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Solution Manual for Calculus I with Precalculus, 3rd Edition

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    iii CONTENTS Chapter P Prerequisites............................................................................................ 1 Chapter 1 Functions and Their Graphs................................................................. 89 Chapter 2 Polynomial and Rational Functions................................................... 173 Chapter 3 Limits and Their Properties................................................................ 277 Chapter 4 Differentiation..................................................................................... 320 Chapter 5 Applications of Differentiation .......................................................... 407 Chapter 6 Integration ........................................................................................... 524 Chapter 7 Exponential and Logarithmic Functions ........................................... 606 Chapter 8 Exponential and Logarithmic Functions and Calculus..................... 675 Chapter 9 Trigonometric Functions.................................................................... 725 Chapter 10 Analytic Trigonometry....................................................................... 813 Chapter 11 Trigonometric Functions and Calculus ............................................. 885 Chapter 12 Topics in Analytic Geometry............................................................. 944 Chapter 13 Additional Topics in Trigonometry................................................. 1054

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    C H A P T E R P Prerequisites Section P.1 Solving Equations................................................................................... 2 Section P.2 Solving Inequalities .............................................................................. 17 Section P.3 Graphical Representation of Data........................................................ 36 Section P.4 Graphs of Equations ............................................................................. 44 Section P.5 Linear Equations in Two Variables ..................................................... 55 Review Exercises .......................................................................................................... 70 Chapter Test ............................................................................................................... 83 Problem Solving ........................................................................................................... 85

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    2 C H A P T E R P Prerequisites Section P.1 Solving Equations 1. equation 2. 0 ax b  3. extraneous 4. factoring; extracting square roots; completing the square; Quadratic Formula 5. 4 1 4 4 x x   is an identity by the Distributive Property. It is true for all real values at x . 6. 6 3 5 2 10 x x      is conditional. There are real values of x for which the equation is not true. 7. 4 1 2 4 4 2 2 4 2 2 x x x x x x       This is an identity by simplification. It is true for all real values of x . 8. 2 2 2 3 2 6 4 x x x x     is an identity by simplification. It is true for all real values of x . 9. 1 4 3 1 1 x x x    is conditional. There are real values of x for which the equation is not true. 10. 5 3 24 x x  is conditional. There are real values of x for which the equation is not true (for example, 1 x ). 11. 11 15 11 11 15 11 4 x x x     12. 7 19 7 19 7 19 7 19 19 19 12 x x x x x x x          13. 7 2 25 7 7 2 25 7 2 18 2 18 2 2 9 x x x x x          14. 7 2 23 7 2 2 23 2 7 21 7 21 7 7 3 x x x x x     15. 8 5 3 20 8 3 5 3 3 20 5 5 20 5 5 5 20 5 5 25 5 25 5 5 5 x x x x x x x x x x x           16. 7 3 3 17 7 3 3 3 3 17 3 3 4 20 5 x x x x x x x x           17. 4 2 5 7 6 2 7 6 2 6 7 6 6 2 5 7 2 2 5 7 2 5 5 5 5 1 y y y y y y y y y y y y y              18. 5 8 3 3 5 1 1 3 9 5 5 1 3 9 4 5 3 9 5 9 4 5 5 9 8 5 x x x x x x x x x x x x                 19. 3 2 3 8 5 6 9 8 5 5 9 8 5 5 5 9 8 5 5 9 8 x x x x x x x x x x x x                z No solution. No solution. No solution.

