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Solution Manual for Algebra for College Students, 8th Edition

Solution Manual for Algebra for College Students, 8th Edition offers step-by-step solutions to help you understand tough concepts with ease.

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

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    TABLE OF CONTENTS for INSTRUCTOR SOLUTIONS A LGEBRA FOR C OLLEGE S TUDENTS Chapter 1: Algebra, Mathematical Models, and Problem Solving 1 Chapter 2: Functions and Linear Functions 67 Chapter 3: Systems of Linear Functions 131 Chapter 4: Inequalities and Problem Solving 251 Chapter 5: Polynomials, Polynomial Functions, and Factoring 331 Chapter 6: Rational Expressions, Functions, and Equations 423 Chapter 7: Radicals, Radical Functions, and Rational Exponents 587 Chapter 8: Quadratic Equations and Functions 677 Chapter 9: Exponential and Logarithmic Functions 839 Chapter 10: Conic Sections and Systems of Nonlinear Equations 943 Chapter 11: More on Polynomial and Rational Functions 1051 Chapter 12: Sequences, Induction, and Probability 1135

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 1 1.1 Check Points 1. a.  eight times a number five more 8 5 8 5 x x         b.  the quotient of a decreased by number and seven twice the number 2 2 7 7 x x x x        2. replace with 10 23 0.12 23 0.12(10) 23 1.2 21.8 x x          At age 10, the average neurotic level is 21.8. 3. replace with 13 2 2 2 8 6( 3) 8 6(13 3) 8 6(10) 8 6(100) 8 600 608 x x                 4. a. 2010 is 10 years after 2000. replace with 10 2 2 46 541 17,650 46(10) 541(10) 17,650 46(100) 541(10) 17,650 4600 5410 17,650 27,660 x D x x D               The formula indicates that the mean student-loan debt for college students who graduated in 2010 was $27,660. b. The model value, $27,660, is more than the actual data value, $26,682. Thus, the mathematical model overestimates by $978. 5. a. true; Because the number 13 is an element of the set of integers. b. true; Because 6 is not an element of {7, 8, 9, 10}, the statement is true. 6. a. 8  is less than 2;  true b. 7 is greater than 3;  true c. 1  is less than or equal to 4;  false d. 5 is greater than or equal to 5; true e. 2 is greater than or equal to 14;  true 7. a.   2 5 x x    b.   1 3.5 x x   c.   1 x x   1.1 Concept and Vocabulary Check 1. variable 2. expression 3. b th to the n th power; base; exponent 4. formula; modeling; models 5. natural 6. whole 7. integers 8. rational 9. irrational 10. rational; irrational 11. left 12. 2; 5; 2; 5 13. greater than 14. less than or equal to

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 2 1.1 Exercise Set 1. 5 x  2. 6 x  3. 4 x  4. 9 x  5. 4 x 6. 2 x 7. 2 10 x  8. 5 4 x  9. 1 6 2 x  10. 1 3 2 x  11. 4 2 x  12. 5 3 x  13. 3 5 x  14. 6 10 x  15. 7 5(10) 7 50 57     16.   8 6 5 8 30 38     17. 6(3) 8 18 8 10     18.   8 3 4 24 4 20     19.     2 1 1 1 1 3 3 9 9 3 1 1     20.     2 1 1 1 1 2 2 4 4 2 1 1     21. 2 7 6(7) 3 49 42 3 7 3 10         22.   2 8 7 8 4 64 56 4 8 4 12         23. 3 3 4 5(9 7) 4 5(2) 4 5(8) 4 40 44          24.       3 3 6 5 8 6 6 5 2 6 5 8 6 40 46          25. 2 8 3(8 2) 64 3(6) 64 18 46        26.     2 8 4 8 3 64 4 5 64 20 44        27. {1, 2, 3, 4} 28. {1, 2, 3} 29. {–7, –6, –5, –4} 30. {–6, –5, –4, –3} 31. {8, 9, 10, . . .} 32. {10, 11, 12, . . .} 33. {1, 3, 5, 7, 9} 34. {1, 3, 5, 7} 35. true; Seven is an integer. 36. true; Nine is an integer. 37. true; Seven is a rational number. 38. true; Nine is a rational number. 39. false; Seven is a rational number. 40. false; Nine is not an irrational number. 41. true; Three is not an irrational number. 42. true; Five is not an irrational number. 43. false; 1 2 is a rational number. 44. false; 1 4 is a rational number. 45. true; 2 is not a rational number. 46. true; π is not a rational number. 47. false; 2 is a real number.

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    Section 1.1 Algebraic Expressions, Real Numbers, and Interval Notation 3 48. false; π is a real number. 49. –6 is less than –2; true 50. –7 is less than –3; true 51. 5 is greater than –7; true 52. 3 is greater than –8; true 53. 0 is less than –4; false. 0 is greater than –4. 54. 0 is less than –5; false. 0 is greater than –5. 55. –4 is less than or equal to 1; true 56. –5 is less than or equal to 1; true 57. –2 is less than or equal to –6; false. –2 is greater than –6. 58. –3 is less than or equal to –7; false. –3 is greater than –7. 59. –2 is less than or equal to –2; true 60. –3 is less than or equal to –3; true 61. –2 is greater than or equal to –2; true 62. –3 is greater than or equal to –3; true 63. 2 is less than or equal to 1 2  ; false. 2 is greater than 1 2  . 64. 4 is less than or equal to 1 2  ; false. 4 is greater than 1 2  . 65.   1 6 x x   66.   2 4 x x    67.   5 2 x x    68.   4 3 x x    69.   3 1 x x    70.   2 5 x x    71.   2 x x  72.   3 x x  73.   3 x x   74.   5 x x   75.   3 x x  76.   2 x x  77.   5.5 x x  78.   3.5 x x  79. true 80. true 81. false;     3 1, 2,3, 4  . 82. false;     4 1, 2, 3, 4,5  .

