CONTENTS
1 Algebra and Equations .................................................................................................. 1
2 Graphs, Lines, and Inequalities .................................................................................. 51
3 Functions and Graphs................................................................................................ 100
4 Exponential and Logarithmic Functions ................................................................. 169
5 Mathematics of Finance............................................................................................. 208
6 Systems of Linear Equations and Matrices ............................................................. 245
7 Linear Programming ................................................................................................. 326
8 Sets and Probability ................................................................................................... 427
9 Counting, Probability Distributions, and Further Topics in Probability ............. 459
10 Introduction to Statistics ........................................................................................... 496
11 Differential Calculus .................................................................................................. 530
12 Applications of the Derivative................................................................................... 611
13 Integral Calculus ........................................................................................................ 670
14 Multivariate Calculus ................................................................................................ 732
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
, Chapter 1 Algebra and Equations
Section 1.1 The Real Numbers
q+r 3 + (−5) −2
1. True. This statement is true, since every integer 13. = = = −2
can be written as the ratio of the integer and 1. q + p 3 + (−2) 1
5
For example, 5 = . 3q 3(3) 9 9
1 14. = = =
3 p − 2r 3(−2) − 2(−5) −6 + 10 4
2. False. For example, 5 is a real number, and
10 15. Let r = 1.5.
5= which is not an irrational number. APR = 12r = 12(1.5) = 18%
2
3. Answers vary with the calculator, but 16. Let r = 1.67.
2, 508, 429, 787 APR = 12r = 12(1.67) = 20.04%
is the best.
798, 458, 000
17. Let APR = 9.
4. −5 + 0 = −5 APR = 12r
This illustrates the identity property of addition. 9 = 12r
9 3
5. 6(t + 4) = 6t + 6 ⋅ 4 = =r
12 4
This illustrates the distributive property. r = .75%
6. 3 + (–3) = (–3) + 3 18. Let APR = 19.5.
This illustrates the commutative property of APR = 12r
addition. 19.5 = 12r
19.5
7. 0 + (–7) = –7 + 0 =r
This illustrates the commutative property of 12
addition. r = 1.625%
8. 8 + (12 + 6) = (8 + 12) + 6 19. 3 − 4 ⋅ 5 + 5 = 3 − 20 + 5 = −17 + 5 = −12
This illustrates the associative property of
addition. 20. (4 − 5) ⋅ 6 + 6 = −1 ⋅ 6 + 6 = −6 + 6 = 0
9. Answers vary. One possible answer: The sum of 21. 8 − 4 2 − (−12)
a number and its additive inverse is the additive
identity. The product of a number and its Take powers first.
multiplicative inverse is the multiplicative 8 – 16 – (–12)
identity. Then add and subtract in order from left to right.
8 – 16 + 12 = –8 + 12 = 4
10. Answers vary. One possible answer: When using
the commutative property, the order of the 22. 8 − (−4) 2 − (−12)
addends or multipliers is changed, while the Take powers first.
grouping of the addends or multipliers is 8 – 16 – (–12)
changed when using the associative property. Then add and subtract in order from left to right.
8 – 16 + 12 = –8 + 12 = 4
For Exercises 11–14, let p = –2, q = 3 and r = –5.
⎣ 2 + 5 (3)⎤⎦ = −3 [ −2 + 15]
11. −3 ( p + 5q ) = −3 ⎡− ⎣ ( )
23. − (3 − 5) − ⎡ 2 − 3 2 − 13 ⎤
⎦
= −3 (13) = −39 Take powers first.
–(3 – 5) – [2 – (9 – 13)]
Work inside brackets and parentheses.
12. 2 (q − r ) = 2 (3 + 5) = 2 (8) = 16 – (–2) – [2 – (–4)] = 2 – [2 + 4]
= 2 – 6 = –4
1
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
,2 CHAPTER 1 ALGEBRA AND EQUATIONS
2(3 − 7) + 4(8) 36. 3 4 = .75
24.
4(−3) + (−3)(−2)
Work above and below fraction bar. Do 37. 3.14 < π
multiplications and work inside parentheses.
38. 1 3 > .33
2(−4) + 32 −8 + 32 24
= = = = −4
−12 + 6 −12 + 6 −6 39. a lies to the right of b or is equal to b.
2(−3) + ( −32) − 2
25.
(− 16 ) 40. b + c = a
64 − 1 41. c < a < b
Work above and below fraction bar. Take roots.
42. a lies to the right of 0
2(−3) + ( −32) − ( −24)
8 −1 43. (–8, –1)
Do multiplications and divisions. This represents all real numbers between –8 and
−6 − 32 + 12 –1, not including –8 and –1. Draw parentheses at
–8 and –1 and a heavy line segment between
8 −1 them. The parentheses at –8 and –1 show that
Add and subtract. neither of these points belongs to the graph.
