2 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
C H A P T E R P Prerequisites
Section P.1 Review of Real Numbers and Their Properties
1. irrational
2. origin
3. absolute value
4. composite
5. terms
6. Zero-Factor Property
7. 7 22 3
9, , 5, , 2, 0, 1, 4, 2, 11
(a) Natural numbers: 5, 1, 2
(b) Whole numbers: 0, 5, 1, 2
(c) Integers: 9, 5, 0, 1, 4, 2, 11
(d) Rational numbers: 7 22 3
9, , 5, , 0, 1, 4, 2, 11
(e) Irrational numbers: 2
8. 7 53 4
5, 7, , 0, 3.12, , 3, 12, 5
(a) Natural numbers: 12, 5
(b) Whole numbers: 0, 12, 5
(c) Integers: 7, 0, 3, 12, 5
(d) Rational numbers: 7 53 4
7, , 0, 3.12, , 3, 12, 5
(e) Irrational numbers: 5
9. 2.01, 0.666 . . ., 13, 0.010110111 . . ., 1, 6
(a) Natural numbers: 1
(b) Whole numbers: 1
(c) Integers: 13, 1, 6
(d) Rational numbers: 2.01, 0.666 . . ., 13, 1, 6
(e) Irrational numbers: 0.010110111 . . .
10. 12 15 2
25, 17, , 9, 3.12, , 7, 11.1, 13
(a) Natural numbers: 25, 9, 7, 13
(b) Whole numbers: 25, 9, 7, 13
(c) Integers: 25, 17, 9, 7, 13
(d) Rational numbers:
125
25, 17, , 9, 3.12, 7, 11.1, 13
(e) Irrational numbers: 12
11. (a)
(b)
(c)
(d)
12. (a)
(b)
(c)
(d)
13. 4 8
14. 163
1
15. 5 26 3
16. 8 37 7
17. (a) The inequality 5x denotes the set of all real numbers less than or equal to 5.
(b)
(c) The interval is unbounded.
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Section P.1 Review of Real Numbers and Their Properties 3
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18. (a) The inequality 0x denotes the set of all real numbers less than zero.
(b)
(c) The interval is unbounded.
19. (a) The interval 4, denotes the set of all real
numbers greater than or equal to 4.
(b)
(c) The interval is unbounded.
20. (a) , 2 denotes the set of all real numbers less
than 2.
(b)
(c) The interval is unbounded.
21. (a) The inequality 2 2x denotes the set of all real numbers greater than 2 and less than 2.
(b)
(c) The interval is bounded.
22. (a) The inequality 0 6x denotes the set of all real numbers greater than zero and less than or equal to 6.
(b)
(c) The interval is bounded.
23. (a) The interval 5, 2 denotes the set of all real
numbers greater than or equal to 5 and less than 2.
(b)
(c) The interval is bounded.
24. (a) The interval 1, 2 denotes the set of all real
numbers greater than 1 and less than or equal to 2.
(b)
(c) The interval is bounded.
25. 0; 0,y
26. 25; , 25y
27. 10 22; 10, 22t
28. 3 5; 3, 5k
29. 65; 65,W
30. 2.5% 5%; 2.5%, 5%r
31. 10 10 10
32. 0 0
33. 3 8 5 5 5
34. 4 1 3 3
35. 1 2 1 2 1
36. 3 3 3 3 6
37. 5 5 5
15 5 5
38. 3 3 3 3 9
39. If 2,x then 2x is negative.
So, 2 2
1.2 2
x x
x x
40. If 1,x then 1x is positive.
So, 1 1
1.1 1
x x
x x
41. 4 4 because 4 4 and 4 4.
42. 5 5 since 5 5.
43. 6 6 because 6 6 and
6 6 6.
44. 2 2 because 2 2.
45. 126, 75 75 126 51d
46. 126, 75 75 126
51
d
47. 5 5 52 2 2, 0 0d
48. 51 11 11 14 4 4 4 2,d
49. 16 16 128112 1125 75 75 5 75
,d
50. 9.34, 5.65 5.65 9.34 14.99d
51. , 5 5d x x and , 5 3,d x so 5 3.x
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4 Chapter P Prerequisites
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52. , 10 10 ,d x x and , 10 6,d x so
10 6.x
53. ,d y a y a and , 2,d y a so 2.y a
54. 60 23 37 F
Receipts, R Expenditures, E R E
55. $1880.1 $2292.8 1880.1 2292.8 $412.7 billion
56. $2406.9 $2655.1 2406.9 2655.1 $248.2 billion
57. $2524.0 $2982.5 2524.0 2982.5 $458.5 billion
58. $2162.7 $3456.2 2162.7 3456.2 $1293.5 billion
59. 7 4x
Terms: 7 , 4x
Coefficient: 7
60. 36 5x x
Terms: 36 , 5x x
Coefficient: 6, 5
61. 34 52
xx
Terms: 34 , , 52
xx
Coefficients: 1
4,2
62. 23 3 1x
Terms: 23 3 , 1x
Coefficients: 3 3
63. 4 6x
(a) 4 1 6 4 6 10
(b) 4 0 6 0 6 6
64. 9 7x
(a) 9 7 3 9 21 30
(b) 9 7 3 9 21 12
65. 2 5 4x x
(a) 2
1 5 1 4 1 5 4 10
(b) 2
1 5 1 4 1 5 4 0
66. 1
1
x
x
(a) 1 1 2
1 1 0
Division by zero is undefined.