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 4 83. true 84. true 85. false; The value of { x | x is an integer between 3  and 0} = { 2, 1}   , not { 3, 2, 1,0}    . 86. false; The value of { x | x is an integer between 4  and 0} = { 3, 2, 1}    , not { 4, 3, 2, 1, 0}     . 87. false; Twice the sum of a number and three is represented by   2 3 x  , not 2 3 x  . 88. false; Three times the sum of a number and five is represented by   3 5 x  , not 3 5 x  . 89. 4.6 0.02 4.6 0.02(20) 4.2 R x      The average resistance to happiness at age 20 is 4.2. 90. 4.6 0.02 4.6 0.02(30) 4.0 R x      The average resistance to happiness at age 30 is 4.0. 91. [4.6 0.02(30)] [4.6 0.02(50)] 4.0 3.6 0.4       The difference between the average resistance to happiness at age 30 and at age 50 is 0.4. 92. [4.6 0.02(20)] [4.6 0.02(70)] 4.2 3.2 1.0       The difference between the average resistance to happiness at age 20 and at age 70 is 0.4. 93.     2 2 32 8.7 0.3 32 8.7 4 0.3 4 32 34.8 4.8 62 S x x           According to the formula, 67% of American adults used smartphones to go online in 2013. The formula underestimated the actual value by 1%. 94.     2 2 32 8.7 0.3 32 8.7 3 0.3 3 32 26.1 2.7 55.4 S x x           According to the formula, 55.4% of American adults used smartphones to go online in 2012. The formula overestimated the actual value by 0.4%. 95. 5 (50 32) 9 C   5 (18) 10 9   10 ° C is equivalent to 50 ° F. 96. 5 5 5 ( 32) (86 32) (54) 30 9 9 9 C F       30 ° C is equivalent to 86 ° F. 97. 2 2 4 60 16 4 60(2) 16(2) 4 120 16(4) 4 120 64 124 64 60 h t t                Two seconds after it was kicked, the ball’s height was 60 feet. 98. 2 2 4 60 16 4 60(3) 16(3) 4 180 16(9) 4 180 144 184 144 40 h t t                Three seconds after it was kicked, the ball’s height was 40 feet. 99. – 116. Answers will vary. 117. does not make sense; Explanations will vary. Sample explanation: Many models work for a while and then no longer are valid beyond a certain point. 118. does not make sense; Explanations will vary. Sample explanation: Though this value is beyond the capabilities of a calculator, it still exists. This particular expression can be obtained via several software applications. 119. makes sense 120. does not make sense; Explanations will vary. Sample explanation: The model can be used to estimate the number in 2000 by letting 0. x 

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    Section 1.2 Operations with Real Numbers and Simplifying Algebraic Expression 5 121. false; Changes to make the statement true will vary. A sample change is: Every integer is a rational number. 122. false; Changes to make the statement true will vary. A sample change is: Some integers are not whole numbers. 123. true 124. true 125. Evaluate the two expressions.     2 4 20 2 24 48    2 4 20 8 20 28      Since the bird lover purchases 1 7 of the birds, the expression has to be a multiple of 7. Since 48 in not a multiple of 7 and 28 is a multiple of 7, we know that the correct expression is 2 4 20.   126.   2 3 3 5 45     127.     8 2 4 3 10     or   8 2 4 3 10     128. 26 is not a perfect square and 26 cannot be simplified. Consider the numbers closest to 26, both smaller and larger, which are perfect squares. The first perfect square smaller than 26 is 25. The first perfect square larger than 26 is 36. We know that the square root of 26 will lie between these numbers. We have 36 26 25.      If we simplify, we have 6 26 5.      Therefore, 26  lies between –6 and –5. 129. –5 and 5 are both a distance of five units from zero on a real number line. 130. 4 16 3(2) 16 3(16) 16 48 64 8 12 (10 6) 12 (4) 8 8           131. 2(3 5) 2(3(4) 5) 2(12 5) 2(17) 34 x        6 10 6(4) 10 24 10 34 x       1.2 Check Points 1. a. 6 6   because 6  is 6 units from 0. b. 4.5 4.5  because 4.5 is 4.5 units from 0. c. 0 0  because 0 is 0 units from 0. 2. a. 10 ( 18) 28      b. 0.2 0.9 0.7    c. 3 1 6 5 1 5 2 10 10 10        3. a. If 8, x   then ( 8) 8. x      b. If 1 3 , x  then 1 3 . x    4. a. 7 10 7 ( 10) 3       b. 4.3 ( 6.2) 4.3 6.2 10.5      c. 4 1 4 1 3 5 5 5 5 5               5. a. 2 ( 5) ( 5)( 5) 25      b. 2 5 (5 5) 25       c. 3 ( 4) ( 4)( 4)( 4) 64        d. 4 3 3 3 3 3 81 5 5 5 5 5 625                                      6. a. 32 8 4    b. 2 5 2 4 8 3 4 3 5 15                     7. 2 2 3 5 12 2( 4) 3 25 12 2(16) 3 25 6(16) 3 25 96 22 96 74                  

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 6 8. 3 4 3( 2) 2 (6 9) 4 3( 8) 2 ( 3) 4 24 2 3 20 5 4                 9. Commutative Property of Addition: 4 9 9 4 x x    Commutative Property of Multiplication: 4 9 4 9 x x     10. a. 6 (12 ) (6 12) 18 x x x        b. 7(4 ) ( 7 4) 28 x x x       11. 4(7 2) 28 8 x x      12. 2 2 2 2 2 2 3 14 11 (14 ) (3 11 ) (14 1) (3 11) 15 14 x x x x x x x x x x x x              13. 8(2 5) 4 16 40 4 16 4 40 12 40 x x x x x x x           14. 6 4[7 ( 2)] 6 4[7 2] 6 4[9 ] 6 36 4 42 4 x x x x x                1.2 Concept and Vocabulary Check 1. negative number 2. 0 3. positive number 4. positive number 5. positive number 6. negative number 7. positive number 8. divide 9. subtract 10. absolute value; 0; a 11. a ; a  12 . 0; inverse; 0; identity 13. b a  14. ( ) ab c 15. ab ac  16. simplified 1.2 Exercise Set 1. 7 7   2. 10 10   3. 4 4  4. 13 13  5. 7.6 7.6   6. 8.3 8.3   7. 2 2 π π  8. 3 3 π π  9. 2 2   10. 3 3   11. 2 2 5 5     12. 7 7 10 10    

