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A P P E N D I X A
Review of Fundamental Concepts of Algebra
Appendix A.1
Review of Real Numbers and Their Properties . . . 1136
Appendix A.2
Exponents and Radicals . . . . . . . . . . . . . . . 1143
Appendix A.3
Polynomials and Factoring
Appendix A.4
Rational Expressions
Appendix A.5
Solving Equations . . . . . . . . . . . . . . . . . . 1172
Appendix A.6
Linear Inequalities in One Variable . . . . . . . . . 1194
Appendix A.7
Errors and the Algebra of Calculus . . . . . . . . . 1204
. . . . . . . . . . . . . 1150
. . . . . . . . . . . . . . . . 1164
A P P E N D I X A
Review of Fundamental Concepts of Algebra
Appendix A.1
■
Review of Real Numbers and Their Properties
You should know the following sets.
(a) The set of real numbers includes the rational numbers and the irrational numbers.
(b) The set of rational numbers includes all real numbers that can be written as the ratio pq of two integers,
where q 0.
(c) The set of irrational numbers includes all real numbers which are not rational.
(d) The set of integers: . . . , 3, 2, 1, 0, 1, 2, 3, . . .
(e) The set of whole numbers: 0, 1, 2, 3, 4, . . .
(f) The set of natural numbers: 1, 2, 3, 4, . . .
■
The real number line is used to represent the real numbers.
■
Know the inequality symbols.
■
(a) a < b means a is less than b.
(b) a ≤ b means a is less than or equal to b.
(c) a > b means a is greater than b.
(d) a ≥ b means a is greater than or equal to b.
Interval Notation
Inequality Notation
a, b
a ≤ x ≤ b
a, b
Graph
a
b
a
b
a
b
a
b
a < x < b
a, b
a ≤ x < b
a, b
a < x ≤ b
a, x ≥ a
Type
x
Bounded and Closed
x
Bounded and Open
x
Bounded
x
Bounded
x
Unbounded
x
Unbounded
x
Unbounded
x
Unbounded
x
Unbounded
a
a, x > a
a
, b
x ≤ b
b
, b
x < b
b
, < x <
a, if a ≥ 0
.
a,
if a < 0
■
You should know that a ■
Know the properties of absolute value.
(a) a ≥ 0
1136
(b) a a
(c) ab a b
(d)
a
a
,b0
b
b
—CONTINUED—
Appendix A.1
Review of Real Numbers and Their Properties
■
The distance between a and b on the real line is da, b b a a b .
■
You should be able to identify the terms in an algebraic expression.
■
You should know and be able to use the basic rules of algebra.
■
Commutative Property
(a) Addition: a b b a
■
(b) Multiplication: a
Associative Property
(a) Addition: a b c a b c
■
(b) Multiplication: abc abc
Identity Property
(a) Addition: 0 is the identity; a 0 0 a a.
■
bba
(b) Multiplication: 1 is the identity; a 1 1 a a .
Inverse Property
(a) Addition: a is the additive inverse of a; a a a a 0.
(b) Multiplication: 1a is the multiplicative inverse of a, a 0; a1a 1aa 1 .
■
Distributive Property
(a) ab c ab ac
■
■
■
■
(b) a bc ac bc
Properties of Negation
(a) 1a a
(b) a a
(c) ab ab ab
(d) ab ab
(e) a b a b a b
Properties of Equality
(a) If a b, then a ± c b ± c.
(b) If a b, then ac bc.
(c) If a ± c b ± c, then a b.
(d) If ac bc and c 0, then a b.
Properties of Zero
(a) a ± 0 a
(b) a 0 0
(d) a0 is undefined.
(e) If ab 0, then a 0 or b 0.
(c) 0 a 0a 0, a 0
Properties of Fractions b 0, d 0
(a) Equivalent Fractions: ab cd if and only if ad bc.
(b) Rule of Signs: ab ab ab and ab ab
(c) Equivalent Fractions: ab acbc, c 0
(d) Addition and Subtraction
1. Like Denominators: ab ± cb a ± cb
2. Unlike Denominators: ab ± cd ad ± bcbd
(e) Multiplication: ab cd acbd
(f) Division: ab cd ab dc adbc if c 0.
Vocabulary Check
1. rational
2. irrational
3. absolute value
4. composite
5. prime
6. variables; constants
7. terms
8. coefficient
9. zero-factor property
1137
1138
Appendix A
Review of Fundamental Concepts of Algebra
7
2
1. 9, 2, 5, 3, 2, 0, 1, 4, 2, 11
7
5
2. 5, 7, 3, 0, 3.12, 4, 3, 12, 5
(a) Natural numbers: 5, 1, 2
(a) Natural numbers: 12, 5
(b) Whole numbers: 0, 5, 1, 2
(b) Whole numbers: 0, 12, 5
(c) Integers: 9, 5, 0, 1, 4, 2, 11
(d) Rational numbers: 9,
72,
5,
2
3,
(c) Integers: 7, 0, 3, 12, 5
7
5
(d) Rational numbers: 7, 3, 0, 3.12, 4, 3, 12, 5
0, 1, 4, 2, 11
(e) Irrational numbers: 2
(e) Irrational numbers: 5
4. 2.3030030003 . . . , 0.7575, 4.63, 10, 75, 4
3. 2.01, 0.666 . . . , 13, 0.010110111 . . . , 1, 6
(a) Natural numbers: 1
(a) Natural numbers: 4
(b) Whole numbers: 1
(b) Whole numbers: 4
(c) Integers: 13, 1, 6
(c) Integers: 75, 4
(d) Rational numbers: 2.01, 0.666 . . . , 13, 1, 6
(d) Rational numbers: 0.7575, 4.63, 75, 4
(e) Irrational numbers: 0.010110111 . . .