(b) 1 1 0
01 1 2
67. 1
6 1, 66
h hh
Multiplicative Inverse Property
68. 3 3 0x x
Additive Inverse Property
69. 2 3 2 2 3x x
Distributive Property
70. 2 0 2z z
Additive Identity Property
71. 3 3 Associative Property of Multiplication
3 Commutative Property of Multiplication
x y x y
x y
72. 1 17 7
7 12 7 12 Associative Property of Multiplication
1 12 Multiplicative Inverse Property
12 Multiplicative Identity Property
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Section P.2 Solving Equations 5
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73. 5 5 15 10 9 31 48 12 6 24 24 24 24 8
74. 4 18 2
6 6 3
75. 2 8 3 5
3 4 12 12 12
x x x x x
76. 5 2 5 1 5
6 9 3 9 27
x x x
77. (a) Because 0, 0.A A
The expression is negative.
(b) Because , 0.B A B A
The expression is negative.
(c) Because 0, 0.C C
The expression is positive.
(d) Because , 0.A C A C
The expression is positive.
78. The types of real numbers that may be used in a range of prices are nonnegative rational numbers because a price is neither negative nor irrational. For example, a price
of 1.80 or $ 2 would not be reasonable. The types of real numbers to describe a range of lengths are nonnegative and rational because lengths are often not in whole unit amounts but in parts, such as 2.3 centimeters
or 3
4inch.
79. False. Because 0 is nonnegative but not positive, not every nonnegative number is positive.
80. False. Two numbers with different signs will always have a product less than zero.
81. (a)
(b) (i) As n approaches 0, the value of 5 n increases
without bound (approaches infinity).
(ii) As n increases without bound (approaches infinity), the value of 5 n approaches 0.
Section P.2 Solving Equations
1. equation
2. 0ax b
3. extraneous
4. factoring; square roots; completing; square; Quadratic Formula
5. 11 15
11 11 15 11
4
x
x
x
6. 7 19
7 19
7 19
7 19 19 19
12
x
x x x
x
x
x
7. 7 2 25
7 7 2 25 7
2 18
2 182 2
9
x
x
x
x
x
8. 3 5 2 7
3 2 5 2 2 7
5 7
5 5 7 5
12
x x
x x x x
x
x
x
9. 4 2 5 7 6
4 5 2 7 6
2 7 6
6 2 7 6 6
5 2 7
5 2 2 7 2
5 5
5 5
5 51
y y y
y y y
y y
y y y y
y
y
y
y
y
10. 0.25 0.75 10 3
0.25 7.5 0.75 3
0.50 7.5 3
0.50 4.5
9
x x
x x
x
x
x
n 0.0001 0.01 1 100 10,000
5 n 50,000 500 5 0.05 0.0005
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6 Chapter P Prerequisites
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11. 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
No solution
12. 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.
13. 3 4
48 3
9 324
24 2423
424
23 24 244
24 23 23
96
23
x x
x x
x
x
x
or 3 4
48 3
3 424 24 4
8 3
9 32 96
23 96
96
23
x x
x x
x x
x
x
The second method is easier. The fractions are eliminated in the first step.
14. 5 1 14 2 2
5 1 14 4
4 2 2
5 1 14 4 4 4
4 2 2
5 2 4 2
4
xx
xx
xx
x x
x
15. 5 4 25 4 3
3 5 4 2 5 4
15 12 10 8
5 20
4
xx
x x
x x
x
x
16. 10 3 1
5 6 2
2 10 3 1 5 6
20 6 5 6
15 0
0
x
x
x x
x x
x
x
17. 13 5
10 4
10 13 4 5
10 13 4 5
6 18
3
x xx x
x xx x
x
x
18.1 2
0 Multiply both sides by 5 .5
1 5 2 0
3 5 0
3 5
53
x xx x
x x
x
x
x
19.4
2 04 4
42 0
41 2 0
3 0
xx x
xx
Contradiction; no solution
20.