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    Section 1.2 Operations with Real Numbers and Simplifying Algebraic Expression 7 13. 3 ( 8) 11      14. 5 ( 10) 15      15. 14 10 4     16. 15 6 9     17. 6.8 2.3 4.5     18. 7.9 2.4 5.5     19. 11 3 11 9 2 15 5 15 15 15                   20. 7 4 7 4 2 10 5 10 5 2 7 8 1 10 10 10                                   21. 2 3 2 3 9 4 9 4 8 27 35 36 36 36                         22. 3 4 3 4 5 7 5 7 21 20 41 35 35 35                         23. 3.7 ( 4.5) 8.2      24. 6.2 ( 5.9) 12.1      25. 0 ( 12.4) 12.4     26. 0 ( 15.3) 15.3     27. 12.4 ( 12.4) 0    28. 15.3 ( 15.3) 0    29. 11 11 x x     30. 13 13 x x     31. 5 5 x x     32. 9 9 x x     33. 0 0 x x    34. 2 2 x x     35.   3 15 3 15 12       36.   4 20 4 20 16       37. 8 ( 10) 8 10 18      38. 7 ( 13) 7 13 20      39. 20 ( 5) 20 5 15         40. 30 ( 10) 30 10 20         41. 1 1 1 1 1 2 1 4 2 4 2 4 4 4                      42. 1 2 1 2 1 2 2 10 5 10 5 10 5 2 1 4 3 10 10 10                                     43. 2.3 ( 7.8) 2.3 7.8 5.5        44. 4.3 ( 8.7) 4.3 8.7 4.4        45.   0 2 0 2 2      46.   0 3 0 3 3      47. 9( 10) 90    48. 8( 10) 80    49.     3 11 33    50.     7 11 77    51.   15 15 1 13 13   

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 8 52.   11 11 1 13 13    53. 2 0 0    54. 3 0 0    55.           4 2 1 8 1 8        56.           5 3 2 15 2 30        57. 2( 3)( 1)( 2)( 4) ( 6)( 1)( 2)( 4) (6)( 2)( 4) ( 12)( 4) 48                 58.                    3 2 1 5 3 6 1 5 3 6 5 3 30 3 90                 59.      2 10 10 10 100      60.      2 8 8 8 64      61.     2 10 10 10 100      62.     2 8 8 8 64      63.       3 2 2 2 2 8        64.       3 3 3 3 3 27        65.        4 1 1 1 1 1 1        66.        4 4 4 4 4 4 256        67. Since a product with an odd number of negative factors is negative,   33 1 1.    68. A product with an odd number of negative factors is negative.   35 1 1    69. 3 1 1 1 1 1 2 2 2 2 8                                 70. 3 1 1 1 1 1 4 4 4 4 64                                 71. 12 3 4    72. 30 6 5    73. 90 45 2    74. 55 11 5    75. 0 0 4.6   76. 0 0 5.3   77. 4.6 0  is undefined. 78. 5.3 0  is undefined. 79. 1 7 1 9 9 2 9 2 7 14                     80. 1 3 1 5 5 2 5 2 3 6                     81. 2 6 5 30 6 15 5 1 2 2                      82. 2 8 9 72 8 36 9 1 2 2                      83. 4( 5) 6( 3) 20 ( 18) 20 18 2             84. 8( 3) 5( 6) 24 ( 30) 24 30 6            85. 2 2 3( 2) 4( 3) 3(4) 4(9) 12 36 24          86. 2 2 5( 3) 2( 2) 5(9) 2(4) 45 8 37        

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    Section 1.2 Operations with Real Numbers and Simplifying Algebraic Expression 9 87. 2 2 8 16 2 4 3 64 16 4 4 3 64 4 4 3 64 16 3 48 3 45                    88. 2 2 10 100 5 2 3 100 100 25 2 3 100 4 2 3 100 8 3 92 3 89                    89.     2 2 2 2 2 2 5 2 3 5 2 9 9 ( 2) 3 ( 2) 10 9 9 2 1 11 1 121                   90.       2 2 2 10 2 3 4 5 3 4 5 12 17 36 12 3 2 12 6 6             91.         8 3 2(2 5) 4(8 6) 8 3 2( 3) 4(2) 8 3 6 8 8 3 2 8 6 14                    92.           8 3 2(5 7) 5(4 2) 8 3 2( 2) 5(2) 8 3 4 10 8 3 4 10 8 3 6 8 18 26                            93.     2 2 4 3 4 12 8 8 5 8 3 3 3             94.     6 4 5 3 24 15 9 9 9 10 1 1             95.       2 2 2 5 6 2 3 7 1 2 4 89 3 25 89 3 5 1 2 4 89 75 1 8 7 1 14 14 2                     96. 2 2 2 12 3 5 2 3 12 3 5 4 9 7 3 36 7 3 6 12 3 5 13 10 36 12 3 5(13) 4 5(13) 26 26 20(13) 260 10 26 26                             97. 15 3 ( 1) 12 2 3 15 4 12 2 3 15 2 12 2 3 15 2 6 3 15 2 18 13 18 31                           98.   17 5 2 12 2 3 17 7 12 2 3 17 7 12 2 3 17 7 6 3 17 7 18 10 18 28                           99.             2 2 2 2 20 1 10 5 1 2 20 1 10 6 2 20 1 100 36 2 20 1 64 2 20 1 8 2 20 1 16 37                            100.         2 2 2 24 3 5 2 1 3 24 3(3) 1 3 24 9 2 24 3 4 8 4 2                          101. Commutative Property of Addition 4 10 10 4 x x    Commutative Property of Multiplication 4 10 4 10 x x     102. Commutative Property of Addition 5 30 30 5 x x    Commutative Property of Multiplication 5 30 5 30 x x     103. Commutative Property of Addition 7 5 5 7 x x     Commutative Property of Multiplication 7 5 7 5 x x    