(e) Irrational numbers: 2.3030030003 . . . , 10
1
6. 25, 17, 12
5 , 9, 3.12, 2 , 7, 11.1, 13
5. , 13, 63 , 12
2, 7.5, 1, 8, 22
(a) Natural numbers:
6
3
(a) Natural numbers: 25, 9, 7, 13
(since it equals 2), 8
(b) Whole numbers: 63, 8
(c) Integers:
6
3,
(b) Whole numbers: 25, 9, 7, 13
(c) Integers: 25, 17, 9, 7, 13
1, 8, 22
(d) Rational numbers:
13, 63,
7.5, 1, 8, 22
(d) Rational numbers:
25, 17, 12
5 , 9, 3.12, 7, 11.1, 13
1
(e) Irrational numbers: , 2
2
1
(e) Irrational numbers: 2 7.
5
8
0.625
8.
1
3
0.3
9.
41
333
0.123
−8
14. 3.5 < 1
15.
3
2
< 7
− 3.5
−4
−3
−2
−1
0
5
6
>
16. 1 <
−7
−6
−5
−4
16
3
3
2
1
1
17.
0.54
13. 4 > 8
12. 6 < 2.5
11. 1 < 2.5
6
11
10.
2
3
16
3
2
3
4
5
6
0
7
1
2
3
4
5
6
18. 87 < 37
− 87
2 5
3 6
0
−2
1
19. (a) The inequality x ≤ 5 denotes the set of all real
numbers less than or equal to 5.
(b)
x
0
1
2
3
4
5
6
(c) The interval is unbounded.
−1
− 37
0
20. (a) The inequality x ≥ 2 denotes the set of all real
numbers greater than or equal to 2.
(b)
x
−4
−3
−2
−1
0
(c) The interval is unbounded.
Appendix A.1
21. (a) The inequality x < 0 denotes the set of all negative
real numbers.
(b)
−1
0
1
(b)
3
4
5
6
2
4
5
6
24. (a) , 2 denotes the set of all real numbers less than
2.
x
0
7
1
2
3
4
(c) The interval is unbounded.
(c) The interval is unbounded.
25. (a) The inequality 2 < x < 2 denotes the set of all real
numbers greater than 2 and less than 2.
(b)
3
(b)
x
2
1
(c) The interval is unbounded.
23. (a) The interval 4, denotes the set of all real numbers
greater than or equal to 4.
1
x
0
2
(c) The interval is unbounded.
1139
22. (a) The inequality x > 3 denotes the set of all real
numbers greater than 3.
(b)
x
−2
Review of Real Numbers and Their Properties
26. (a) The inequality 0 ≤ x ≤ 5 denotes the set of all real
numbers greater than or equal to zero and less than
or equal to 5.
x
−2
−1
0
1
2
(b)
x
0
1
2
3
4
5
(c) The interval is bounded.
(c) The interval is bounded.
27. (a) The inequality 1 ≤ x < 0 denotes the set of all
negative real numbers greater than or equal to 1.
(b)
(b)
x
−1
(b)
2
3
4
5
2
3
4
5
6
30. (a) The interval 1, 2 denotes the set of all real
numbers greater than 1 and less than or equal to 2.
(b)
x
1
1
(c) The interval is bounded.
29. (a) The interval 2, 5 denotes the set of all real
numbers greater than or equal to 2 and less than 5.
0
x
0
0
(c) The interval is bounded.
−2 −1
28. (a) The inequality 0 < x ≤ 6 denotes the set of all real
numbers greater than zero and less than or equal to 6.
x
−2
6
(c) The interval is bounded.
−1
0
1
2
(c) The interval is bounded.
31. 2 < x ≤ 4
32. 6 ≤ y < 0
33. y ≥ 0
34. y ≤ 25
35. 10 ≤ t ≤ 22
36. 3 ≤ k < 5
37. W > 65
38. 2.5% ≤ r ≤ 5%
39. 10 10 10
42. 4 1 3 3
45.
5
5
5
1
5
5
5
41. 3 8 5 5 5
44. 3 3 3 3 6
40. 0 0
43. 1 2 1 2 1
46. 3 3 33 9
47. If x < 2, then x 2 is negative.
Thus
x 2 x 2 1.
x2
x2
1140
Appendix A
Review of Fundamental Concepts of Algebra
49. 3 > 3 since 3 > 3.
48. If x > 1, then x 1 is positive.
x1
x1
Thus,
1.
x1
x1
51. 5 5 since 5 5.
52. 6 < 6 since 6 6
and 6 6 6.
54. 2 > 2 since 2 2.
55. d126, 75 75 126 51
50. 4 4 since 4 4 and
4 4.
53. 2 2 since 2 2.
56. d126, 75 75 126
51
57. d 52, 0 0 52 52
58. d14, 11
4 11
4
112
59. d16
5 , 75 14 52
112
75
128
16
5 75
60. d9.34, 5.65 5.65 9.34 14.99
61. Budgeted Expense, b
$112,700
Actual Expense, a
a b
0.05b
$113,356
$656
0.05112,700 $5635
Since $656 < $5635 but $656 > $500, the actual expense does not pass the “budget variance test.”
62. Budgeted Expense, b
$9400
Actual Expense, a
a b
0.05b
$9772
$372
0.059400 $470
Since $372 < $470 and $372 < $500, the actual expense does pass the “budget variance test.”