− 12 − 32 + 12 − 14 −7
2
= 2 = = −1
7 7 7
44. [–1, 10]
6 2 − 3 25 This represents all real numbers between –1 and
26.
6 2 + 13 10, including –1 and 10. Draw brackets at –1 and
Take powers and roots. 10 and a heavy line segment between them.
36 − 3(5) 36 − 15 21
= = =3
36 + 13 49 7
45. [–2, 2)
2040 189 4587 This represents all real numbers between –2 and
27. , , 27, , 6.735, 47
523 37 691 2, including –2, not including 2.
Draw a bracket at –2, a parenthesis at 2, and a
187 385 heavy line segment between them.
28. , 2.9884, 85 , π , 10,
63 117
29. 12 is less than 18.5.
12 < 18.5 46. (−2, 3]
This represents all real numbers x such that
30. –2 is greater than –20. –2 < x ≤ 3. Draw a heavy line segment from –2
–2 > –20 to 3. Use a parenthesis at –2 since it is not part of
the graph. Use a bracket at 3 since it is part of
31. x is greater than or equal to 5.7. the graph.
x ≥ 5.7
32. y is less than or equal to –5.
y ≤ −5
47. (−2, ∞ )
33. z is at most 7.5. This represents all real numbers x such that
z ≤ 7.5 x > –2. Start at –2 and draw a heavy line
segment to the right. Use a parenthesis at –2
34. w is negative. since it is part of the graph.
w<0
35. −6 < −2
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
, SECTION 1. 1 THE REAL NUMBERS 3
48. (–∞, –2] 58. a. Shaquille O’Neal’s height in inches is
This represents all real numbers less than or 7(12) + 1 = 85 in.
equal to –2. Draw a bracket at –2 and a heavy .455W .455(300)
ray to the left. B= =
(.0254 H ) 2
(.0254(85)) 2
136.5
= ≈ 29.3
(2.159) 2
49. 3; 2000, 2006, and 2007
b. No, Shaquille O’Neal’s body mass index
50. 6; 1998, 1999, 2002, 2003, 2004, 2005 falls above the desirable range.
51. 7; 1998, 1999, 2001, 2002, 2003, 2004, 2005 59. A wind at 10 miles per hour with a 30°
temperature has a wind-chill factor of 21°.
52. 3; 2000, 2006, 2007 A wind at 30 miles per hour with a –10°
temperature has a wind-chill factor of –39°.
53. 0
21 − (−39) = 21 + 39 = 60°
54. 5; 1998, 1999, 2002, 2003, 2005
60. A wind at 20 miles per hour with a –20°
55. a. Steffi Graf’s height in inches is temperature has a wind-chill factor of –48°.
5(12) + 9 = 69 in. A wind at 5 miles per hour with a 30°
.455W .455(119) temperature has a wind-chill factor of 25°.
B= =
(.0254 H ) 2
(.0254(69)) 2 −48 − 25 = −73 = 73°
54.145
= ≈ 17.6 61. A wind at 25 miles per hour with a –30°
(1.7526) 2 temperature has a wind-chill factor of –64°.
A wind at 15 miles per hour with a –30°
b. No, Steffi Graf’s body mass index falls temperature has a wind-chill factor of –58°.
below the desirable range.
−64 − (−58) = −64 + 58 = −6 = 6°
56. a. Jackie Joyner-Kersee’s height in inches is
5(12) + 10 = 70 in. 62. A wind at 40 miles per hour with a 40°
.455W .455(153) temperature has a wind-chill factor of 27°. A
B= = wind at 25 miles per hour with a –30°
(.0254 H ) 2 (.0254(70)) 2 temperature has a wind-chill factor of –64°.
=
69.615
≈ 22.0 27 − (−64) = 27 + 64 = 91 = 91°
(1.778) 2
63. 8 − −4 = 8 − (4) = 4
b. Yes, Jackie Joyner-Kersee’s body mass
index falls within the desirable range.