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 10 104. Commutative Property of Addition 3 7 7 3 x x     Commutative Property of Multiplication 3 7 3 7 x x     105. 4 (6 ) (4 6) 10 x x x        106. 12 (3 ) (12 3) 15 x x x        107. 7(3 ) ( 7 3) 21 x x x       108. 10(5 ) ( 10 5) 50 x x x       109. 1 1 ( 3 ) 3 3 3 y y y              110. 1 1 ( 4 ) 4 4 4 y y y              111. 3(2 5) 3 2 3 5 6 15 x x x        112. 5(4 7) 5 4 5 7 20 35 x x x        113.   7(2 3) 7 2 7 3 14 21 x x x           114.   9(3 2) 9 3 9 2 27 18 x x x           115.   (3 6) 1 3 1 6 3 6 x x x           116.     (6 3) 1 6 1 3 6 3 x x x          117.   7 5 7 5 12 x x x x     118.   8 10 8 10 18 x x x x     119.   2 2 2 2 6 6 1 5 x x x x     120.   2 2 2 2 9 9 1 8 x x x x     121.     2 2 2 2 2 2 6 10 4 2 6 4 10 2 6 4 10 2 10 12 x x x x x x x x x x x x              122.     2 2 2 2 9 5 3 4 9 3 5 4 12 9 x x x x x x x x          123.   8(3 5) 6 8 3 8 5 6 24 40 6 24 6 40 24 6 40 18 40 x x x x x x x x x x                   124.   7(4 5) 8 7 4 7 5 8 28 35 8 28 8 35 20 35 x x x x x x x x                125.       5(3 2) 7 2 5 3 5 2 1 7 1 2 15 10 7 2 15 7 10 2 15 7 12 8 12 y y y y y y y y y y                         126.         4(5 3) 6 3 4 5 4 (3) 1 6 1 3 20 12 6 3 20 6 15 14 15 y y y y y y y y                    127.     7 4 3 4 5 7 4 3 4 5 7 12 16 20 16 25 y y y y                  128.          6 5 8 2 4 6 5 8 2 4 6 5 12 2 6 5 12 5 2 6 60 10 10 54 y y y y y y                         129.     2 2 2 2 2 2 2 2 2 2 2 2 18 4 6 2 5 18 4 6 12 5 18 4 6 7 18 4 6 7 18 6 4 7 18 6 11 12 11 x x x x x x x x x x x x                                      

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    Section 1.2 Operations with Real Numbers and Simplifying Algebraic Expression 11 130.   2 2 2 2 2 2 14 5 7 2 4 14 5 7 14 4 14 5 7 10 x x x x x x                            2 2 2 2 2 2 14 5 7 10 14 7 5 10 14 7 15 7 15 x x x x x x              131.   4 4 4 x x x x        132.   8 8 2 8 x x x x x        133.   6 5 30 x x    134.   10 4 40 x x    135. 5 2 3 x x x   136.   6 2 6 2 8 x x x x x      137.   8 3 6 8 3 6 5 6 x x x x x        138.   8 3 6 8 3 18 3 10 x x x         139. 21 ( 29) 8     140. 4 ( 10) 6     141. 21 ( 29) 21 29 50      142. 4 ( 10) 4 10 14      143. 3 ( 10) 3 10 7        The approval rating of France exceeds the approval rating of China by 7. 144. 3 ( 29) 3 29 26        The approval rating of France exceeds the approval rating of Iran by 26. 145. 10 ( 3) 4 9 3 3 3         The average approval rating of China, France, and Israel is 3.  146. 29 ( 10) 21 18 3 3 6         The average approval rating of Iran, China, and the UK is 6.  147.   2 2 1.2 1.6( 40) 1.2 6 1.6(6 40) 116.8 D x x        According to the model, college students spent $116.8 billion in 2013. The model underestimates the actual value displayed in the graph by $0.2 billion. 148.   2 2 1.2 1.6( 40) 1.2 4 1.6(4 40) 89.6 D x x        According to the model, college students spent $89.6 billion in 2011. The model overestimates the actual value displayed in the graph by $2.6 billion. 149. a.   0.05 0.12 10,000 0.05 1200 0.12 1200 0.07 x x x x x        b.         0.05 6000 0.12 10,000 6000 0.05 6000 0.12 4000 300 480 780          1200 0.07 6000 1200 420 780     The total interest will be $780. 150. a. 0.06 0.5(50 ) 0.06 25 0.5 25 0.44 t t t t t        b. 0.06(20) 0.5(50 20) 0.06(20) 0.5(30) 1.2 15 16.2 25 0.44(20) 25 8.8 16.2            The total distance will be 16.2 miles.

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 12 151. – 167. Answers will vary. 168. makes sense 169. makes sense 170. does not make sense; Explanations will vary. Sample explanation: For terms to be considered like terms they must have the same variables and the same powers. 171. does not make sense; Explanations will vary. Sample explanation: When there is no number in front of a variable, the coefficient has a value of 1. 172. false; Changes to make the statement true will vary. A sample change is: 16 4 2 4 2 8      173. false; Changes to make the statement true will vary. A sample change is:     6 2 4 3 6 2 7 6 14 8         174. false; Changes to make the statement true will vary. A sample change is: 5 3( 4) 5 3 12 3 7 x x x        175. false; Changes to make the statement true will vary. A sample change is: 2 x x x     176. true 177. (8 2) 3 4 14     178. 1 2 5 10 9 45 2            179.     2 2 9 4 (1 6) (3 9) 9 4 7 ( 6) 12 12 5 5 6 6 5 5 2 1 3                9 3 36 12 5 5 2 27 36 12 5 3 63 5 4 63 9 7                 180. 10 4 x x  181.   4 4 4 10 2( 5) 10 2(7 5) 10 2(2) 10 2 16 10 32 42 x             182. true; 1 2 is not an irrational number. 183. x 2 4 y x   − 3 2 4 ( 3) 4 9 5 y        − 2 2 4 ( 2) 4 4 0 y       − 1 2 4 ( 1) 4 1 3 y       0 2 4 (0) 4 0 4 y      1 2 4 (1) 4 1 3 y      2 2 4 (2) 4 4 0 y      3 2 4 (3) 4 9 5 y       184. x 2 1 y x   − 3 2 1 ( 3) 1 9 8 y        − 2 2 1 ( 2) 1 4 3 y        − 1 2 1 ( 1) 1 1 0 y       0 2 1 (0) 1 0 1 y      1 2 1 (1) 1 1 0 y      2 2 1 (2) 1 4 3 y       3 2 1 (3) 1 9 8 y       185. x 1 y x   − 4 4 1 3 3 y       − 3 3 1 2 2 y       − 2 2 1 1 1 y       − 1 1 1 0 0 y      0 0 1 1 1 y     1 1 1 2 2 y     2 2 1 3 3 y    