63. Budgeted Expense, b
$37,640
Actual Expense, a
a b
0.05b
$37,335
$305
0.0537,640 $1882
Since $305 < $500 and $305 < $1882, the actual expense passes the “budget variance test.”
64. Budgeted Expense, b
$2575
Actual Expense, a
a b
0.05b
$2613
$38
0.052575 $128.75
$590.9
1990
$1253.2
2000
$1788.8
96
48
236.4 (s)
73.8 (d)
144
Year
2000
1980
192
1990
$195.6
240
1980
1970
92.5 92.2 $0.3 surplus
192.8 195.6 $2.8 deficit
517.1 590.9 $73.8 deficit
1032.0 1253.2 $221.2 deficit
2025.2 1788.8 $236.4 surplus
(b)
0.3 (s)
2.8 (d)
$92.2
Surplus or Deficit
(in billions)
1970
1960
Expenditures
(in billions)
1960
Year
Surplus or deficit
(in billions)
65. (a)
221.2 (d)
Since $38 < $500, and $38 < $128.75, the actual expense passes the “budget variance test.”
Appendix A.1
Review of Real Numbers and Their Properties
66. Total: 2213 3290 4666 5665 9784 25,618
Under 35
8.64%
2213
0.0863857 8.63857%
25,618
Under 35:
8.63857% of 360 31.1
3290
0.1284253 12.84253%
25,618
35–44:
35−44
12.84%
65 and older
38.19%
45−54
18.21%
55−64
22.11%
12.84253% of 360 46.2
4666
0.18213756 18.213756%
25,618
45–54:
18.213756% of 360 65.6
5665
0.221133578 22.1133578%
25,618
55–64:
22.1133578% of 360 79.6
65 and older:
9784
0.38191896 38.191896%
25,618
38.191896% of 360 137.5%
68. dx, 10 x 10 , and dx, 10 ≥ 6, thus,
x 10 ≥ 6.
70. d y, a y a and d y, a ≤ 2, thus y a ≤ 2.
67. dx, 5 x 5 and dx, 5 ≤ 3, thus x 5 ≤ 3.
69. d y, 0 y 0 y and d y, 0 ≥ 6, thus y ≥ 6.
71. d326, 351 351 326 25 miles
73. 7x 4
72. d48, 82 82 48 34
75. 3x2 8x 11
74. 6x 3 5x
Terms: 7x, 4
Terms: 6x 3, 5x
Terms: 3x2, 8x, 11
Coefficient: 7
Coefficients: 6, 5
Coefficients: 3, 8
76. 3
3x2 1
77. 4x 3 Terms: 3
3x2 , 1
Coefficients: 3
3
x
5
2
x
Terms: 4x 3, , 5
2
Coefficients: 4,
79. 4x 6
78. 3x 4 x2
4
Terms: 3x 4, 1
2
Coefficients: 3, 1
4
80. 9 7x
(a) 41 6 4 6 10
(a) 9 73 9 21 30
(b) 40 6 0 6 6
(b) 9 73 9 21 12
81. x2 3x 4
x2
4
82. x2 5x 4
(a) 22 32 4 4 6 4 14
(a) 12 51 4 1 5 4 10
(b) 22 32 4 4 6 4 2
(b) 12 51 4 1 5 4 0
1141
1142
83.
Appendix A
Review of Fundamental Concepts of Algebra
x1
x1
(a)
84.
11 2
11 0
Division by zero is undefined
(b)
86. 2
x
x2
85. x 9 9 x
(a)
2
2 1
22 4 2
(b)
2
2
2 2 0
1 1
0
0
1 1 2
Division by 0 is undefined.
12 1
87.
89. 2x 3 2x 6
1
h 6 1, h 6
h 6
Associative Property of Addition
Associative Property of Multiplication
712
Associative Property of Multiplication
12
Multiplicative Inverse Property
5
8
1
3
16 16 16 2
99.
5
8
98.
5
10
4
9
3
12
16 15
24 24 24 24 8
100.
x
8x
3x
5x
2x
3
4 12 12 12
104.
n
1
0.5
0.01
0.0001
0.000001
5n
5
10
500
50,000
5,000,000
(b) The value of 5n approaches infinity as n
approaches 0.
107. False. If a < b, then
6 4 64 2
7 7
7
7
10
11
6
60
12
13
59
33
13
66 66 66 66 66
102. 6
1
4
101. 12 4 12 1 12 4 48
105. (a)
Distributive Property
Multiplicative Identity Property
97.
103.
95. 3t 4 3 t 3 4
Commutative Property of Multiplication
12
Multiplicative Identity Property
93. x y 10 x y 10
Right Distributive Property
1
91. 1 1 x 1 x
Additive Identity Property
92. z 5x z x 5 x
96. 17 7 12 17
Additive Inverse Property
90. z 2 0 z 2
Distributive Property
3xy
88. x 3 x 3 0
Multiplicative Inverse Property
Multiplicative Inverse Property
94. x3y x 3y
Commutative Property of Addition
1 1
> , where a b 0.
a b
5x
6
106. (a)
48 6 12 3
2
9
5x
3
1
5x
9 27
n
1
10
100
10,000
100,000
5n
5
0.5
0.05
0.0005
0.00005
(b) The value of 5n approaches 0 as n increases
without bound.
108. False. The denominators cannot be added when adding
fractions.
Appendix A.2
(b) u v ≤ u v
Exponents and Radicals
1143
109. (a) u v u v if u is positive and v is negative or vice versa.
They are equal when u and v have the same sign. If they differ in sign, u v is less than u v .