2 2
7 84 Multiply both sides by 2 1 2 1 .
2 1 2 17 2 1 8 2 1 4 2 1 2 1
14 7 16 8 16 4
6 11
11
6
xx x
x xx x x x x
x x x x
x
x
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Section P.2 Solving Equations 7
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21.2 1 2
Multiply both sides by 4 2 .4 2 4 2
2 1 2 2 4
2 2 2 8
2 3 10
12 3
4
x xx x x x
x x
x x
x
x
x
A check reveals that 4x is an extraneous solution—it makes the denominator zero. There is no real solution.
22.4 6 15
Multiply both sides by 1 3 1 .1 3 1 3 1
4 6 151 3 1 1 3 1 1 3 1
1 3 1 3 14 3 1 6 1 15 1
12 4 6 6 15 15
18 2 15 15
3 13
13
3
x xx x x
x x x x x xx x x
x x x
x x x
x x
x
x
23.2
1 1 103 3 9
1 1 10Multiply both sides by 3 3 .
3 3 3 3
1 3 1 3 10
2 10
5
x x x
x xx x x x
x x
x
x
24. 2
1 3 42 3 6
1 3 4Multiply both sides by 3 2 .
2 3 3 2
3 3 2 4
3 3 6 4
4 3 4
4 7
7
4
x x x x
x xx x x x
x x
x x
x
x
x
25. 26 3 0
3 2 1 0
x x
x x
12
3 0 or 2 1 0
0 or
x x
x x
26. 29 1 0
3 1 3 1 0
x
x x
13
13
3 1 0
3 1 0
x x
x x
27. 2 2 8 0
4 2 0
x x
x x
4 0 or 2 0
4 or 2
x x
x x
28. 2 10 9 0
9 1 0
x x
x x
9 0 9
1 0 1
x x
x x
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8 Chapter P Prerequisites
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29. 2 10 25 0
5 5 0
5 0
5
x x
x x
x
x
30. 24 12 9 0
2 3 2 3 0
x x
x x
322 3 0x x
31. 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
32. 2
2
2
2
8 12
8 12 0
1 8 12 1 0
8 12 0
6 2 0
x x
x x
x x
x x
x x
6 0 6
2 0 2
x x
x x
33. 2
2
2
34
34
8 20 0
4 8 20 4 0
3 32 80 0
3 20 4 0
x x
x x
x x
x x
203
3 20 0 or 4 0
or 4
x x
x x
34. 2
2
18 16 0
8 128 0
16 8 0
x x
x x
x x
16 0 16
8 0 8
x x
x x
35. 2 49
7
x
x
36. 2 32
32 4 2
x
x
37. 2
2
3 81
27
3 3
x
x
x
38. 2
2
9 36
4
4 2
x
x
x
39. 2
12 16
12 4
12 4
16 or 8
x
x
x
x x
40. 2
9 24
9 24
9 2 6
x
x
x
41. 2
2 1 18
2 1 18
2 1 3 2
1 3 22
x
x
x
x
42. 2 2
7 3
7 3
x x
x x
7 3 or 7 3
7 3 or 2 4
2
x x x x
x
x
The only solution of the equation is 2.x
43. 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
44. 2
2
2 22
2
2 3 0
2 3
2 1 3 1
1 4
1 4
1 2
3 or 1
x x
x x
x x
x
x
x
x x
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Section P.2 Solving Equations 9
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45. 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
46. 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
47. 2
2
2 2 2
2
9 18 3
12
31
2 1 13
21
3
21
3
21
3
61
3
x x
x x
x x
x
x
x
x
48. 2
2
2
2
2 22
2
7 2 0
2 7 0
2 7 0
2 7
2 1 7 1
1 8
1 2 2
1 2 2
x x
x x
x x
x x
x x
x
x
x
49. 2
2
2
2 22
2
2 5 8 0
2 5 8
54
2
5 5 54
2 4 4
5 894 16
5 894 4
5 89
4 4
5 894
x x
x x
x x
x x
x
x
x
x
50. 2
2
2
2 22
2
3 4 7 0
3 4 7
4 7
3 3
4 2 7 2
3 3 3 3
2 25
3 9
2 5
3 32 5
3 37
1 or 3
x x
x x
x x
x x
x
x
x
x x
51. 22 1 0x x
2
2
42
1 1 4 2 1
2 2
1 3 1, 1
4 2
b b acx
a
52. 22 1 0x x
2
2
42
1 1 4 2 1
2 2
1 1 8
41 3 1
1,4 2
b b acx
a
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10 Chapter P Prerequisites
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53. 2
