Solutions for ALGEBRA AND TRIGONOMETRY 7th Edition by DANIEL S. MILLER

Chapter P
Fundamental Concepts of Algebra

Section P.1 6. a. 1− 2
Check Point Exercises Because 2 ≈ 1.4, the number inside the
absolute value bars is negative. The absolute
1. 8 + 6( x − 3)2 = 8 + 6(13 − 3)2 value of x when x < 0 is –x. Thus,
= 8 + 6(10) 2 (
1 − 2 = − 1 − 2 = 2 −1 )
= 8 + 6(100)
= 8 + 600 b. π −3
= 608 Because π ≈ 3.14, the number inside the
absolute value bars is positive. The absolute
2. a. Since 2016 is 16 years after 2000, substitute 16 value of a positive number is the number itself.
for x. Thus,
M
T = − x 2 + 361x + 3193 π − 3 = π − 3.
= −(16) 2 + 361(16) + 3193
= 8713 x
c.
EL
The average cost of tuition and fees at public x
U.S. colleges for the school year ending in Because x > 0, x = x.
2016 was $8713.
x x
b. The formula underestimates the actual answer Thus, = =1
x x
AN
by $65.
7. −4 − (5) = −9 = 9
3. The elements common to {3, 4, 5, 6, 7} and
{3, 7, 8, 9} are 3 and 7. The distance between –4 and 5 is 9.
{3, 4,5, 6, 7} ∩ {3, 7,8,9} = {3, 7}
8. 7(4 x 2 + 3x) + 2(5 x 2 + x)
IE
4. The union is the set containing all the elements of = 7(4 x 2 + 3 x) + 2(5 x 2 + x)
either set.
{3, 4,5, 6, 7} ∪ {3, 7,8,9} = {3, 4,5, 6, 7,8,9} = 28 x 2 + 21x + 10 x 2 + 2 x
??
= 38 x 2 + 23 x
­ π ½
5. ® −9, − 1.3, 0, 0.3, , 9, 10 ¾
9. 6 + 4[7 − ( x − 2)]
¯ 2 ¿
= 6 + 4[7 − x + 2)]
??
a. Natural numbers: 9 because 9 =3 = 6 + 4[9 − x]
= 6 + 36 − 4 x
b. Whole numbers: 0, 9 = 42 − 4 x

c. Integers: −9, 0, 9
Concept and Vocabulary Check P.1
d. Rational numbers: −9, − 1.3, 0, 0.3, 9
C1. expression
π C2. b to the nth power; base; exponent
e. Irrational numbers: , 10
2
C3. formula; modeling; models
π
f. Real numbers: −9, − 1.3, 0, 0.3, , 9, 10 C4. intersection; A ∩ B
2
C5. union; A ∪ B



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,Chapter P Fundamental Concepts of Algebra


C6. natural 10.
3
6 + 5 (8 − 6) = 6 + 5 ( 2)
3


C7. whole = 6 + 5 (8)

C8. integers = 6 + 40 = 46

C9. rational 11. 82 − 3(8 − 2) = 64 − 3(6)
= 64 − 18 = 46
C10. irrational
12. 82 − 4 ( 8 − 3) = 64 − 4 ( 5 ) = 64 − 20 = 44
C11. rational; irrational

C12. absolute value; x, x 5( x + 2) 5(10 + 2)
13. =
2 x − 14 2(10) − 14
C13. b + a ; ba 5(12)
=
6
C14. a + (b + c) ; ( ab)c
M
= 5⋅2
C15. ab + ac = 10

C16. 0; inverse; 0; identity 7( x − 3) 7(9 − 3) 7(6)
14. = = = 7 ⋅ 3 = 21
EL
2 x − 16 2(9) − 16 2
C17. inverse; 1; identity
2x + 3y
C18. simplified 15. ; x = −2, y = 4
x +1
AN
C19. a 2 ( −2 ) + 3 ( 4 ) −4 + 12 8
= = = = −8
−2 + 1 −1 −1
Exercise Set P.1
2x + y
16. ; x = −2 and y = 4
1. 7 + 5(10) = 7 + 50 = 57 xy − 2 x
IE
2 ( −2 ) + 4 −4 + 4 0
= = =0
2. 8 + 6 ( 5 ) = 8 + 30 = 38 ( −2 )( 4 ) − 2 ( −2 ) −8 + 4 4

6(3) − 8 = 18 − 8 = 10
??
3. 5 5
17. C= (50 − 32) = (18) = 10
9 9
4. 8 ( 3) − 4 = 24 − 4 = 20 50°F is equivalent to 10°C.

5 5 5
??
5. 82 + 3(8) = 64 + 24 = 88 18. C= ( F − 32) = (86 − 32) = (54) = 30
9 9 9
6. 62 + 5 ( 6 ) = 36 + 30 = 66 86°F is equivalent to 30°C.