110. Yes. y is nonnegative if y ≥ 0. y is positive if y > 0.
111. The only even prime number is 2, because its factors are
itself and 1.
112.
113. (a) Since A > 0, A < 0. The expression is negative.
Negative
integers
. . . , −3, −2, −1,
Irrational
numbers
Integers
. . . , −3, −2, −1,
0, 1, 2, 3, . . .
Real
numbers
Rational
numbers
Noninteger
fractions
(positive and
negative)
Whole
numbers
0, 1, 2, . . .
(b) Since B < A, B A < 0. The expression is
negative.
Natural
numbers
1, 2, 3, . . .
Zero
114. (a) Since C < 0, C > 0. The expression is positive.
(b) Since A > C, A C > 0. The expression is
positive.
Appendix A.2
■
115. Yes, if a is a negative number, then a is positive. Thus,
a a if a is negative.
Exponents and Radicals
You should know the properties of exponents.
(a) a1 a
(b) a0 1, a 0
(c) a ma n a mn
(d) aman amn, a 0
(e) an 1an 1an, a 0
(f) (a m)n a mn
(g) (ab)n anbn
(h) (ab)n a nb n, b 0
(i) (abn ba n, a 0, b 0
(j) a2 a 2 a2
■
You should be able to write numbers in scientific notation, c 10n, where 1 ≤ c < 10 and n is an integer.
■
You should be able to use your calculator to evaluate expressions involving exponents.
■
You should know the properties of radicals.
n am (a) n a , a > 0
m
(c)
n a
n
b
ab, b 0
n
n a
(e) a
n
n a
(b) (d)
n b n ab
m n
a mn
a
n an a .
(f) For n even, n an a.
For n odd, n a
(g) a1n ■
n a
n a m, a ≥ 0
(h) a mn You should be able to simplify radicals.
(a) All possible factors have been removed from the radical sign.
(b) All fractions have radical-free denominators.
(c) The index for the radical has been reduced as far as possible.
■
You should be able to use your calculator to evaluate radicals.
m
1144
Appendix A
Review of Fundamental Concepts of Algebra
Vocabulary Check
1. exponent; base
2. scientific notation
3. square root
4. principal nth root
5. index; radicand
6. simplest form
7. conjugates
8. rationalizing
9. power; index
1. 85 8 8 8 8 8
2. 27 2222222
3. 4.94.94.94.94.94.9 4.96
4. 1010101010 105
5. (a) 32
3 33 27
6. (a)
(b) 3 33 34 81
(b)
8. (a) 23 322 23 2
26
(b)
3
5
3
5
3
2
9. (a)
34 64 81 5184
33
13 3
5
3 5
3
5
4 32
4 22
22 31
52
1
32
(b) 32 9
3 44
31 4 44 1 35
34 41
332 523
32 1 4 4 31 16
3
32
32
1
25 32
11. (a) 21 31 1 1 3 2 5
2 3 6 6 6
(b) 212 212 22 4
12. (a) 31 22 1 1
4
3
7
3 4 12 12 12
(b) 322 34 13. 4352 6425 1600
1
1
34 81
36 729
2.125
73 343
14. 84103 0.244
15.
17. When x 2,
18. When x 4,
32 24.
3
20. When x 3,
3
7x2 742 2x3
7
7
.
42 16
4x2 4 12 414 1.
43
4334 5184
34
6x0 6100 61 6.
22. When x 2,
23 227 54.
3
527 135.
23. When x 12,
16.
19. When x 10,
21. When x 3,
5x 53
3
2
43
35 243
43
64
(b) 3225 (b) 20 1
3x3
7. (a) 330 1
1
1
32
32 2 34
3
9
32 2
1
10. (a)
55
53 125
52
3x 4 324
316 48.
1
24. When x 3,
25. (a) 5z3 53z3 125z3
1
5x3 5 13 5 27
275 .
3
(b) 5x 4x2 5x 42 5x6
Appendix A.2
26. (a) 3x2 32x2 9x2
(b) 4x30 40
27. (a) 6y 22y02 6y 22
x0 1,
x0
(b)
28. (a) z33z4 13z33z4
1 3
29. (a)
z34 3z7
(b)
25y 8 5 84 5 4
(b)
y
y
10y 4 2
2
30. (a)
(b)
1
r4
r 46 r2 2
r6
r
3
4
y
3
y
4
43
3
y
32. (a) 2x 50 1, x 0
1
z 24
34. (a) 4y28y 4 48 y42 32y 2
36. (a)
x3y4
5
x2
x3
xn
xn
3
(b)
3
ab ab
3
5xy 3 3
4
a3
b3
7x 2
7
7x 23 7x1 x3
x
4
12x y3 4
x y31 x y2
9x y
3
3
1
1
2x22 4x4
(b) 2x22 a33
10
x
1
(b)
8x6
2 x 3
4x3
10
x
32n 3n2n 33n
2
125x9
y12
a3
(b)
35. (a) 3n
x2n
1
x2n3n x1 x3n
x
3
12 6y 24 24y 2
3x 5