2
2 2 0
2 2 0
x x
x x
2
2
42
2 2 4 1 2
2 1
2 2 31 3
2
b b acx
a
54. 2 10 22 0x x
2
2
42
10 10 4 1 22
2 1
10 100 882
10 2 35 3
2
b b acx
a
55. 22 3 4 0x x
2
2
42
3 3 4 2 4
2 2
3 41 3 414 4 4
b b acx
a
56. 2
2
3 1 0
3 1 0
x x
x x
2
2
42
3 3 4 1 1
2 1
3 13 3 13
2 2 2
b b acx
a
57. 2
2
12 9 3
9 12 3 0
x x
x x
2
2
42
12 12 4 9 3
2 9
12 6 7 2 7
18 3 3
b b acx
a
58. 2
2
9 37 6
9 6 37 0
x x
x x
2
2
42
6 6 4 9 37
2 9
6 6 38 1 3818 3 3
b b acx
a
59. 29 30 25 0x x
2
2
42
30 30 4 9 25
2 9
30 0 5
18 3
b b acx
a
60. 2
2
28 49 4
49 28 4 0
x x
x x
2
2
42
28 28 4 49 4
2 49
28 0 2
98 7
b b acx
a
61. 2
2
8 5 2
2 8 5 0
t t
t t
2
2
42
8 8 4 2 5
2 2
8 2 6 62
4 2
b b act
a
62. 225 80 61 0h h
2
2
42
80 80 4 25 61
2 25
80 6400 6100
50
8 10 35 50
8 35 5
b b ach
a
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Section P.2 Solving Equations 11
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63.2
2
5 2
12 25 0
y y
y y
2
2
42
12 12 4 1 25
2 1
12 2 116 11
2
b b acy
a
64. 2
2
2
6 2
12 36 2
14 36 0
z z
z z z
z z
2
2
42
14 14 4 1 36
2 1
14 527 13
2
b b acz
a
65. 25.1 1.7 3.2 0x x
2
1.7 1.7 4 5.1 3.2
2 5.1
0.976, 0.643
x
66. 20.005 0.101 0.193 0x x
2
2
42
0.101 0.101 4 0.005 0.193
2 0.005
0.101 0.006341
0.01
2.137, 18.063
b b acx
a
67. 2422 506 347 0x x
2
506 506 4 422 347
2 422
1.687, 0.488
x
68. 23.22 0.08 28.651 0x x
2
2
4
2
0.08 0.08 4 3.22 28.651
2 3.22
0.08 369.0312.995, 2.971
6.44
b b acx
a
69. 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
70. 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
71. 2
3 81 Extract square roots.
3 9
x
x
3 9 or 3 9
6 or 12
x x
x x
72. 2
14 49 0 Extract square roots.
7 0
7 0
7
x x
x
x
x
73. 2
2
2 22
2
114
114
1 11 12 4 2
1 122 4
1 122 4
12
0 Complete the square.
3
x x
x x
x x
x
x
x
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12 Chapter P Prerequisites
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74. 2
22
2
34
3 3 92 4 4
32
32
32
3 0 Complete the square.
3
3
3
3
x x
x x
x
x
x
75. 2 2
22
12
1 Extract square roots.
1
1
For 1 :
0 1 No solution
For 1 :
2 1
x x
x x
x x
x x
x x
x
x
76. 2
2
2
3 4 2 7 Quadratic Formula
0 2 3 11
3 3 4 2 11
2 2
3 974
3 974 4
x x
x x
x
77. 4 2
2 2
2
2
6 14 0
2 3 7 0
2 0 0
213 7 0
3
x x
x x
x x
x x
78. 3
2
53
53
36 100 0
4 9 25 0
4 3 5 3 5 0
4 0 0
3 5 0
3 5 0
x x
x x
x x x
x x
x x
x x
79. 3 2
2
2
5 3 45 0
5 6 9 0
5 3 0
5 0 0
3 0 3
x x x
x x x
x x
x x
x x
80. 3 2
2
2
3 3 0
3 3 0
3 1 0
3 3 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
81. 3 12 0
3 12
3 144
48
x
x
x
x
82. 10 4 0
10 0
10 16
26
x
x
x
x
83. 3
3
2 5 3 0
2 5 3
2 5 27
2 32
16
x
x
x
x
x
84. 3
3
1243
3 1 5 0
3 1 5
3 1 125
3 124
x
x
x
x
x
85.
2
2
26 11 4
4 26 11
16 8 26 11
3 10 0
5 2 0
5 0 5
2 0 2
x x
x x
x x x
x x
x x
x x
x x
86.
2
2
31 9 5
31 9 5
31 9 25 10
0 6
0 3 2
0 3 3
2 2
x x
x x
x x x
x x
x x
x x
x x x
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Section P.2 Solving Equations 13
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87.