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    Chapter P.1 Solving Equations 3 20. 9 10 5 2 2 5 9 10 5 4 10 9 10 9 10 x x x x x x x x         The solution is the set of all real numbers. 21. 3 4 4 8 3 3 4 24 24 4 8 3 9 32 96 23 96 23 23 96 23 x x x x x x x x  § ·  ¨ ¸ © ¹      22. 3 3 5 2 10 3 10 10 3 5 2 10 2 5 30 3 6 30 5 x x x x x x x x x x x   § · § ·   ¨ ¸ ¨ ¸ © ¹ © ¹     23. 3 1 2 4 3 1 2 4 3 1 2 4 6 5 5 24 0 4 5 24 4 0 4 5 4 24 4 0 6 5 24 0 6 30 24 0 5 6 z z z z z z z z z z z z    ª º    ¬ ¼            24. 0.60 0.40 100 50 0.60 40 0.40 50 0.20 10 50 x x x x x x     25. 8 2 2 8 2 4 8 4 8 4 x x x x x x x x         z  Contradiction; no solution 26. 8 2 3 2 1 2 5 8 16 6 3 2 10 2 13 2 10 13 10 x x x x x x x x           Contradiction; no solution 27. 100 4 5 6 6 3 4 100 4 5 6 12 12 12 6 3 4 4 100 4 3 5 6 72 400 16 15 18 72 31 310 10 x x x x x x x x x x      § · § ·  ¨ ¸ ¨ ¸ © ¹ © ¹         28. 17 32 100 17 32 100 17 32 100 49 2 100 49 98 1 2 y y y y y y y y y y y y y y y y y y           29. 5 4 2 5 4 3 3 5 4 2 5 4 15 12 10 8 5 20 4 x x x x x x x x       30. 15 6 4 3 15 6 7 9 7 9 7 9 7 x x x x x x x    31. 2 3 2 2 2 3 2 2 2 2 3 6 2 4 2 0 z z z z z z z   § ·    ¨ ¸  © ¹    32. 1 2 0 Multiply both sides by 5 . 5 1 5 2 0 3 5 0 3 5 5 3 x x x x x x x x x       Chapter P.1 Solvin Chapter P.1 Solvi

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    4 Chapter P Prerequisites 33. 4 2 0 Multiply both sides by 4 . 4 4 4 2 4 0 4 2 8 0 3 12 0 3 12 4 x x x x x x x x x x x               A check reveals that 4 x  is an extraneous solution because it makes the denominator zero. There is no real solution. 34. 2 2 7 8 4 Multiply both sides by 2 1 2 1 . 2 1 2 1 7 2 1 8 2 1 4 2 1 2 1 14 7 16 8 16 4 6 11 11 6 x x x x x x x x x x x x x x x x                  35. 2 3 4 1 3 3 3 4 1 Multiply both sides by 3 . 3 3 3 4 3 3 4 12 3 9 3 x x x x x x x x x x x x x x x x            A check reveals that 3 x is an extraneous solution because it makes the denominator zero. There is no solution. 36. 3 5 6 2 Multiply both sides by 3 . 3 3 6 3 2 3 5 6 18 2 3 15 4 18 3 15 3 x x x x x x x x x x x x x x x x               A check reveals that 3 x  is an extraneous solution because it makes the denominator zero. There is no solution. 37. 2 2 2 2 2 5 3 4 4 5 6 9 4 9 6 9 2 0 0 x x x x x x x x x x            38. 2 2 2 2 2 1 4 1 4 4 1 4 4 4 1 4 x x x x x x x        This is a contradiction. So, the equation has no solution. 39. 2 2 3 8 x x  General form: 2 2 8 3 0 x x   40. 2 2 2 13 3 7 0 13 3 14 49 0 13 3 42 147 0 x x x x x         General form: 2 3 42 134 0 x x    r equisites requisites

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    Chapter P.1 Solving Equations 5 41. 2 2 1 5 3 10 18 3 10 90 x x x x   General form: 2 3 90 10 0 x x   42. 2 2 2 2 2 2 5 1 2 5 1 4 2 1 0 1 4 2 1 1 0 x x x x x x x x x x             General form: 2 4 2 1 0 x x   43. 2 6 3 0 3 2 1 0 x x x x   1 2 3 0 or 2 1 0 0 or x x x x   44. 2 9 4 0 3 2 3 2 0 x x x    2 2 3 3 3 2 0 or 3 2 0 x x x x    45. 2 2 8 0 4 2 0 x x x x     4 0 or 2 0 4 or 2 x x x x    46. 2 10 9 0 9 1 0 x x x x     9 0 9 1 0 1 x x x x  Ÿ  Ÿ 47. 2 12 35 0 7 5 0 x x x x     7 0 or 5 0 7 or 5 x x x x   48. 2 4 12 9 0 2 3 2 3 0 x x x x     3 2 2 3 0 x x  Ÿ  49. 2 3 5 2 0 3 1 2 0 x x x x     1 2 3 0 or 1 2 0 3 or x x x x    50. 2 2 3 2 2 19 33 2 19 33 0 2 3 11 0 2 3 0 11 0 11 x x x x x x x x x x       Ÿ   Ÿ 51. 2 2 4 12 4 12 0 6 2 0 x x x x x x      6 0 or 2 0 6 or 2 x x x x    52. 2 2 1 8 16 0 8 128 0 16 8 0 x x x x x x       16 0 16 8 0 8 x x x x  Ÿ  Ÿ  53. 2 2 2 2 0 0 0 x ax a x a x a x a      54. 2 2 0 0 0 x a b x a b x a b x a b x a b   ª   ºª   º ¬ ¼¬ ¼ ª   ºª   º ¬ ¼¬ ¼ 0 0 x a b x a b x a b x a b   Ÿ     Ÿ   55. 2 49 7 x x r 56. 2 32 32 4 2 x x r r 57. 2 2 3 81 27 3 3 x x x r 58. 2 2 9 36 4 4 2 x x x r r Chapter P.1 Solvin Chapter P.1 Solvi