3x 53 3x 2
x3
33. (a) 2x23 4x31 (b) z 23z 21 z 24 (b)
1145
31. (a) x 50 1, x 5
34
64 81
5184
34 7
y4
y
y
Exponents and Radicals
ab ba ba ba ab
3
2
2
3
5
2
3
5
37. 57,300,000 5.73 107 square miles
a0
b3 b33 b0 1
38. 9,460,000,000,000 9.460 1012 kilometers
39. 0.0000899 8.99 105 gram per cubic centimeter
40. 0.00003937 3.937 105 inch
41. 4.568 109 4,568,000,000 ounces
42. 1.5 107 15,000,000 degrees Celsius
43. 1.6022 1019 0.00000000000000000016022 coulomb
44. 9.0 105 0.00009 meter
45. (a) 25 108 5 104 50,000
3 8 1015 2 105 200,000
(b) 46. (a) 1.2 1075 103 6.0 104 60,000
(b)
6.0 108
2.0 1011
3.0 103
47. (a) 750 1 (b)
0.11
365
800
954.448
67,000,000 93,000,000
30,769,230,769.2
0.0052
3.077 1010
1146
Appendix A
Review of Fundamental Concepts of Algebra
48. (a) 9.3 10636.1 104 4.907 1017
(b)
2.414 1046
1.479
1.68 1055
3
6.3 104 39.791
(b) 50. (a) 2.65 10413 0.064
51. (a) 9 3
(b) 9 104 0.03
53. (a) 3235 (b)
16
81
34
64
1
55. (a) (b)
49. (a) 4.5 109 67,082.039
1
1
5
3235
32
1
81
16
13
32
(b)
34
3
81
16
3
3
27
3
3
2
3
32 25
(b) 36 32 216
3
2
54. (a) 10032 100 3
27
8
(b)
3 64 4
6413 25
3
8
1
1
(2)3 8
4
278 3 27 3
52. (a) 2713 56. (a)
94
125
27 (b) 5 32
5 32 2
2
12
49
13
1
125
43
12
103 4
9
27
125
13
1
1000
2
3
3 27
3
3
5
5
3 125
12543
3 125
125134 4
54 625
57. (a) 57 7.550
59. (a) 12.41.8 0.011
3 452 12.651
58. (a) 5 27 3 27 35
(b) (b) 532.5 0.005
6 125 2.236
(b) 7.225
60. (a)
(b)
7 4.13.2
3.495
2
13
3
32
3
2
133
3 4
61. (a) 433 41 4
3
5 96x5 5 32x5
(b) 5.906
3
2
(b)
24 3
8
1627 75
(b) 4
64. (a)
42 22
3
3
4 3x24 4 34x8
(b) 3x 2
3
3 23
x
3 2
2 2
3 3
3
3
65. (a) 72x3 36x2
22
2
32 x y22
3y 26x
(b)
32a4 222 2 a22
b2
b2
4a2
b
2
3 16x5 3 8x3
67. (a) 2x2
(b)
3
182
z2
z
18
zz
4 3x4y2 3x4y 214
68. (a) 314 x y12
75x2
y4
25x2
3
y4
18z
2
3 2x2
2x
(b) 75x2y4 2x
6x2x
3 53
52
3
3
3
8
3 2
3
66. (a) 54xy4 6
3 36 6
5 3
2x
2 315
63. (a) 8 4
62. (a) 12
5 x 3
y2
4 3y
x (b)
5 160x8z4
160x8z415
25 5x5 x3 z415
2x5x3z415
5 5x3z4
2x
Appendix A.2
69. (a) 250 128 225
2 124 2 252 1222 102 242 342
(b) 1032 618 1016
70. (a) 427 75 432
Exponents and Radicals
2 69 2 104 2 632 402 182 222
3 52 3
71. (a) 5x 3x 2x
(b) 29y 10y 23y 10y
4 33 53
123 53
6y 10y 4y
73
3 16 3
3 54 3 2
(b) 3 2
23 3
33
3 2 3
3 2
2
3
3 2 9
3 2
2
3 2
11
72. (a) 849x 14100x 872
x 14102 x
8 7x 14 10x 56x 140x
84x
(b) 348x2 775x2 33
42 x2 73 52 x2
12 x3 35x3 23x3
3 4 x 3 7 5 x 3
73. (a) 3x 1 10x 1 13x 1
(b) 780x 2125x 716
5x 225 5x 745x 255x 285x 105x 185x
74. (a) x3 7 5x3 7 4x3 7
(b) 11245x3 945x3 115
72 x x2 95 32 x x2
11 7x5x 9 3x5x
77x5x 27x5x
50x5x
75. 5 3 3.968 and
76.
5 3 8 2.828
113 3
77. 32 22 9 4
11
13 3.606
Thus, 5 3 > 5 3.
Thus, 5 > 32 22.
78. 32 42 9 16
79.
25 5
1
3
1
3
3
3
3
3
80.
82.
Thus, 5 32 42.
81.
5
10
5
10
10
10
510 10
10
2
2
5 3 2 5 3 25 3 2 5 3 5 3
2
5 3 5 3 5 3 5 2 32
25 3
22
11
3
5 6
3
5 6
5 6
5 6
35 6 35 6 35 6 36 5 56
1
1147
1148
83.
85.
86.
Appendix A
8
2
4
2
5 3
7 3
4
2 22
3
Review of Fundamental Concepts of Algebra
2
5 3
3
7 3
4
2
1
2
2
5 3
5 3
7 3
7 3
2
84.
2
2
3
3
64 4, Given
88. 6413 4, Answer
5 32 2,
89. Answer
3215 2, Given
90. 144 12, Answer
14412 12, Given
3 216 6,
91. Given
21613 6, Answer
5 243 3,
92. Answer
24315 3, Given
4 813 27,
93. Given
8134 27, Answer
4 165 32,
94. Answer
1654 32, Given
2x232 232x232
212x4
212x4
x 121323 x3 x43y23 x43y23
13 13 x 33y13 xy13
xy13
x y
98.
