Solution Manual for Algebra for College Students, 8th Edition

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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
EIGHTH EDITION
Robert Blitzer
Miami Dade College

TABLE OF CONTENTS for INSTRUCTOR SOLUTIONS
ALGEBRA FOR COLLEGE STUDENTS
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

Chapter 1
Algebra, Mathematical Models, and Problem Solving
1
1.1 Check Points
1. a.

eight times a number five more
8 5 8 5x x  

b.

the quotient of a decreased bynumber 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 5x x  
b.  1 3.5x x 
c.  1x x  
1.1 Concept and Vocabulary Check
1. variable
2. expression
3. bth to the nth 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. 5x 
2. 6x 
3. 4x 
4. 9x 
5. 4x
6. 2x
7. 2 10x 
8. 5 4x 
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.    
21 1 1 1
3 3 9 9
3 1 1   
20.    
21 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 6x x 
66.  2 4x x  
67.  5 2x x  
68.  4 3x x  
69.  3 1x x  
70.  2 5x x  
71.  2x x 
72.  3x x 
73.  3x x  
74.  5x x  
75.  3x x 
76.  2x x 
77.  5.5x x 
78.  3.5x 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 3x  , not 2 3x  .
88. false; Three times the sum of a number and five is
represented by  3 5x  , not 3 5x  .
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 4x x  
Commutative Property of Multiplication:
4 9 4 9x x   
10. a. 6 (12 ) (6 12) 18x x x      
b. 7(4 ) ( 7 4) 28x x x     
11. 4(7 2) 28 8x 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       

Loading page 11...

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 22
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
3612 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 2589 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 3 367 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.    
 
 
 
 
22
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 4x x  
Commutative Property of Multiplication
4 10 4 10x x   
102. Commutative Property of Addition
5 30 30 5x x  
Commutative Property of Multiplication
5 30 5 30x x   
103. Commutative Property of Addition
7 5 5 7x x   
Commutative Property of Multiplication
7 5 7 5x x   

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Chapter 1 Algebra, Mathematical Models, and Problem Solving
10
104. Commutative Property of Addition
3 7 7 3x x   
Commutative Property of Multiplication
3 7 3 7x x   
105. 4 (6 ) (4 6) 10x x x      
106. 12 (3 ) (12 3) 15x x x      
107. 7(3 ) ( 7 3) 21x x x     
108. 10(5 ) ( 10 5) 50x 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 15x x x      
112. 5(4 7) 5 4 5 7 20 35x x x      
113.  7(2 3) 7 2 7 3
14 21
x x
x
      
  
114.  9(3 2) 9 3 9 2 27 18x x x         
115.  (3 6) 1 3 1 6 3 6x x x         
116.    (6 3) 1 6 1 3 6 3x x x        
117.  7 5 7 5 12x x x x   
118.  8 10 8 10 18x x x x   
119.  2 2 2 2
6 6 1 5x x x x   
120.  2 2 2 2
9 9 1 8x 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
    
 
      
     
   
   
    

Loading page 13...

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 4x x x x      
132.  8 8 2 8x x x x x      
133.  6 5 30x x  
134.  10 4 40x x  
135. 5 2 3x x x 
136.  6 2 6 2 8x x x x x    
137.  8 3 6 8 3 6 5 6x x x x x      
138.  8 3 6 8 3 18 3 10x 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.

Loading page 14...

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 7x x x      
175. false; Changes to make the statement true will vary.
A sample change is: 2x 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 4x
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
4y x 
−3 2
4 ( 3) 4 9 5y       
−2 2
4 ( 2) 4 4 0y      
−1 2
4 ( 1) 4 1 3y      
0 2
4 (0) 4 0 4y     
1 2
4 (1) 4 1 3y     
2 2
4 (2) 4 4 0y     
3 2
4 (3) 4 9 5y      
184. x 2
1y x 
−3 2
1 ( 3) 1 9 8y       
−2 2
1 ( 2) 1 4 3y       
−1 2
1 ( 1) 1 1 0y      
0 2
1 (0) 1 0 1y     
1 2
1 (1) 1 1 0y     
2 2
1 (2) 1 4 3y      
3 2
1 (3) 1 9 8y      
185. x 1y x 
−4 4 1 3 3y      
−3 3 1 2 2y      
−2 2 1 1 1y      
−1 1 1 0 0y     
0 0 1 1 1y    
1 1 1 2 2y    
2 2 1 3 3y    

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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 2Y in the table is
(c), 2 2y 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 2y x  , but all
are not on any of the others.
32. The equation that corresponds to 1Y in the table is
(b), 2
1y 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 1Y and 2Y intersect at the points
( 2, 4) and (1,1) .
38. The values of 1Y and 2Y are the same when
2x   and 1x  .
39. 2 4y x 
40. 4 2y x 

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Section 1.3 Graphing Equations
19
41. 2
3y x 
42. 2 2y 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 4y 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 numberx 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 6x  
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}.

Loading page 27...

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 numberx 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 numberx 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 numberx 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 1x   :
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 2x   :
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 3x   and
4y   :
 
 
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 2x   and
3y   :
 
 
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.  
 
33
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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Solution Manual for Algebra for College Students, 8th Edition

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