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    Section 1.3 Graphing Equations 13 1.3 Check Points 1. 2. Make a table: 2 2 2 2 2 2 2 2 1 ( , ) 3 1 ( 3) 8 ( 3, 8) 2 1 ( 2) 3 ( 2, 3) 1 1 ( 1) 0 ( 1, 0) 0 1 (0) 1 (0,1) 1 1 (1) 0 (1, 0) 2 1 (2) 3 (2, 3) 3 1 (3) 8 (3, 8) x y x x y y y y y y y y = − − = − − = − − − − = − − = − − − − = − − = − = − = = − = = − = − − = − = − − 3. Make a table: 1 ( , ) 4 4 1 3 3 ( 4,3) 3 3 1 2 2 ( 3, 2) 2 2 1 1 1 ( 2,1) 1 1 1 0 0 ( 1,0) 0 0 1 1 1 (0,1) 1 1 1 2 2 (1, 2) 2 2 1 3 3 (2,3) x y x x y y y y y y y y                                              4. a. The drug concentration is increasing from 0 to 3 hours. b. The drug concentration is decreasing from 3 to 13 hours. c. The drug’s maximum concentration is 0.05 milligram per 100 milliliters, which occurs after 3 hours. d. None of the drug is left in the body. 5. The minimum x -value is –100, the maximum x - value is 100, and the distance between consecutive tick marks is 50. The minimum y -value is –100, the maximum y -value is 100, and the distance between consecutive tick marks is 10. 1.3 Concept and Vocabulary Check 1. x -axis 2 . y -axis 3 . origin 4. quadrants; four 5. x -coordinate; y -coordinate 6 . solution; satisfies

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 14 1.3 Exercise Set 1. – 10. 11. x   , x y − 3   3,5  − 2   2, 0  − 1   1, 3   0   0, 4  1   1, 3  2   2,0 3   3,5 12. x   , x y − 3   3, 0  − 2   2, 5   − 1   1, 8   0   0, 9  1   1, 8  2   2, 5  3   3, 0 13. x   , x y − 3   3, 5   − 2   2, 4   − 1   1, 3   0   0, 2  1   1, 1  2   2,0 3   3,1

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    Section 1.3 Graphing Equations 15 14. x   , x y − 3   3, 1   − 2   2, 0  − 1   1,1  0   0, 2 1   1,3 2   2, 4 3   3,5 15. x   , x y − 3   3, 5   − 2   2, 3   − 1   1, 1   0   0,1 1   1,3 2   2,5 3   3, 7 16. x   , x y − 3   3, 10   − 2   2, 8   − 1   1, 6   0   0, 4  1   1, 2  2   2,0 3   3, 2 17. x   , x y − 3 3 3, 2        − 2   2,1  − 1 1 1, 2        0   0,0 1 1 1, 2        2   2, 1  3 3 3, 2       

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 16 18. x   , x y − 3 7 3, 2        − 2   2,3  − 1 5 1, 2        0   0, 2 1 3 1, 2       2   2,1 3 1 3, 2       19. x   , x y − 3   3, 4  − 2   2,3  − 1   1, 2  0   0,1 1   1, 2 2   2,3 3   3, 4 20. x   , x y − 3   3, 2  − 2   2,1  − 1   1, 0  0   0, 1  1   1,0 2   2,1 3   3, 2 21. x   , x y − 3   3, 6  − 2   2, 4  − 1   1, 2  0   0,0 1   1, 2 2   2, 4 3   3, 6

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    Section 1.3 Graphing Equations 17 22. x   , x y − 3   3, 6   − 2   2, 4   − 1   1, 2   0   0,0 1   1, 2  2   2, 4  3   3, 6  23. ( , ) 3 ( 3, 9) 2 ( 2, 4) 1 ( 1, 1) 0 (0,0) 1 (1, 1) 2 (2, 4) 3 (3, 9) x x y             24. ( , ) 9 3 ( 3, ) 2 2 ( 2, 2) 1 1 ( 1, ) 2 0 (0,0) 1 1 (1, ) 2 2 (2, 2) 9 3 (3, ) 2 x x y             25. ( , ) 3 ( 3, 27) 2 ( 2, 8) 1 ( 1, 1) 0 (0,0) 1 (1,1) 2 (2,8) 3 (3, 27) x x y         

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 18 26. x   , x y − 3   3, 28   − 2   2, 9   − 1   1, 2   0   0, 1  1   1,0 2   2,7 3   3, 26 27. [–5, 5, 1] by [–5, 5, 1] This matches graph c. 28. [–10, 10, 2] by [– 4, 4, 2] This matches graph d. 29. [–20, 80, 10] by [–30, 70, 10] This matches graph b. 30. [–40, 40, 20] by [–1000, 1000, 100] This matches graph a. 31. The equation that corresponds to 2 Y in the table is (c), 2 2 y x   . We can tell because all of the points ( 3,5)  , ( 2, 4)  , ( 1,3)  , (0, 2) , (1,1) , (2,0) , and (3, 1)  are on the line 2 y x   , but all are not on any of the others. 32. The equation that corresponds to 1 Y in the table is (b), 2 1 y x  . We can tell because all of the points ( 3,9)  , ( 2, 4)  , ( 1,1)  , (0,0) , (1,1) , (2, 4) , and (3,9) are on the graph 2 y x  , but all are not on any of the others. 33. No. It passes through the point (0, 2) . 34. Yes. It passes through the point (0,0) . 35. (2,0) 36. (0, 2) 37. The graphs of 1 Y and 2 Y intersect at the points ( 2, 4)  and (1,1) . 38. The values of 1 Y and 2 Y are the same when 2 x   and 1 x  . 39. 2 4 y x   40. 4 2 y x  