512 5x52 512 5x52
5x32
532x32
1
, x > 0
x3
4 32 324 312 3
99. (a) 32 (3212)12
4 32 4 16
3214 x
512x52
51x , x > 0
532x32
5
6 x3 x36 x12 x
100. (a) 6
3
x 14 x 146 x 123 x 12
(b) (b)
96.
2
232x3
23212 x34 21x1 212x4
x
x3 x 12 x 12 x1
x 32 x1 x 32 x 3
101. (a)
2
Rational Exponent Form
912 3, Answer
97.
3
2
2 32
53
2
35 3 35 3 87. 9 3, Given
2
2
1
79
47 3 47 3
27 3
Radical Form
95.
4
(b) 3x24 3x 2
102. (a)
243x 1 243x 1
1212
4 2
2 2
243x 114
4 2x 2x)1412 2x18 8 2x
4
243x 1
(b)
4
3
81x 1
3
3x 1
4
10a b 10a b 3
7
7
13 12
10a7b16
6
10a
a6 b
6
a
10ab
Appendix A.2
103. T 2
322
2
161
2
4
Exponents and Radicals
1149
3
104. Size 0.03v; For v :
4
34
Size 0.03
1
0.03 1.57 seconds
2
0.03
3
4
3
2
0.026 inch
105. t 0.031252 12 h52
, 0 ≤ h ≤ 12
(a)
h (in centimeters)
106. Time (b) As h approaches 12, t approaches
t (in seconds)
0
0
1
2.93
2
5.48
3
7.67
4
9.53
5
11.08
6
12.32
7
13.29
8
14.00
9
14.50
10
14.80
11
14.93
12
14.96
0.031252 8.643 14.96 seconds.
Distance
93,000,000 miles
Rate
11,180,000 miles per minute
107. True. When dividing variables, you subtract exponents.
8.32 minutes, or 8 minutes 19.1 seconds
109. 1 108. False. When a power is raised to a power, you multiply
the exponents: a n k a nk.
110. (a) 3 is also raised to the negative one power so, 3x1 am
amm a0, a 0
am
1
.
3x
(b) When two powers have the same base, the exponents are added, y 3
y 2 y 5.
(c) When a power is raised to a power, exponents are multiplied, a2b34 a8b12.
(d) The square of a binomial contains a cross product term, a b2 a2 2ab b2.
(e) If x < 0, then 4x2 > 0 but 2x < 0, 4x2 2 x .
(f) Radicals can only be added together if they have the same radicand and index: 2 2 22.
1150
Appendix A
Review of Fundamental Concepts of Algebra
25 54 0.8
2
111. When any positive integer is squared, the units digit is
0, 1, 4, 5, 6, or 9. Therefore, 5233 is not an integer.
112.
2
5
5
5
25
0.8944
5
Since 0.8 0.8944,
2
5
2
2
5
5
5
and squaring is not equivalent to rationalizing the
denominator.
Appendix A.3
■
Polynomials and Factoring
Given a polynomial in x, an x n a n1 x n1 . . . a1x a0, where an 0, and n is a nonnegative integer, you
should be able to identify the following.
(a) Degree: n
(b) Terms: an x n, a n1 x n1, . . . , a1 x, a0
(c) Coefficients: a n, a n1, . . . , a1, a0
(d) Leading coefficient: an
(e) Constant term: a0
■
You should be able to add and subtract polynomials.
■
You should be able to multiply polynomials by the Distributive Properties.
■
You should be able to multiply two binomials by the FOIL Method.
■
You should know the special binomial products.
(b) u ± v2 u2 ± 2uv v2
(a) (u v)(u v) u2 v2
(c) u ± v3 u3 ± 3u2v 3uv2 ± v3
■
You should be able to factor out all common factors, the first step in factoring.
■
You should be able to factor the following special polynomial forms.
(a) u2 v2 u vu v
(b) u2 ± 2uv v2 u ± v2
(c) u3 ± v3 u ± vu2 uv v2
■
You should be able to factor by grouping.
■
You should be able to factor some trinomials by grouping.
Vocabulary Check
1. n; an; a0
2. descending
3. monomial; binomial; trinomial
4. like terms
5. First terms; Outer terms; Inner terms; Last terms
6. factoring
7. completely factored
1. (d) 12 is a polynomial of degree
zero.
2. (e) 3x 5 2x 3 x is a
polynomial of degree five.
3. (b) 1 2x3 2x3 1
is a binomial with leading
coefficient 2.
4. (a) 3x2 is a monomial of positive
degree.
2
5. (f) 3x4 x2 10 is a trinomial
2
with leading coefficient 3.
6. (c) x 3 3x2 3x 1 is a
third-degree polynomial with
leading coefficient 1.
Appendix A.3
7. 2x3; 2x3 5;
2x3 4x 2 3x 20, etc.
(Answers will vary.)
10. 20x 3 5 (Answers will vary.)
8. 6x 5 3x 1
(Answers will vary.)
1
11. (a) Standard form: 2 x 5 14x
1
2
14. (a) Standard form: 7x
(b) Degree: 1
(c) Monomial
16. (a) Standard form: 25y 2 y 1
(b) Degree: 0
(c) Monomial
(c) Trinomial
19. (a) Standard form: 4x 5 6x 4 1
(b) Degree: 1
Leading coefficient: 4
Leading coefficient: 2
21. (a) Standard form: 4x3y
(b) Degree: 4 (add the exponents
on x and y)
Leading coefficient: 4
(c) Binomial
(c) Trinomial
Leading coefficient: 1
(c) Binomial
20. (a) Standard form: 2x 3
(b) Degree: 5
18. (a) Standard form: t 2 9
(b) Degree: 2
Leading coefficient: 3
Leading coefficient: 25
Leading coefficient: 1
(c) Binomial
17. (a) Standard form: 3
(b) Degree: 2
15. (a) Standard form: x5 1
(b) Degree: 5
Leading coefficient: 7
(c) Trinomial
Leading coefficient: 2
(c) Trinomial
(c) Binomial
Leading coefficient: 3
12. (a) Standard form: 2x 2 x 1
(b) Degree: 2
Leading coefficient:
(b) Degree: 4
1151
9. 15x 4 1; 3x 4 7x2;
5x4 6x, etc.