64. −9 − −12 = 9 − (12) = −3
57. a. Tiger Wood’s height in inches is
6(12) + 2 = 74 in. 65. − −4 − −1 − 14 = − (4) − −15
B=
.455W
=
.455(180) = − (4) − 15 = −19
(.0254 H ) 2
(.0254(74)) 2
81.9 66. − 6 − −12 − 4 = − (6) − −16 = −6 − (16) = −22
= ≈ 23.2
(1.8796) 2
67. 5 −5
b. Yes, Tiger Wood’s body mass index falls 5 __ 5
within the desirable range. 5=5
68. − −4 4
−4 4
−4<4
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
1 Algebra and Equations .................................................................................................. 1
2 Graphs, Lines, and Inequalities .................................................................................. 51
3 Functions and Graphs................................................................................................ 100
4 Exponential and Logarithmic Functions ................................................................. 169
5 Mathematics of Finance............................................................................................. 208
6 Systems of Linear Equations and Matrices ............................................................. 245
7 Linear Programming ................................................................................................. 326
8 Sets and Probability ................................................................................................... 427
9 Counting, Probability Distributions, and Further Topics in Probability ............. 459
10 Introduction to Statistics ........................................................................................... 496
11 Differential Calculus .................................................................................................. 530
12 Applications of the Derivative................................................................................... 611
13 Integral Calculus ........................................................................................................ 670
14 Multivariate Calculus ................................................................................................ 732
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
, Chapter 1 Algebra and Equations
Section 1.1 The Real Numbers
q+r 3 + (−5) −2
1. True. This statement is true, since every integer 13. = = = −2
can be written as the ratio of the integer and 1. q + p 3 + (−2) 1
5
For example, 5 = . 3q 3(3) 9 9
1 14. = = =
3 p − 2r 3(−2) − 2(−5) −6 + 10 4
2. False. For example, 5 is a real number, and
10 15. Let r = 1.5.
5= which is not an irrational number. APR = 12r = 12(1.5) = 18%
2
3. Answers vary with the calculator, but 16. Let r = 1.67.
2, 508, 429, 787 APR = 12r = 12(1.67) = 20.04%
is the best.
798, 458, 000
17. Let APR = 9.
4. −5 + 0 = −5 APR = 12r
This illustrates the identity property of addition. 9 = 12r
9 3
5. 6(t + 4) = 6t + 6 ⋅ 4 = =r
12 4
This illustrates the distributive property. r = .75%
6. 3 + (–3) = (–3) + 3 18. Let APR = 19.5.
This illustrates the commutative property of APR = 12r
addition. 19.5 = 12r
19.5
7. 0 + (–7) = –7 + 0 =r
This illustrates the commutative property of 12
addition. r = 1.625%
8. 8 + (12 + 6) = (8 + 12) + 6 19. 3 − 4 ⋅ 5 + 5 = 3 − 20 + 5 = −17 + 5 = −12
This illustrates the associative property of
addition. 20. (4 − 5) ⋅ 6 + 6 = −1 ⋅ 6 + 6 = −6 + 6 = 0
9. Answers vary. One possible answer: The sum of 21. 8 − 4 2 − (−12)
a number and its additive inverse is the additive
identity. The product of a number and its Take powers first.
multiplicative inverse is the multiplicative 8 – 16 – (–12)
identity. Then add and subtract in order from left to right.
8 – 16 + 12 = –8 + 12 = 4
10. Answers vary. One possible answer: When using
the commutative property, the order of the 22. 8 − (−4) 2 − (−12)
addends or multipliers is changed, while the Take powers first.
grouping of the addends or multipliers is 8 – 16 – (–12)
changed when using the associative property. Then add and subtract in order from left to right.
8 – 16 + 12 = –8 + 12 = 4
For Exercises 11–14, let p = –2, q = 3 and r = –5.
⎣ 2 + 5 (3)⎤⎦ = −3 [ −2 + 15]
11. −3 ( p + 5q ) = −3 ⎡− ⎣ ( )
23. − (3 − 5) − ⎡ 2 − 3 2 − 13 ⎤
⎦
= −3 (13) = −39 Take powers first.
–(3 – 5) – [2 – (9 – 13)]
Work inside brackets and parentheses.
12. 2 (q − r ) = 2 (3 + 5) = 2 (8) = 16 – (–2) – [2 – (–4)] = 2 – [2 + 4]
= 2 – 6 = –4
1
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
,2 CHAPTER 1 ALGEBRA AND EQUATIONS
2(3 − 7) + 4(8) 36. 3 4 = .75
24.
4(−3) + (−3)(−2)
Work above and below fraction bar. Do 37. 3.14 < π
multiplications and work inside parentheses.
38. 1 3 > .33
2(−4) + 32 −8 + 32 24
= = = = −4
−12 + 6 −12 + 6 −6 39. a lies to the right of b or is equal to b.
2(−3) + ( −32) − 2
25.
(− 16 ) 40. b + c = a
64 − 1 41. c < a < b
Work above and below fraction bar. Take roots.
42. a lies to the right of 0
2(−3) + ( −32) − ( −24)
8 −1 43. (–8, –1)
Do multiplications and divisions. This represents all real numbers between –8 and
−6 − 32 + 12 –1, not including –8 and –1. Draw parentheses at
–8 and –1 and a heavy line segment between
8 −1 them. The parentheses at –8 and –1 show that
Add and subtract. neither of these points belongs to the graph.