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    6 Chapter P Prerequisites 59. 2 12 16 12 4 12 4 16 or 8 x x x x x   r r 60. 2 13 25 13 25 13 5 13 5 8, 18 x x x x   r  r  r   61. 2 2 14 2 14 2 14 x x x   r  r 62. 2 5 30 5 30 5 30 x x x   r r 63. 2 2 1 18 2 1 18 2 1 3 2 1 3 2 2 x x x x   r r r 64. 2 2 2 3 27 0 2 3 27 2 3 27 2 3 3 3 2 3 3 3 3 3 3 2 x x x x x x     r  r  r  r 65. 2 2 7 3 7 3 x x x x    r  7 3 or 7 3 7 3 or 2 4 2 x x x x x x       z The only solution to the equation is 2. x 66. 2 2 5 4 5 4 x x x x    r  9 2 5 4 5 4 or 5 4 or 5 4 2 9 x x x x x x x x          z   The only solution to the equation is 9 2 . x  67. 2 2 2 2 2 2 4 32 0 4 32 4 2 32 2 2 36 2 6 2 6 4 or 8 x x x x x x x x x x x         r  r  68. 2 2 2 2 2 2 6 2 0 6 2 6 3 2 3 3 7 3 7 3 7 x x x x x x x x x           r  r 69. 2 2 2 2 2 2 12 25 0 12 25 12 6 25 6 6 11 6 11 6 11 x x x x x x x x x           r  r 70. 2 2 2 2 2 8 14 0 8 14 8 4 14 16 4 2 4 2 4 2 x x x x x x x x x           r  r r equisites requisites

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    Chapter P.1 Solving Equations 7 71. 2 2 2 2 2 2 2 2 8 4 0 4 8 0 4 8 0 4 8 4 2 8 2 2 12 2 12 2 2 3 x x x x x x x x x x x x x              r r 72. 2 2 2 2 2 2 4 14 3 9 4 2 14 4 3 3 9 9 18 2 3 9 2 3 2 3 2 3 9 12 14 2 2 2 x x x x x x x x x x          r r 73. 2 2 2 2 2 2 2 2 5 8 0 2 5 8 5 4 2 5 5 5 4 2 4 4 5 89 4 16 5 89 4 4 5 89 4 4 5 89 4 x x x x x x x x x x x x     § · § ·    ¨ ¸ ¨ ¸ © ¹ © ¹ § ·  ¨ ¸ © ¹  r  r  r 74. 2 2 2 2 2 2 99 4 99 1 1 2 4 4 100 1 2 4 1 2 1 2 9 1 11 2 2 2 4 4 99 0 25 25 5 , x x x x x x x x x x           r r  75. 2 2 2 2 2 2 2 5 15 7 0 7 3 0 5 7 3 5 3 7 3 3 2 5 2 3 17 2 20 3 17 2 20 3 17 2 2 5 3 17 2 2 5 3 85 2 10 15 85 10 x x x x x x x x x x x x x x        § · § ·    ¨ ¸ ¨ ¸ © ¹ © ¹ § ·  ¨ ¸ © ¹  r r  r r r 76. 2 2 2 2 2 2 2 2 3 9 5 0 5 3 0 3 5 3 3 3 5 3 3 2 3 2 3 5 9 2 3 4 3 7 2 12 3 7 2 12 3 7 2 2 3 3 7 2 2 3 3 21 2 6 9 21 6 x x x x x x x x x x x x x x x       § · § ·     ¨ ¸ ¨ ¸ © ¹ © ¹ § ·    ¨ ¸ © ¹ § ·  ¨ ¸ © ¹  r  r  r  r  r Chapter P.1 Solvin Chapter P.1 Solvi