(Answers will vary.)
(b) Degree: 5
13. (a) Standard form: 3x 4 2x 2 5
Polynomials and Factoring
(c) Monomial
Standard form: 3x 2x 8
24. 2x3 x 3x1 is not a
polynomial because it includes a
term with a negative exponent.
x2 2x 3
is a polynomial.
2
27. y2 y 4 y3 is a polynomial.
23. 2x 3x3 8 is a polynomial.
22. (a) Standard form:
x 5y 2x2y 2 xy 4
3
(b) Degree: 6
Leading coefficient: 1
(c) Trinomial
25.
3x 4
4
3 3 4x1 is not
x
x
a polynomial because it includes a
term with a negative exponent.
26.
1
3
Standard form: x2 x 2
2
28. y 2 y 4 is not a polynomial because of the square root.
Standard form: y 4 y3 y2
29. 6x 5 8x 15 6x 5 8x 15
6x 8x 5 15
2x 10
30. 2x2 1 x2 2x 1 2x2 1 x2 2x 1
31. x3 2 4x3 2x x3 2 4x3 2x
2x2 x2 2x 1 1
4x3 x3 2x 2
x2 2x
3x3 2x 2
1152
Appendix A
Review of Fundamental Concepts of Algebra
32. 5x2 1 3x2 5 5x2 1 3x2 5
5x2 3x2 1 5
2x2 4
33. 15x2 6 8.3x3 14.7x2 17 15x2 6 8.3x3 14.7x2 17
8.3x3 15x2 14.7x2 6 17
8.3x3 29.7x2 11
34. 15.2x 4 18x 19.1 13.9x 4 9.6x 15 15.2x 4 18x 19.1 13.9x 4 9.6x 15
15.2x 4 13.9x 4 18x 9.6x 19.1 15
1.3x 4 8.4x 34.1
35. 5z 3z 10z 8 5z 3z 10z 8
5z 3z 10z 8
5z 3z 10z 8
12z 8
36. y 3 1 y 2 1 3y 7 y 3 1 y 2 1 3y 7
y 3 1 y 2 1 3y 7
y 3 y 2 3y 1 1 7
y 3 y 2 3y 7
37. 3xx2 2x 1 3xx2 3x2x 3x 1
3x3 6x2 3x
39. 5z3z 1 5z3z 5z1
15z2
5z
41. 1 x34x 14x x34x
38. y 24y 2 2y 3 y 24y 2 y 22y y 23
4y 4 2y 3 3y 2
40. 3x5x 2 3x5x 3x2
15x 2 6x
42. 4x3 x 3 4x3 4xx 3
4x 4x 4
12x 4x 4
4x 4 4x
4x 4 12x
43. 2.5x2 33x 2.5x23x 33x
7.5x3 9x
45. 4x18 x 3 4x18 x 4x3
12 x2 12x
44. 2 3.5y2y 3 22y 3 3.5y2y 3
4y3 7y 4 7y 4 4y 3
46. 2y4 78 y 2y4 2y 78 y
8y 74 y 2
74 y2 8y
47. x 3x 4 x2 4x 3x 12 FOIL
x2 7x 12
48. x 5x 10 x2 10x 5x 50
x2 5x 50
FOIL
Appendix A.3
49. 3x 52x 1 6x2 3x 10x 5 FOIL
Polynomials and Factoring
50. 7x 24x 3 28x2 21x 8x 6
6x 7x 5
2
51. Multiply: x2 0x3 1
0x3
FOIL
28x 29x 6
2
x2
1153
52. Multiply: x2 x2 x4 1
x4 0x3 x2
x4 0x3 x2 9x
x4 0x3 x2 0x 1
x4 x4 0x3 x2 0x 1 x4 x2 1
0x3 3x
3x
3x3
3x3
2
2
2x2
9x2 6x
2x2 6x 4
13x2 0x 4 x 4 13x 2 4
53. x 10x 10 x2 102 x2 100
54. 2x 32x 3 2x2 32 4x2 9
55. x 2yx 2y x2 2y2 x2 4y2
56. 2x 3y2x 3y 2x2 3y2 4x2 9y2
57. 2x 32 2x2 22x3 32
58. 4x 52 4x2 24x5 52
4x2 12x 9
16x2 40x 25
59. 2x 5y2 2x2 22x5y 5y2
60. 5 8x2 52 258x 8x2
4x2 20xy 25y2
25 80x 64x2
61. x 13 x3 3x21 3x12 13
62. x 23 x3 3x22 3x22 23
x3 3x2 3x 1
x3 6x2 12x 8
63. 2x y3 2x3 32x2y 32xy2 y3
64. 3x 2y3 3x3 33x22y 33x2y2 2y3
8x3 12x2y 6xy2 y3
27x3 54x2y 36xy2 8y3
65. 4x3 32 4x32 24x33 32
66. 8x 32 8x2 28x3 32
16x6 24x3 9
64x2 48x 9
67. m 3 nm 3 n m 32 n2
68. x y 1x y 1 x y2 12
x 2 2xy y 2 1
m2 6m 9 n2
m2 n2 6m 9
69. x 3 y2 x 32 2yx 3 y2
70. x 1 y2 x 12 2x 1y y2
x2 6x 9 2xy 6y y2
x2 2x 1 2xy 2y y2
x2 2xy y2 6x 6y 9
x2 2xy y2 2x 2y 1
71. 2r2 52r2 5 2r 22 52 4r 4 25
72. 3a3 4b23a3 4b2 3a32 4b22
9a6 16b4
73.