2 2
5 1
1 5
1 5
1 2 5 5
4 2 5
2 5
4 5
9
x x
x x
x x
x x x
x
x
x
x
88.
2 2
2
2
2 1 2 3 1
2 1 1 2 3
2 1 1 2 3
4 1 1 2 2 3 2 3
2 2 2 3
2 3
2 3
2 3 0
3 1 0
3 0 3
1 0 1, extraneous
x x
x x
x x
x x x
x x
x x
x x
x x
x x
x x
x x
89. 3 2
3 2
3
5 8
5 8
5 64
5 4 9
x
x
x
x
90. 2 3
2 3
2 9
2 9
2 729
2 27 29, 25
x
x
x
x
91. 3 22
32 2
32 2
2
2
5 27
5 27
5 27
5 9
14
14
x
x
x
x
x
x
92. 3 22
2 2 3
2
2
2
22 27
22 27
22 9
31 0
1 1 4 1 31 1 125 1 5 5
2 1 2 2
x x
x x
x x
x x
x
93. 2 5 11
2 5 11 8
2 5 11 3
x
x x
x x
94.
53
3 2 7
3 2 7
3 2 7
3 2 7 3
x
x x
x
x x
95. 2 6 3 18x x x
First equation: Second 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
2
2
6 3 18
0 9 18
0 3 6
0 3 3
6 6
x x x
x x
x x
x x
x x x
The solutions of the original equation are 3x and 6.x
NOT FOR SALESection P.2 SolvinSection P.2 Solvi
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14 Chapter P Prerequisites
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96. 25 15x x x
First equation: Second equation:
2
2
15 15
16 15 0
1 15 0
1 0 1
15 0 15
x x x
x x
x x
x x
x x
2
2
15 15
14 15 0
1 15 0
1 0 1
15 0 15
x x x
x x
x x
x x
x x
Only 15x and 1x are solutions of the original equation. 1x is extraneous
97. Let 18:y
0.432 10.44
18 0.432 10.44
28.44 0.432
28.44
0.43265.8
y x
x
x
x
x
So, the height of the female is about 65.8 inches or 5 feet 6 inches.
98. Let 21:y
0.449 12.15
21 0.449 12.15
33.15 0.449
33.15
0.44973.8
y x
x
x
x
x
Because the height of the male is about 73.8 inches or 6 feet 2 inches, it is possible the femur belongs to the missing man.
99. False—See Example 14 on page 123.
100. 2 3 1 2 5
2 6 1 2 5
2 5 2 5
x x
x x
x x
False. The equation is an identity, so every real number is a solution.
101. True. There is no value to satisfy this equation.
10 10 0
10 10
10 10
10 10
x x
x x
x x
102. (a) The formula for volume of the glass cube is V = Length × Width × Height. The volume of water in the cube is the length × width × height of the water.
So, the volume is 23 3x x x x x .
(b) Given the equation 2 3 320x x . The
dimensions of the glass cube can be found by solving for x.
So, the capacity of the cube is equal to 3V x .
103. Equivalent equations are derived from the substitution principle and simplification techniques. They have the same solution(s).
2 3 8x and 2 5x are equivalent equations.
Section P.3 The Cartesian Plane and Graphs of Equations
1. Cartesian
2. Distance Formula
3. Midpoint Formula
4. solution or solution point
5. graph
6. intercepts
7. y-axis
8. circle; , ;h k r
9. : 2, 6 , : 6, 2 , : 4, 4 , : 3, 2A B C D
10. 3 52 2
: , 4 , : 0, 2 , : 3, , : 6, 0A B C D
11.
NOT FOR SALErequisitesrequisites
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Section P.3 The Cartesian Plane and Graphs of Equations 15
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12.
13. 3, 4
14. 12, 0
15. 0x and 0y in Quadrant IV.
16. 4x and 0y in Quadrant II.
17. ,x y is in the second Quadrant means that ,x y is
in Quadrant III.
18. , , 0x y xy means x and y have the same signs.
This occurs in Quadrant I or III.
19.
Year 3 2003t
20.