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    Section 1.3 Graphing Equations 19 41. 2 3 y x   42. 2 2 y x   43. ( , ) 3 ( 3,5) 2 ( 2,5) 1 ( 1,5) 0 (0,5) 1 (1,5) 2 (2,5) 3 (3,5) x x y       44. ( , ) 3 ( 3, 1) 2 ( 2, 1) 1 ( 1, 1) 0 (0, 1) 1 (1, 1) 2 (2, 1) 3 (3, 1) x x y              45.     ( , ) 1 2 2, 2 1 1, 1 1 1 , 2 2 2 1 1 , 3 3 3 1 1 ,3 3 3 1 1 , 2 2 2 1 1,1 1 2 2, 2 x x y                                                

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 20 46.     ( , ) 1 2 2, 2 1 1,1 1 1 , 2 2 2 1 1 ,3 3 3 1 1 , 3 3 3 1 1 , 2 2 2 1 1, 1 1 2 2, 2 x x y                                                 47. The greatest percentage of the U.S. population that used the internet was 84%, in 2013. 48. The least percentage of the U.S. population that used the internet was 70%, in 2011. 49. The percentage of the U.S. population that used the internet remained constant in 2009 and 2010 at 71%. 50. The percentage of the U.S. population that used the internet increased most rapidly between 2011 and 2012. It increased by 9%. 51. The percentage of the U.S. population that used the internet decreased most rapidly between 2008 and 2009. It decreased by 3%. 52. Between 2007 to 2013, the increase was 9%. 53. At age 8, women have the least number of awakenings, averaging about 1 awakening per night. 54. At age 65, men have the greatest number of awakenings, averaging about 8 awakenings per night. 55. The difference between the number of awakenings for 25-year-old men and women is about 1.9. 56. The difference between the number of awakenings for 18-year-old men and women is about 1.1. 57. graph a 58. graph d 59. graph b 60. graph c 61. graph b 62. graph a 63. graph c 64. graph b 65. – 72. Answers will vary. 73. makes sense 74. does not make sense; Explanations will vary. Sample explanation: Most graphing utilities do not display numbers on the axes. 75. makes sense

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    Section 1.4 Solving Linear Equations 21 76. does not make sense; Explanations will vary. Sample explanation: There may or may not be a mathematical model that perfectly describes the graph’s data. 77. false; Changes to make the statement true will vary. A sample change is: If the product of a point’s coordinates is positive, the point could be in quadrant I or III. 78. false; Changes to make the statement true will vary. A sample change is: When a point lies on the x -axis, y = 0. 79. true 80. false; Changes to make the statement true will vary. A sample change is: Substituting the coordinates of (2,5) into 3 2 4 y x    gives 3(5) 2(2) 4    which simplifies to 11 4   which is false. 81. The four hour day costs $6 and the five hour day costs $9. Thus the total cost for the two days is $15. 82. Your car was parked more than six hours, but not exceeding eight hours. 83. 14.3 14.3   84.             12 13 17 9 6 10 12 4 9 4 12 4 9 4 16 13 3                                   85.       6 5 4 3 10 6 20 15 10 6 20 15 10 14 25 x x x x x x               86. 4 3 5 6 4( 9) 3 5( 9) 6 36 3 45 6 39 39 x x                 The statement is true for 9. x   87. 13 3( 2) 13 3 6 7 3 x x x        88. 3 1 10 2 10 3 1 1 2 5(3 1) 15 5 x x x x               1.4 Check Points 1. 4 5 29 4 5 5 29 5 4 24 4 24 4 4 6 x x x x x          The solution set is {6}. Check: 4 5 29 4(6) 5 29 24 5 29 29 29 x        2. 2 12 6 4 5 3 12 11 4 3 11 4 12 8 8 8 8 8 8 1 x x x x x x x x x x x                     The solution set is {–1}. Check: 2 12 6 4 5 2( 1) 12 ( 1) 6( 1) 4 5( 1) 2 12 1 6 4 5 15 15 x x x x                        

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 22 3. 2( 3) 17 13 3( 2) 2 6 17 13 3 6 2 23 7 3 2 3 7 23 5 30 5 30 5 5 6 x x x x x x x x x x x                    The solution set is {6}. Check: 2( 3) 17 13 3( 2) 2(6 3) 17 13 3(6 2) 2(3) 17 13 3(8) 6 17 13 24 11 11 x x                    4. 5 3 5 7 4 14 x x           5 3 5 28 28 7 4 14 28 5 28 3 28 5 1 7 1 4 1 14 4 5 7 3 2 5 4 20 7 21 10 11 1 10 11 10 1 11 11 11 11 11 11 1 x x x x x x x x x x x x x                                                      The solution set is {1}. Check: 5 3 5 7 4 14 1 5 1 3 5 7 4 14 6 2 5 7 4 14 24 14 10 28 28 28 10 10 28 28 x x                5. 4 7 4( 1) 3 4 7 4 4 3 4 7 4 1 7 1 x x x x x x               This equation is an inconsistent equation and thus has no solution. The solution set is { }. 6. 7 9 9( 1) 2 7 9 9 9 2 7 9 7 9 9 9 x x x x x x x x             This equation is an identity and all real numbers are solutions. The solution set is   is a real number x x or ( , )   or  . 7. 394 3123 11, 397 394 3123 11,397 3123 394 8274 394 8274 394 394 394 21 T x x x x x x          The average cost of tuition and fees at public colleges will reach $11,397 in the school year ending 21 years after 2000, or 2021. 1.4 Concept and Vocabulary Check 1. linear 2. equivalent 3. b c  4. bc 5. apply the distributive property 6. least common denominator; 12 7. inconsistent;  8. identity; ( , )   1.4 Exercise Set 1. 5 3 18 5 3 3 18 3 5 15 5 15 5 5 3 x x x x x          The solution set is {3}.