− 12 − 32 + 12 − 14 −7
2
= 2 = = −1
7 7 7
44. [–1, 10]
6 2 − 3 25 This represents all real numbers between –1 and
26.
6 2 + 13 10, including –1 and 10. Draw brackets at –1 and
Take powers and roots. 10 and a heavy line segment between them.
36 − 3(5) 36 − 15 21
= = =3
36 + 13 49 7
45. [–2, 2)
2040 189 4587 This represents all real numbers between –2 and
27. , , 27, , 6.735, 47
523 37 691 2, including –2, not including 2.
Draw a bracket at –2, a parenthesis at 2, and a
187 385 heavy line segment between them.
28. , 2.9884, 85 , π , 10,
63 117
29. 12 is less than 18.5.
12 < 18.5 46. (−2, 3]
This represents all real numbers x such that
30. –2 is greater than –20. –2 < x ≤ 3. Draw a heavy line segment from –2
–2 > –20 to 3. Use a parenthesis at –2 since it is not part of
the graph. Use a bracket at 3 since it is part of
31. x is greater than or equal to 5.7. the graph.
x ≥ 5.7
32. y is less than or equal to –5.
y ≤ −5
47. (−2, ∞ )
33. z is at most 7.5. This represents all real numbers x such that
z ≤ 7.5 x > –2. Start at –2 and draw a heavy line
segment to the right. Use a parenthesis at –2
34. w is negative. since it is part of the graph.
w<0
35. −6 < −2
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
, SECTION 1. 1 THE REAL NUMBERS 3
48. (–∞, –2] 58. a. Shaquille O’Neal’s height in inches is
This represents all real numbers less than or 7(12) + 1 = 85 in.
equal to –2. Draw a bracket at –2 and a heavy .455W .455(300)
ray to the left. B= =
(.0254 H ) 2
(.0254(85)) 2
136.5
= ≈ 29.3
(2.159) 2
49. 3; 2000, 2006, and 2007
b. No, Shaquille O’Neal’s body mass index
50. 6; 1998, 1999, 2002, 2003, 2004, 2005 falls above the desirable range.
51. 7; 1998, 1999, 2001, 2002, 2003, 2004, 2005 59. A wind at 10 miles per hour with a 30°
temperature has a wind-chill factor of 21°.
52. 3; 2000, 2006, 2007 A wind at 30 miles per hour with a –10°
temperature has a wind-chill factor of –39°.
53. 0
21 − (−39) = 21 + 39 = 60°
54. 5; 1998, 1999, 2002, 2003, 2005
60. A wind at 20 miles per hour with a –20°
55. a. Steffi Graf’s height in inches is temperature has a wind-chill factor of –48°.
5(12) + 9 = 69 in. A wind at 5 miles per hour with a 30°
.455W .455(119) temperature has a wind-chill factor of 25°.
B= =
(.0254 H ) 2
(.0254(69)) 2 −48 − 25 = −73 = 73°
54.145
= ≈ 17.6 61. A wind at 25 miles per hour with a –30°
(1.7526) 2 temperature has a wind-chill factor of –64°.
A wind at 15 miles per hour with a –30°
b. No, Steffi Graf’s body mass index falls temperature has a wind-chill factor of –58°.
below the desirable range.
−64 − (−58) = −64 + 58 = −6 = 6°
56. a. Jackie Joyner-Kersee’s height in inches is
5(12) + 10 = 70 in. 62. A wind at 40 miles per hour with a 40°
.455W .455(153) temperature has a wind-chill factor of 27°. A
B= = wind at 25 miles per hour with a –30°
(.0254 H ) 2 (.0254(70)) 2 temperature has a wind-chill factor of –64°.
=
69.615
≈ 22.0 27 − (−64) = 27 + 64 = 91 = 91°
(1.778) 2
63. 8 − −4 = 8 − (4) = 4
b. Yes, Jackie Joyner-Kersee’s body mass
index falls within the desirable range.
64. −9 − −12 = 9 − (12) = −3
57. a. Tiger Wood’s height in inches is
6(12) + 2 = 74 in. 65. − −4 − −1 − 14 = − (4) − −15
B=
.455W
=
.455(180) = − (4) − 15 = −19
(.0254 H ) 2
(.0254(74)) 2
81.9 66. − 6 − −12 − 4 = − (6) − −16 = −6 − (16) = −22
= ≈ 23.2
(1.8796) 2
67. 5 −5
b. Yes, Tiger Wood’s body mass index falls 5 __ 5
within the desirable range. 5=5
68. − −4 4
−4 4
−4<4
Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.