21. 2 2
2 1 2 1
2 2
2 2
3 2 6 6
5 12
25 144
13 units
d x x y y
22. 2 2
2 1 2 1
2 2
2 2
0 8 20 5
8 15
64 225
289
17 units
d x x y y
23. 2 2
2 1 2 1
2 2
2 2
5 1 1 4
6 5
36 25
61 units
d x x y y
Month, x Temperature, y
1 –39
2 –39
3 –29
4 –5
5 17
6 27
7 35
8 32
9 22
10 8
11 –23
12 –34
Year, x Number of Stores, y
2003 4906
2004 5289
2005 6141
2006 6779
2007 7262
2008 7720
2009 8416
2010 8970
Tem
pera
ture
(in
°F)
Month (1 ↔ January)
x
−40
10
−30
−20
−10
20
30
2 6 8 10 12
40
0
y
NOT FOR SALESection P.3 The Cartesian ction P.3 The Cartesian Plane and Graphs Plane and Graphs
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16 Chapter P Prerequisites
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24. 2 2
2 1 2 1
22
2 2
3.9 9.5 8.2 2.6
13.4 10.8
179.56 116.64
296.2
17.21 units
d x x y y
25. (a) 1, 0 , 13, 5
2 2
2 2
Distance 13 1 5 0
12 5 169 13
13, 5 , 13, 0
Distance 5 0 5 5
1, 0 , 13, 0
Distance 1 13 12 12
(b) 2 2 25 12 25 144 169 13
26. (a) The distance between 1, 1 and 9, 1 is 10.
The distance between 9, 1 and 9, 4 is 3.
The distance between 1, 1 and 9, 4 is
2 2
9 1 4 1 100 9 109.
(b) 2
2 210 3 109 109
27. 2 2
1
2 22
2 23
4 2 0 1 4 1 5
4 1 0 5 25 25 50
2 1 1 5 9 36 45
d
d
d
2 2 2
5 45 50
28. 2 2
1
2 22
2 23
3 1 5 3 16 4 20
5 3 1 5 4 16 20
5 1 1 3 36 4 40
d
d
d
2 2 2
20 20 40
29. 2 2
1
2 22
2 23
1 2
1 3 3 2 4 25 29
3 2 2 4 25 4 29
1 2 3 4 9 49 58
d
d
d
d d
30. 2 2
1
2 22
2 23
1 2
4 2 9 3 4 36 40
2 4 7 9 36 4 40
2 2 3 7 16 16 32
d
d
d
d d
31. (a)
(b) 2 2(5 ( 3)) (6 6) 64 8d
(c) 6 6 5 ( 3)
, (6, 1)2 2
32. (a)
(b) 2 2
9 1 7 1 64 36 10d
(c) 9 1 7 1
, 5, 42 2
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Section P.3 The Cartesian Plane and Graphs of Equations 17
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33. (a)
(b) 2 2
5 1 4 2
36 4 2 10
d
(c) 1 5 2 4
, 2, 32 2
34. (a)
(b) 2 2
1 5 41
2 2 3
1 829
9 3
d
(c) 5 2 1 2 4 3 1 7
, 1,2 2 6
35. 2 2120 150
36,900
30 41
192.09
d
The plane flies about 192 kilometers.
36. 2 2
2 2
42 18 50 12
24 38
2020
2 505
45
d
The pass is about 45 yards.
37. 2002 2010 19,564 35,123
midpoint ,2 2
2006, 27,343.5
In 2006, the sales for the Coca-Cola Company were about $27,343.5 million.
38. 1 2 1 2midpoint ,2 2
2008 2010 1.89 2.83,
2 2
2009, 2.36
x x y y
In 2009, the earnings per share for Big Lots, Inc. were about $2.36.
39. (a) ?
0, 2 : 2 0 4
2 2
Yes, the point is on the graph.
(b) ?
?
5, 3 : 3 5 4
3 9
3 3
Yes, the point is on the graph.
40. (a) ?
?
1, 5 : 5 4 1 2
5 4 1
5 3
No, the point is not on the graph.
(b) ?
?
6, 0 : 0 4 6 2
0 4 4
0 0
Yes, the point is on the graph.
41. (a) ?2
?
2, 0 : 2 3 2 2 0
4 6 2 0
0 0
Yes, the point is on the graph.
(b) ?2
?
2, 8 : 2 3 2 2 8
4 6 2 8
12 8
No, the point is not on the graph.
42. (a) ?
1, 2 : 2 1 2 3 0
3 0
No, the point is not on the graph.
(b) ?
?
1, 1 : 2 1 1 3 0
2 1 3 0
0 0
Yes, the point is on the graph.
NOT FOR SALESection P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs
INSTRUCTOR USE ONLY about $27,343.5 million. about $27,343.5 million.
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18 Chapter P Prerequisites
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43. (a) ?2 2
?
3, 2 : 3 2 20
9 4 20
13 20
No, the point is not on the graph.
(b) ?2 2
?
4, 2 : 4 2 20
16 4 20
20 20
Yes, the point is on the graph.
44. (a) ?3 2
?
?
?
16 1613 3 3
1613 3
8 163 3
8 16243 3 3
16 163 3
2, : 2 2 2
8 2 4
8
Yes, the point is on the graph.
(b) ?3 2
?
?