19. h = 4 + 60t − 16t 2 = 4 + 60(2) − 16(2) 2
7. 7 2 − 6(7) + 3 = 49 − 42 + 3 = 7 + 3 = 10
= 4 + 120 − 16(4) = 4 + 120 − 64
8. 8 − 7 ( 8 ) + 4 = 64 − 56 + 4 = 8 + 4 = 12
2 = 124 − 64 = 60
Two seconds after it is kicked, the ball’s height is
60 feet.
9. 4 + 5(9 − 7)3 = 4 + 5(2)3
= 4 + 5(8) = 4 + 40 = 44




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, Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers


20. h = 4 + 60t − 16t 2 36. a. 49
= 4 + 60(3) − 16(3) 2
b. 0, 49
= 4 + 180 − 16(9)
= 4 + 180 − 144 c. −7, 0, 49
= 184 − 144 = 40
Three seconds after it is kicked, the ball’s height is d. −7, − 0.6, 0, 49
40 feet.
e. 50
21. {1, 2,3, 4}  {2, 4,5} = {2, 4}
f. −7, − 0.6, 0, 49, 50
22. {1,3, 7}  {2,3,8} = {3}
23. {s, e, t}  {t, e, s} = {s, e, t} 37. a. 64

b. 0, 64
M
24. {r , e, a, l}  {l , e, a, r} = {r , e, a, l}
c. −11, 0, 64
25. {1,3,5, 7}  {2, 4, 6,8,10} = { }
The empty set is also denoted by ∅. 5
EL
d. −11, − , 0, 0.75, 64
6
26. {1,3,5, 7}  {−5, −3, −1} = { } or ∅
e. 5, π
27. {a, b, c, d}  ∅ = ∅
AN
5
f. −11, − , 0, 0.75, 5, π , 64
28. {w, y, z}  ∅ = ∅ 6

29. {1, 2,3, 4} * {2, 4,5} = {1, 2,3, 4,5} 38. a. 4

30. {1,3, 7,8} * {2,3,8} = {1, 2,3, 7,8} b. 0, 4
IE
31. {1, 3,5, 7} * {2, 4, 6,8,10} c. −5, 0, 4
= {1, 2,3, 4,5, 6, 7,8,10}
??
d. −5, − 0.3, 0, 4
32. {0,1,3,5} * {2, 4,6} = {0,1, 2,3, 4,5, 6}
e. 2
33. {a, e, i, o, u} * ∅ = {a, e, i, o, u}
??
f. −5, − 0.3, 0, 2, 4
34. {e, m, p, t , y} * ∅ = {e, m, p, t , y}
39. 0
35. a. 100
1
40. Answers will vary. An example is .
b. 0, 100 2

c. −9, 0, 100 41. Answers will vary. An example is 2.
4 42. Answers will vary. An example is −2.
d. −9, − , 0, 0.25, 9.2, 100
5
43. true; –13 is to the left of –2 on the number line.
e. 3
44. false; –6 is to the left of 2 on the number line.
4
f. −9, − , 0, 0.25, 3,9.2, 100
5 45. true; 4 is to the right of –7 on the number line.



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, Chapter P Fundamental Concepts of Algebra


46. true; –13 is to the left of –5 on the number line. 68. The distance is 4 − 15 = −11 = 11 .
47. true; −π = −π
69. The distance is −2 − 5 = −7 = 7 .
48. true; –3 is to the right of –13 on the number line.
70. The distance is −6 − 8 = −14 = 14 .
49. true; 0 is to the right of –6 on the number line.

50. true; 0 is to the right of –13 on the number line. 71. The distance is −19 − (−4) = −19 + 4 = −15 = 15 .

51. 300 = 300 72. The distance is −26 − (−3) = −26 + 3 = −23 = 23 .

52. −203 = 203 73. The distance is
−3.6 − (−1.4) = −3.6 + 1.4 = −2.2 = 2.2 .
53. 12 − π = 12 − π
74. The distance is
M
54. 7−π = 7−π −5.4 − (−1.2) = −5.4 + 1.2 = −4.2 = 4.2 .

75. 6 + (–4) = (–4) + 6;
55. 2 −5 = 5− 2 commutative property of addition
EL
56. 5 − 13 = 13 − 5 76. 11 ⋅ (7 + 4) = 11 ⋅ 7 + 11 ⋅ 4;
distributive property of multiplication over addition
−3 −3
AN
57. = = −1 77. 6 + (2 + 7) = (6 + 2) + 7;
−3 3 associative property of addition

78. 6 ⋅ (2 ⋅ 3) = 6 ⋅ (3 ⋅ 2);
−7 −7
58. = = −1 commutative property of multiplication
−7 7
79. (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3);
IE
59. − 3 − −7 = 3 − 7 = − 4 = 4 commutative property of addition

80. 7 ⋅ (11 ⋅ 8) = (11 ⋅ 8) ⋅ 7;
60. −5 − −13 = 5 − 13 = −8 = 8
??
commutative property of multiplication

61. x + y = 2 + (−5) = −3 = 3 81. 2 (–8 + 6) = –16 + 12;
distributive property of multiplication over addition
62. x − y = 2 − (−5) = 7 = 7
??
82. −8(3 + 11) = −24 + (−88) ;
distributive property of multiplication over addition
63. x + y = 2 + −5 = 2 + 5 = 7
1
64. x − y = 2 − −5 = 2 − 5 = −3 83. ( x + 3) = 1; x ≠ −3 ,
x+3
inverse property of multiplication
y −5 −5
65. = = = −1
y −5 5 84. ( x + 4 ) + ª¬ − ( x + 4 )º¼ = 0;
inverse property of addition
x y 2
2 5 −5
66. + = + = + = 1 + (−1) = 0 85. 5(3x + 4) − 4 = 5 ⋅ 3x + 5 ⋅ 4 − 4
x y 2 −5 2 −5
= 15 x + 20 − 4
67. The distance is 2 − 17 = −15 = 15 . = 15 x + 16




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