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    8 Chapter P Prerequisites 77. 2 2 1 0 x x   2 2 4 2 1 1 4 2 1 2 2 1 3 1 , 1 4 2 b b ac x a  r   r    r  78. 2 25 20 3 0 x x   2 2 4 2 20 20 4 25 3 2 25 20 400 300 50 20 10 3 1 , 50 5 5 b b ac x a  r    r   r  r 79. 2 2 2 2 0 2 2 0 x x x x      2 2 4 2 2 2 4 1 2 2 1 2 2 3 1 3 2 b b ac x a  r   r     r r  80. 2 10 22 0 x x   2 2 4 2 10 10 4 1 22 2 1 10 100 88 2 10 2 3 5 3 2 b b ac x a  r    r   r  r r 81. 2 14 44 0 x x   2 2 4 2 14 14 4 1 44 2 1 14 2 5 7 5 2 b b ac x a  r   r   r  r 82. 2 2 6 4 6 4 0 x x x x    2 2 4 2 6 6 4 1 4 2 1 6 36 16 2 6 2 13 2 3 13 b b ac x a  r   r    r   r  r 83. 2 2 12 9 3 9 12 3 0 x x x x      2 2 4 2 12 12 4 9 3 2 9 12 6 7 2 7 18 3 3 b b ac x a  r   r     r r  84. 2 2 4 4 4 0 1 0 x x x x     2 2 4 2 1 1 4 1 1 2 1 1 1 4 2 1 5 2 2 b b ac x a  r    r    r  r 85. 2 9 24 16 0 x x   2 2 4 2 24 24 4 9 16 2 9 24 0 18 4 3 b b ac x a  r   r   r  r equisites requisites

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    Chapter P.1 Solving Equations 9 86. 2 16 40 5 0 x x   2 2 4 2 40 40 4 16 5 2 16 40 1600 320 32 40 16 5 32 5 5 4 2 b b ac x a  r    r   r  r r 87. 2 2 28 49 4 49 28 4 0 x x x x     2 2 4 2 28 28 4 49 4 2 49 28 0 2 98 7 b b ac x a  r   r      r  88. 2 2 3 1 0 3 1 0 x x x x     2 2 4 2 3 3 4 1 1 2 1 3 13 3 13 2 2 2 b b ac x a  r   r    r  r 89. 2 2 8 5 2 2 8 5 0 t t t t     2 2 8 8 4 2 5 4 8 2 6 6 2 2 2 2 4 2 b b ac t a  r     r   r r   90. 2 25 80 61 0 h h   2 2 4 2 80 80 4 25 61 2 25 80 6400 6100 50 8 10 3 5 50 8 3 5 5 b b ac h a  r   r   r   r  r 91. 2 2 5 2 12 25 0 y y y y    2 2 4 2 12 12 4 1 25 2 1 12 2 11 6 11 2 b b ac y a  r    r   r r 92. 2 2 2 2 5 14 8 7 25 20 196 8 49 25 28 196 0 49 25 1372 9604 0 x x x x x x x x x § ·  ¨ ¸ © ¹       2 2 4 2 1372 1372 4 25 9604 2 25 1372 921,984 50 686 196 6 25 b b ac x a  r    r   r r 93. 2 0.1 0.2 0.5 0 x x   2 2 4 2 0.2 0.2 4 0.1 0.5 2 0.1 1.449, 3.449 b ac x b a x x   r  r   |  Chapter P.1 Solvin Chapter P.1 Solvi

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    10 Chapter P Prerequisites 94. 2 0.005 0.101 0.193 0 x x    2 2 4 2 0.101 0.101 4 0.005 0.193 2 0.005 0.101 0.006341 0.01 2.137, 18.063 b b ac x a  r   r      r  | 95. 2 422 506 347 0 x x   2 506 506 4 422 347 2 422 1.687, 0.488 x x r    |  96. 2 3.22 0.08 28.651 0 x x    2 2 4 2 0.08 0.08 4 3.22 28.651 2 3.22 0.08 369.031 2.995, 2.971 6.44 b b ac x a  r    r     r |   97. 2 2 2 2 2 2 2 1 0 Complete the square. 2 1 2 1 1 1 1 2 1 2 1 2 x x x x x x x x x         r r 98. 2 2 11 33 0 Factor. 11 3 0 3 0 x x x x x x    0 or 3 0 3 x x x   99. 2 3 81 Extract square roots. 3 9 x x   r 3 9 or 3 9 6 or 12 x x x x     100. 2 2 14 49 0 Extract square roots. 7 0 7 0 7 x x x x x     101. 2 2 2 2 2 2 11 4 11 4 1 11 1 2 4 2 1 12 2 4 1 12 2 4 1 2 0 Complete the square. 3 x x x x x x x x x         r r 102. 2 2 3 4 2 7 Quadratic Formula 0 2 3 11 x x x x     2 3 3 4 2 11 2 2 3 97 4 3 97 4 4 x   r    r r 103. 2 2 2 4 2 4 2 8 Factor. 4 4 0 4 1 0 1 1 0 1 0 or 1 0 1 1 x x x x x x x x x x x           104. 2 2 2 0 Factor. 0 0 0 a x b ax b ax b b ax b x a b ax b x a     Ÿ   Ÿ 105. 4 2 2 2 2 2 50 0 2 25 0 2 5 5 0 x x x x x x x     2 or 5 0 or 5 0 2 0 or 5 or 5 0 x x x x x x    r equisites requisites