13
13
3, 9 : 3 2 3 9
27 2 9 9
9 18 9
27 9
No, the point is not on the graph.
45. 2 5y x
46. 34
1y x
47. 2 3y x x
48. 25y x
x 1 0 1 2 52
y 7 5 3 1 0
(x, y) 1, 7 0, 5 1, 3 2, 1 52, 0
x 2 0 1 43
2
y 52
–1 14
0 12
(x, y) 52
2, 0, 1 14
1, 43, 0 1
22,
x 1 0 1 2 3
y 4 0 –2 –2 0
(x, y) 1, 4 0, 0 1, 2 2, 2 3, 0
x 2 –1 0 1 2
y 1 4 5 4 1
x, y 2, 1 1, 4 0, 5 1, 4 2, 1
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Section P.3 The Cartesian Plane and Graphs of Equations 19
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49. 216 4y x
x-intercepts: 2
2
2
0 16 4
4 16
4
2
x
x
x
x
2, 0 , 2, 0
y-intercept: 2
16 4 0 16
0, 16
y
50. 2
3y x
x-intercept: 2
0 3
0 3
3
x
x
x
3, 0
y-intercept: 2
2
0 3
3
9
0, 9
y
y
y
51. 5 6y x
x-intercept:
65
0 5 6
6 5
x
x
x
65, 0
y-intercept: 5 0 6 6y
0, 6
52. 8 3y x
x-intercept:
83
0 8 3
3 8
x
x
x
83, 0
y-intercept: 8 3 0 8y
0, 8
53. 4y x
x-intercept: 0 4
0 4
4
x
x
x
4, 0
y-intercept: 0 4 2y
0, 2
54. 2 1y x
x-intercept:
12
0 2 1
2 1 0
x
x
x
12, 0
y-intercept: 2 0 1
1
y
There is no real solution. There is no y-intercept.
55. 3 7y x
x-intercept:
73
0 3 7
0 3 7
0
x
x
73, 0
y-intercept: 3 0 7 7y
0, 7
56. 10y x
x-intercept: 0 10
10 0
10
x
x
x
10, 0
y-intercept: 0 10
10 10
y
0, 10
NOT FOR SALESection P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs
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20 Chapter P Prerequisites
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57. 3 22 4y x x
x-intercept: 3 2
2
0 2 4
0 2 2
0 or 2
x x
x x
x x
0, 0 , 2, 0
y-intercept: 3 2
2 0 4 0
0
y
y
0, 0
58. 4 25y x
x-intercepts: 4
4
4 2
0 25
25
5 5
x
x
x
5, 0
y-intercept: 4
0 25 25y
0, 25
59. 2 6y x
x-intercept: 0 6
6
x
x
6, 0
y-intercepts: 2 6 0
6
y
y
0, 6 , 0, 6
60. 2 1y x
x-intercept: 0 1
1
x
x
1, 0
y-intercepts: 2 0 1
1
y
y
0, 1 , 0, 1
61.
62.
63.
64.
65. 2 0x y
2 2
2 2
2 2
0 0 -axis symmetry
0 0 No -axis symmetry
0 0 No origin symmetry
x y x y y
x y x y x
x y x y
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Section P.3 The Cartesian Plane and Graphs of Equations 21
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66. 2 0x y
2 2
2 2
2 2
0 0 No -axis symmetry
0 0 -axis symmetry
0 0 No origin symmetry
x y x y y
x y x y x
x y x y
67. 3
3 3
3 3
3 3 3
No -axis symmetry
No -axis symmetry
Origin symmetry
y x
y x y x y
y x y x x
y x y x y x
68. 4 2
4 2 4 2
4 2 4 2
4 2 4 2
3
3 3 -axis symmetry
3 3 No -axis symmetry
3 3 No origin symmetry
y x x
y x x y x x y
y x x y x x x
y x x y x x
69. 2
2 2
2 2
2 2 2
1
No -axis symmetry11
No -axis symmetry1 1
Origin symmetry1 11
xy
xx x
y y yxx
x xy y x
x xx x x
y y yx xx
70. 2
2 2
2 2
2 2
1
11 1
-axis symmetry11
1 1No -axis symmetry
1 11 1
No origin symmetry11
yx
y y yxx
y y xx x
y yxx
71. 2
2 2
2 2
2 2
10 0
10 0 10 0 No -axis symmetry
10 0 10 0 -axis symmetry
10 0 10 0 No origin symmetry
xy
x y xy y
x y xy x
x y xy
72. 4
4 4 No -axis symmetry
4 4 No -axis symmetry
4 4 Origin symmetry
xy
x y xy y
x y xy x
x y xy
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22 Chapter P Prerequisites
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73. 3 1y x
x-intercept: 13, 0
y-intercept: 0, 1
No symmetry
74. 2 3y x
x-intercept: 32, 0
y-intercept: 0, 3
No symmetry
75. 2 2y x x
x-intercepts: 0, 0 , 2, 0
y-intercept: 0, 0
No symmetry
76. 2 1x y
x-intercept: 1, 0