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    Section 1.4 Solving Linear Equations 23 2. 3 8 50 3 42 14 x x x     The solution set is {14}. 3. 6 3 63 6 3 3 63 3 6 66 6 66 6 6 11 x x x x x          The solution set is {11}. 4. 5 8 72 5 80 16 x x x     The solution set is {16}. 5. 14 5 41 14 5 14 41 14 5 55 5 55 5 5 11 x x x x x                  The solution set is {11}. 6. 25 6 83 6 108 18 x x x        The solution set is {18}. 7.   11 6 5 40 11 6 5 40 5 5 40 5 5 5 40 5 5 35 7 x x x x x x x x               The solution set is {7}. 8.   5 2 8 35 5 2 8 35 3 8 35 3 27 9 x x x x x x x           The solution set is {9}. 9. 2 7 6 2 7 6 x x x x x x         7 6 x   7 7 6 7 13 x x      The solution set is {13}. 10. 3 5 2 13 5 13 8 x x x x       The solution set is {8}. 11. 7 4 16 7 4 16 6 4 16 6 4 4 16 4 6 12 6 12 6 6 2 x x x x x x x x x x x                  The solution set is {2}. 12. 7 42 8 1 43 7 1 43 6 x x x x x        The solution set is {6}. 13. 8 3 11 9 8 8 3 11 8 9 3 3 9 3 9 3 9 9 12 3 12 3 3 3 4 y y y y y y y y y y y                       The solution set is {–4}. 14. 5 2 9 2 2 4 2 4 4 1 y y y y y           The solution set is {–1}. 15.     3 2 7 2 5 3 6 7 2 10 x x x x         3 2 6 7 2 2 10 6 7 10 1 10 1 1 10 1 9 x x x x x x x x                 The solution set is {9}.

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 24 16.     2 1 3 3 1 2 2 3 3 3 2 1 2 3 4 1 3 4 4 1 x x x x x x x x x x x                      The solution set is {–1}. 17.       3 4 4 3 3 2 3 12 4 12 3 2 5 5 x x x x x x x x x x                   The solution set is {–5}. 18.   2 7 5 13 3 2 7 5 13 3 7 3 13 3 4 3 13 4 16 4 x x x x x x x x x                    The solution set is {–4}. 19.     16 3 1 7 16 3 3 7 16 2 4 16 4 2 4 4 12 2 12 2 2 2 6 x x x x x x x x x                  The solution set is {6}. 20.     5 2 2 3 5 5 2 2 3 5 3 2 4 5 2 5 3 x x x x x x x x x x x x                  The solution set is {3}. 21.         7 1 4 3 7 7 4 3 7 7 4 2 3 7 7 8 12 7 7 7 8 7 12 7 12 7 12 12 12 19 x x x x x x x x x x x x x x x x x                               The solution set is {19}. 22.         2 3 4 6 5 6 2 3 4 6 5 30 2 6 5 30 2 12 5 30 12 7 30 42 7 6 x x x x x x x x x x x x x                         The solution set is {6}. 23.     1 2 4 8 16 9 12 2 3 2 4 16 6 8 2 12 6 8 8 12 8 8 20 20 5 8 2 z z z z z z z z z                    The solution set is 5 2       . 24.     3 2 24 8 16 6 9 4 3 18 6 16 4 6 2 6 4 6 2 2 6 2 4 2 z z z z z z z z z                     The solution set is {–2}. 25. 2 3 2 6 6 2 3 2 2 3 12 2 3 3 3 12 12 12 x x x x x x x x x x x x                           The solution set is {12}. 26. 1 5 6 30 30 1 5 6 x x x x                 6 5 30 30 x x x    The solution set is {30}.

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    Section 1.4 Solving Linear Equations 25 27. 20 3 2 6 20 6 3 2 120 2 3 120 2 2 3 2 120 5 120 5 5 5 24 x x x x x x x x x x x x x                          The solution set is {24}. 28. 1 5 2 6 1 30 30 5 2 6 6 15 5 15 0 15 x x x x x x x x                      The solution set is {15}. 29. 3 2 1 5 3 3 2 15 15 1 5 3 9 10 15 9 10 10 10 15 15 15 x x x x x x x x x x x x                           The solution set is {–15}. 30. 3 5 2 4 3 4 4 5 2 4 2 3 20 20 20 x x x x x x x x                       The solution set is {–20}. 31. 3 5 5 10 2 3 5 10 10 5 10 2 x x x x x x                   6 10 25 4 25 4 25 5 25 5 x x x x x x x x x x x                The solution set is {5}. 32.     2 17 2 7 2 2 2 17 14 2 14 7 2 2 28 2 2 7 7 17 28 4 7 119 24 7 119 17 119 7 x x x x x x x x x x x x x x x x                             The solution set is {7}. 33.       3 2 5 6 3 4 3 2 5 12 12 12 6 3 4 2 3 4 2 3 5 2 6 8 3 15 2 6 3 7 6 7 13 13 x x x x x x x x x x x x x                                              The solution set is {13}. 34.     1 1 2 4 6 3 1 1 2 12 12 4 6 3 3 1 2 4 2 3 3 2 8 4 3 3 10 4 7 3 10 x x x x x x x x x x x                                  7 7 1 x x   The solution set is {1}. 35.   3 2 4 3 3 12 12 2 4 3 3 24 4 3 3 24 4 12 3 12 4 3 4 12 4 4 12 12 x x x x x x x x x x x x x x x x                                   The solution set is {–12}.