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    Chapter P.1 Solving Equations 11 106. 3 2 5 2 5 2 20 125 0 5 4 25 0 5 2 5 2 5 0 5 0 0 2 5 0 2 5 0 x x x x x x x x x x x x x     Ÿ  Ÿ   Ÿ 107. 4 2 2 81 0 9 3 3 0 9 0 No real solution 3 0 3 3 0 3 x x x x x x x x x      Ÿ  Ÿ   Ÿ 108. 6 3 3 2 2 2 2 64 0 8 8 0 2 2 4 2 2 4 0 2 0 2 2 4 0 No real solution (by the Quadratic Formula) 2 0 2 2 4 0 No real solution (by the Quadratic Formula) x x x x x x x x x x x x x x x x x           Ÿ   Ÿ  Ÿ    Ÿ 109. 3 3 3 2 2 216 0 6 0 6 6 36 0 6 0 6 6 36 0 No real solution (by the Quadratic Formula) x x x x x x x x x       Ÿ    Ÿ 110. 4 3 2 2 2 2 2 2 4 3 9 24 16 0 9 24 16 0 3 4 0 0 0 3 4 0 x x x x x x x x x x x x      Ÿ  Ÿ 111. 3 2 2 2 3 3 0 3 3 0 3 1 0 3 1 1 0 3 0 3 1 0 1 1 0 1 x x x x x x x x x x x x x x x x x             Ÿ  Ÿ   Ÿ 112. 3 2 2 2 2 2 3 6 0 2 3 2 0 2 3 0 2 0 2 3 0 No real solution x x x x x x x x x x x          Ÿ   Ÿ 113. 4 3 4 3 3 3 2 2 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 No real solution (by the Quadratic Formula) x x x x x x x x x x x x x x x x x x x x x                Ÿ  Ÿ    Ÿ Chapter P.1 Solvin Chapter P.1 Solvi 1 0 No real solution (by the Quadratic Formula) 1 0 No real solution (by the Quadratic Formula) x x   1 0 0 0 x x

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    12 Chapter P Prerequisites 114. 4 3 3 4 3 3 3 2 2 2 16 8 4 2 8 16 0 2 8 2 0 8 2 0 2 2 4 2 0 2 0 2 2 4 0 No real solution (by the Quadratic Formula) 2 0 2 x x x x x x x x x x x x x x x x x x x x x x                 Ÿ   Ÿ  Ÿ  115. 4 2 2 2 4 3 0 3 1 0 3 3 1 1 0 3 0 3 3 0 3 1 0 1 1 0 1 x x x x x x x x x x x x x x x x          Ÿ   Ÿ  Ÿ   Ÿ 116. 4 2 2 2 2 2 36 29 7 0 36 7 1 0 6 7 6 7 1 0 7 6 7 0 6 7 6 7 0 6 1 0 No real solution t t t t t t t t t t t t         Ÿ   Ÿ  Ÿ 117. 6 3 3 3 2 2 2 2 7 8 0 8 1 0 2 2 4 1 1 0 2 0 2 2 4 0 No real solution (by the Quadratic Formula) 1 0 1 1 0 No real solution (by the Quadratic Formula) x x x x x x x x x x x x x x x x x x            Ÿ    Ÿ  Ÿ   Ÿ 118. 6 3 3 3 2 3 3 3 2 2 3 3 3 2 0 2 1 0 2 2 2 1 1 0 2 0 2 x x x x x x x x x x x x     ª º       « » ¬ ¼  Ÿ  2 3 3 2 2 2 2 0 No real solution (by the Quadratic Formula) 1 0 1 1 0 No real solution (by the Quadratic Formula) x x x x x x   Ÿ  Ÿ    Ÿ 119. 2 10 0 2 10 2 100 50 x x x x  120. 36 49 7 6 0 7 6 49 36 x x x x  121. 10 4 0 10 4 10 16 26 x x x x     122. 5 3 0 5 3 5 9 4 x x x x      r equisites requisites 49 49 x x 4 4 x x 