y-intercepts: 0, 1 , 0, 1
x-axis symmetry
77. 3 3y x
x-intercept: 3 3, 0
y-intercept: 0, 3
No symmetry
78. 3 1y x
x-intercept: 1, 0
y-intercept: 0, 1
No symmetry
x –1 0 1 2 3
y 3 0 –1 0 3
x –1 0 3
y 0 ±1 ±2
x –2 –1 0 1 2
y –5 2 3 4 11
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Section P.3 The Cartesian Plane and Graphs of Equations 23
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79. 3y x
x-intercept: 3, 0
y-intercept: none
No symmetry
80. 1y x
x-intercept: 1, 0
y-intercept: 0, 1
No symmetry
81. 6y x
x-intercept: 6, 0
y-intercept: 0, 6
No symmetry
82. 1y x
x-intercepts: 1, 0 , 1, 0
y-intercept: 0, 1
y-axis symmetry
83. Center: 0, 0 ; Radius: 4
2 2 2
2 2
0 0 4
16
x y
x y
84. Center: 7, 4 ; Radius: 7
2 2 2
2 2
7 4 7
7 4 49
x y
x y
85. Center: 1, 2 ; Solution point: 0, 0
2 2 2
2 2 2 2
2 2
1 2
0 1 0 2 5
1 2 5
x y r
r r
x y
86. Center: 3, 2 ; Solution point: 1,1
2 2
22
3 1 2 1
4 3 25 5
r
22 2
2 2
3 2 5
3 2 25
x y
x y
87. Endpoints of a diameter: 0, 0 , 6, 8
Center: 0 6 0 8
, 3, 42 2
2 2 2
2 2 2 2
2 2
3 4
0 3 0 4 25
3 4 25
x y r
r r
x y
x 3 4 7 12
y 0 1 2 3
x –2 0 2 4 6 8 10
y 8 6 4 2 0 2 4
Section P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs
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24 Chapter P Prerequisites
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88. Endpoints of a diameter: 4, 1 , 4,1
2 2
2 2
14 4 1 1
21
8 221
64 42
1 168 2 17 17
2 2
r
Midpoint of diameter (center of circle):
22 2
2 2
4 4 1 1, 0, 0
2 2
0 0 17
17
x y
x y
89. 2 2 25x y
Center: 0, 0 , Radius: 5
90. 22 1 1x y
Center: 0, 1 , Radius: 1
91. 2 2 91 1
2 2 4x y
Center: 1 12 2, , Radius: 3
2
92. 2 2 16
92 3x y
Center: 2, 3 , Radius: 43
93. 500,000 40,000 , 0 8y t t
94. 8000 900 , 0 6y t t
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Section P.3 The Cartesian Plane and Graphs of Equations 25
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95. (a)
(b)
When 85.5,x the resistance is about 1.4 ohms.
(c) When 85.5,x
2
10,3701.4 ohms.
(85.5)y
(d) As the diameter of the copper wire increases, the resistance decreases.
96. (a)
Because the line is close to the points, the model fits the data well.
(b) Graphically: The point 90, 75.4 represents a life expectancy of 75.4 years in 1990.
Algebraically: 220.002 0.5 46.6 0.002 90 0.5 90 46.6 75.4 y t t
So, the life expectancy in 1990 was about 75.4 years.
(c) Graphically: The point 94.6, 76.0 represents a life expectancy of 76 years during the year 1994.
Algebraically: 2
2
2
0.002 0.5 46.6
76.0 0.002 0.5 46.6
0 0.002 0.5 29.4
y t t
t t
t t
Use the quadratic formula to solve.
2
2
4
2
0.5 0.5 4 0.002 29.4
2 0.002
0.5 0.0148
0.004
125 30.4
b b act
a
So, 94.6t or 155.4.t Since 155.4 is not in the domain, the solution is 94.6t , which is the year 1994.
(d) When 115:t
220.002 0.5 46.6 0.002 115 0.5 115 46.6 77.65y t t
The life expectancy using the model is 77.65 years, which is slightly less than the given projection of 78.9 years.
(e) Answers will vary. Sample answer: No. Because the model is quadratic, the life expectancies begin to decrease after a certain point.
x 5 10 20 30 40 50 60 70 80 90 100
y 414.8 103.7 25.9 11.5 6.5 4.1 2.9 2.1 1.6 1.3 1.0
NOT FOR SALESection P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs
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