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 26 36. 2 3 5 3 8 x x         2 3 24 5 24 3 8 120 8 2 3 3 120 8 16 3 9 104 8 3 9 104 5 9 5 95 19 x x x x x x x x x x x                                  The solution set is {–19}. 37. 1 2 5 3 7 x x         1 2 21 21 5 3 7 7 1 105 3 2 7 7 105 3 6 7 3 7 105 3 3 6 10 7 99 10 92 92 46 10 5 x x x x x x x x x x x x x                                    The solution set is 46 5       . 38. 3 3 2 5 2 3 x x x           3 3 2 30 30 5 2 3 6 3 15 3 10 2 18 15 45 10 20 3 45 10 20 45 7 20 25 7 25 7 x x x x x x x x x x x x x x                                The solution set is 25 7       . 39.   5 9 9 1 4 5 9 9 9 4 5 9 5 9 x x x x x x x x            The solution set is   is a real number x x or ( , )   or  . The equation is an identity. 40.   4 7 7 1 3 4 7 7 7 3 4 7 4 7 x x x x x x x x            The solution set is   is a real number x x or ( , )   or  . The equation is an identity. 41.   3 2 7 3 3 6 7 3 3 3 6 7 3 3 6 7 y y y y y y y y             There is no solution. The solution set is   or .  The equation is inconsistent. 42.   4 5 21 4 4 20 21 4 20 21 y y y y        There is no solution. The solution set is   or .  The equation is inconsistent. 43. 10 3 8 3 10 8 3 8 8 3 2 0 0 x x x x x x x x           The solution set is   0 . The equation is conditional. 44. 5 7 2 7 3 7 7 3 0 0 x x x x x        The solution set is   0 . The equation is conditional. 45.     1 6 20 8 2 4 2 3 10 8 2 8 3 2 2 8 2 8 10 z z z z z z z z                 The solution set is   10 .  The equation is conditional.

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    Section 1.4 Solving Linear Equations 27 46.     1 1 6 12 20 30 8 3 5 2 4 4 6 8 2 4 4 2 2 6 3 z z z z z z z z                The solution set is   3 . The equation is conditional. 47.   4 3 2 2 7 2 4 6 6 7 2 2 6 7 2 6 7 x x x x x x x x                There is no solution. The solution set is   or .  The equation is inconsistent. 48.     3 3 2 6 1 3 6 3 6 6 6 6 6 6 x x x x x x x x            The solution set is   is a real number x x or ( , )   or  . The equation is an identity. 49.     3 4 2 6 1 5 12 6 6 6 5 13 6 11 6 y y y y y y y y y y              2 6 6 2 0 0 y y y     The solution set is {0}. The equation is conditional. 50.     9 3 6 5 2 3 9 9 18 15 6 18 24 18 5 18 29 18 18 29 0 0 y y y y y y y y y y y y y                    The solution set is   0 . The equation is conditional. 51. 3( 4) 3(2 2 ) 2 x x x     52. 3(2 5) 5 2 17 x x x     53. 3( 3) 5(2 ) 0.5 x x x      54. 2 5 4(3 1) 2 0.7 x x x       55. Solve: 4( 2) 2 4 2(2 ) 4 8 2 4 4 2 4 6 6 4 2 6 4 2 2 1 x x x x x x x x x x x                      Now, evaluate 2 x x  for 1 x   : 2 2 ( 1) ( 1) 1 ( 1) 1 1 2 x x            56. Solve: 2( 6) 3 2(2 1) 2 12 3 4 2 2 12 7 2 5 12 2 5 10 2 x x x x x x x x x x x                    Now, evaluate 2 x x  for 2 x   : 2 2 ( 2) ( 2) 4 ( 2) 4 2 6 x x            57. Solve for x : 3( 3) 2 6 5 3( 3) 5(2 6) 3 9 10 30 7 9 30 7 21 3 x x x x x x x x x                 Solve for y : 2 10 5 18 7 10 18 7 28 4 y y y y y            Now, evaluate 2 ( ) x xy y   for 3 x   and 4 y   :     2 2 2 ( ) ( 3) 3( 4) ( 4) ( 3) 12 ( 4) 9 (12 4) 9 16 7 x xy y                     

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    Chapter 1 Algebra, Mathematical Models, and Problem Solving 28 58. Solve for x : 13 6 5 2 4 13 6 4(5 2) 13 6 20 8 7 6 8 7 14 2 x x x x x x x x x                 Solve for y : 5 7( 4) 1 5 7 28 1 5 7 29 5 8 29 8 24 3 y y y y y y y y y                  Now, evaluate 2 ( ) x xy y   for 2 x   and 3 y   :     2 2 2 ( ) ( 2) 2( 3) ( 3) ( 2) 6 ( 3) 4 (6 3) 4 9 5 x xy y                      59.       2 2 3 6 3 4 54 9 3 4 54 81 3 4 54 27 4 54 108 54 2 x x x x x x                         The solution set is   2  . 60.     3 3 3 2 4 5 3 8 8 4 2 8 8 4 8 8 8 32 8 24 8 3 x x x x x x                           The solution set is   3 . 61.           3 5 12 8 7 6 3 2 5 5 5 12 8 7 6 3 2 125 5 5 12 8 7 6 3 127 5 5 12 8 7 2 127 5 5 12 8 7 254 5 5 12 8 7 254 5 5 12 12 246 5 246 x x x x x x x x x x x x x x x x x x x x                                                    The final statement is a contradiction, so the equation has no solution. The solution set is  . 62.         2 5 58 10 4 21 3.5 11 10 116 10 4 6 11 10 116 10 4 5 10 116 10 20 116 20 x x x x x x x x                   The final statement is a contradiction, so the equation has no solution. The solution set is  . 63. 0.7 0.4(20) 0.5( 20) 0.7 8 0.5 10 0.2 8 10 0.2 2 10 x x x x x x x           The solution set is {10}. 64. 0.5( 2) 0.1 3(0.1 0.3) 0.5 1 0.1 0.3 0.9 0.5 1 0.3 1 0.2 1 1 0.2 0 0 x x x x x x x x x                The solution set is {0}.

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