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    Section P.1 Solving Equations 13 123. 2 5 3 0 2 5 3 2 5 9 2 4 2 x x x x x      No solution. 124. 3 2 2 0 3 2 2 3 2 4 2 1 1 2 x x x x x       125. 3 3 2 1 8 0 2 1 8 2 1 512 2 513 513 2 x x x x x         126. 3 3 4 3 2 0 4 3 2 4 3 8 4 5 5 4 x x x x x         127. 2 2 5 26 4 5 26 4 5 26 8 16 13 42 0 6 7 0 6 or 7 x x x x x x x x x x x x x            128. 5 2 5 5 2 5 10 x x x x x     129. 3 2 2 3 6 8 6 8 6 4 10 x x x x    130. 3 2 3 2 3 3 3 8 3 8 3 64 3 64 3 4 1 x x x x x       131. 2 3 3 2 3 5 3 5 3 5 5 3 5 5 x x x x    r  r 132. 4 3 2 2 3 4 22 16 22 16 x x x x     2 2 2 2 2 22 8 or 22 8 30 0 14 0 6 5 0 1 1 4 1 14 6 or 5 2 1 1 57 2 x x x x x x x x x x x x x x            r    r 133. 1 2 3 2 1 2 1 2 1 2 2 5 3 1 2 1 0 1 3 2 1 0 1 5 2 0 1 0 1 0 1 5 2 0 , extraneous x x x x x x x x x x x x x     ª   º ¬ ¼    Ÿ  Ÿ  Ÿ 134. 1 3 4 3 2 1 3 4 3 1 3 1 3 3 5 4 1 6 1 0 2 2 1 3 1 0 2 1 2 3 1 0 2 1 5 3 0 2 0 0 1 0 1 5 3 0 x x x x x x x x x x x x x x x x x x x x x    ª º    ¬ ¼  ª   º ¬ ¼   Ÿ  Ÿ  Ÿ S ection P.1 Solvin Section P.1 Solvi

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    14 Chapter P Prerequisites 135. 2 2 3 1 2 3 1 2 2 2 2 2 6 2 6 0 2 3 2 0 3 2 3 0 2 2 0 2 x x x x x x x x x x x x x x x x x  § · § ·  ¨ ¸ ¨ ¸ © ¹ © ¹       Ÿ   Ÿ 136. 2 2 4 3 1 1 2 4 2 3 1 1 2 4 8 3 3 3 2 2 3 0 1 3 0 1 0 1 3 0 3 x x x x x x x x x x x x x x x x x x                   Ÿ  Ÿ  137. 2 2 20 20 0 20 0 5 4 5 0 5 4 0 4 x x x x x x x x x x x x x        Ÿ   Ÿ 138. 2 2 3 4 1 3 4 1 4 3 4 3 0 4 3 1 0 3 4 3 0 4 1 0 1 x x x x x x x x x x x x x x x x x         Ÿ  Ÿ  139. 2 1 5 2 1 5 3 2 1 5 2 x x x x x   Ÿ   Ÿ  140. 11 13 13 1 12 13 1 12 13 1 12 13 1 12 1 x x x x x x   Ÿ     Ÿ  141. 2 3 x x x   First equation: 2 2 3 3 0 3 x x x x x    r Second equation: 2 2 3 2 3 0 1 3 0 1 0 1 3 0 3 x x x x x x x x x x x         Ÿ  Ÿ  Only 3 x and 3 x  are solutions to the original equation. 3 x  and 1 x are extraneous 142. 2 6 3 18 x x x   First equation: 2 2 6 3 18 3 18 0 3 6 0 3 0 3 6 0 6 x x x x x x x x x x x        Ÿ  Ÿ  Second equation: 2 2 6 3 18 0 9 18 0 3 6 0 3 3 0 6 6 x x x x x x x x x x x         Ÿ   Ÿ  The solutions to the original equation are 3 x r and 6. x  r equisites requisites

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