 60 60 24 365 × × × × ⋅ ⋅ ⋅ ⋅ = 10 12 586.5696 10 5.865696 10 = × = × There are about 12 5.9 10 × miles in one light- year. 162. 7 2 5 93, 000, 000 9.3 10 5 10 186, 000 1.86 10 500 seconds 8 min. 20 sec. × = = × × = ≈ It takes about 8 minutes 20 seconds for a beam of light to reach Earth from the Sun. 163. 1 0.333333 ... 0.333 3 = > 1 3 is larger by approximately 0.0003333 ... 164. 2 3 0.666666 ... 0.666 = > 2 3 is larger by approximately 0.0006666 ... 165. No. For any positive number a , the value 2 a is smaller and therefore closer to 0. 166. We are given that 2 1 10 x < < . This implies that 1 10 x < < . Since 10 3.162 x < ≈ and 3.142 x π > ≈ , the number could be 3.15 or 3.16 (which are between 1 and 10 as required). The number could also be 3.14 since numbers such as 3.146 which lie between π and 10 would equal 3.14 when truncated to two decimal places. 167. Answers will vary. 168. Answers will vary. 5 < 8 is a true statement because 5 is further to the left than 8 on a real number line. Section R.3 1. right; hypotenuse 2. 1 2 A bh = 3. 2 C r π = 4. similar 5. c 6. b 7. True. 8. True. 2 2 2 6 8 36 64 100 10 + = + = = 9. False; the surface area of a sphere of radius r is given by 2 4 V r π = . 10. True. The lengths of the corresponding sides are equal. 11. True. Two corresponding angles are equal. 12. False. The sides are not proportional. 13. 2 2 2 2 2 5, 12, 5 12 25 144 169 13 a b c a b c = = = + = + = + =  = 14. 2 2 2 2 2 6, 8, 6 8 36 64 100 10 a b c a b c = = = + = + = + =  = 15. 2 2 2 2 2 10, 24, 10 24 100 576 676 26 a b c a b c = = = + = + = + =  = 16. 2 2 2 2 2 4, 3, 4 3 16 9 25 5 a b c a b c = = = + = + = + =  =

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