12 x 32 12 x2 212 x3 32
1
4x2 3x 9
74.
23 t 52 23 t2 223 t5 52
49 t 2 20
3 t 25
75.
13x 213x 2 13x2 22
19x2 4
1154
Appendix A
Review of Fundamental Concepts of Algebra
76. 2x 15 2x 15 2x2 15 2
77. 1.2x 32 1.2x2 21.2x3 32
1.44x2 7.2x 9
1
4x 2 25
78. 1.5y 32 1.5y2 21.5y3 32
79. 1.5x 41.5x 4 1.5x2 42
2.25y 2 9y 9
2.25x2 16
80. 2.5y 32.5y 3 2.5y2 32
81. 5xx 1 3xx 1 2xx 1
2x2 2x
6.25y 9
2
82. 2x 1x 3 3x 3 2x 2x 3 2x2 6x 2x 6
2x2
FOIL
8x 6
83. u 2u 2u2 4 u2 4u2 4
84. x yx yx2 y2 x2 y2x2 y2
x22 y 22 x4 y4
u4 16
85. x y x y x y 2
2
86. 5 x 5 x 52 x 2
xy
25 x
88. x 3 2 x2 2x3 3 2
87. x 5 x2 2x5 5
2
2
x2 23 x 3
x2 25x 5
89. 3x 6 3x 2
90. 5y 30 5 y 6
91. 2x3 6x 2xx2 3
92. 4x3 6x2 12x 2x2x2 3x 6
93. xx 1 6x 1 x 1x 6
94. 3xx 2 4x 2 x 23x 4
95. x 32 4x 3 x 3x 3 4
x 3x 1
96. 3x 12 3x 1 3x 1 13x 1
97. 12x 4 12x 82
12x 8
3x3x 1
1
1
15
1
98. 3 y 5 3 y 3 3 y 15
99. 12x3 2x2 5x 12x3 42x2 10
2x
12xx2 4x 10
6
2
100. 13 y 4 5y 2 2y 13 y 4 15
3 y 3y
2
2
12
101. 3xx 3 4x 3 3xx 3 3 x 3
3x 3x 6
2
13 y y3 15y 6
2
102. 45 y y 1 2 y 1 45 y y 1 10
5 y 1 5 y 12 y 5
103. x 2 81 x 2 92
x 9x 9
104. x2 49 x2 72
x 7x 7
105. 32y 2 18 216y 2 9
24y2 32
24y 34y 3
Appendix A.3
Polynomials and Factoring
107. 16x2 19 4x2 13 2
106. 4 36y 2 41 9y2
4 2
25 y
108.
64 25 y 82
2
5 y 825 y 8
4x 13 4x 13 412 3y2
1155
2
41 3y1 3y
109. x 12 4 x 12 22
110. 25 z 5 2 52 z 5 2
x 1 2x 1 2
5 z 55 z 5
x 1x 3
5 z 55 z 5
zz 10
111. 9u2 4v2 3u2 2v2
113. x2 4x 4 x2 22x 22
112. 25x2 16y 2 5x2 4y2
3u 2v3u 2v
114. x2 10x 25 x2 2(5x 52 x 52
x 22
5x 4y5x 4y
115. 4t2 4t 1 2t2 22t1 12
2t 12
116. 9x2 12x 4 3x2 23x2 22 3x 22
117. 25y2 10y 1 5y2 25y1 12
5y 12
118. 36y 2 108y 81 94y 2 12y 9
119. 9u2 24uv 16v2 3u2 23u4v 4v2
92y2 22y3 32
3u 4v2
92y 32
120. 4x2 4xy y 2 2x2 22xy y 2
2
x 23 2
2x y2
1
1
122. z2 z 4 z2 2z2 4
4
2
2
121. x2 3x 9 x2 2x3 3 12 2
z 2 1 2
123. x3 8 x3 23
x 2x2 2x 4)
124. x3 27 x3 33 x 3x2 3x 9
125. y3 64 y3 43 y 4 y2 4y 16
126. z3 125 z3 53 z 5z 2 5z 25
127. 8t3 1 2t3 13
2t 14t2 2t 1
128. 27x3 8 3x3 23 3x 29x2 6x 4
129. u3 27v3 u3 3v3
u 3vu2 3uv 9v2
130. 64x 3 y 3 4x3 y 3 4x y16x2 4xy y 2
131. x2 x 2 x 2x 1
132. x2 5x 6 x 2x 3
133. s2 5s 6 s 3s 2
134. t 2 t 6 t 2t 3
135. 20 y y2 y2 y 20
y 5 y 4
1156
Appendix A
Review of Fundamental Concepts of Algebra
136. 24 5z z2 z 2 5z 24 z 8z 3
137. x2 30x 200 x 20x 10
138. x2 13x 42 x 6x 7
139. 3x2 5x 2 3x 2x 1
140. 2x2 x 1 2x 1x 1
141. 5x2 26x 5 5x 1x 5
142. 12x2 7x 1 3x 14x 1
143. 9z2 3z 2 9z2 3z 2
3z 23z 1
144. 5u2 13u 6 5u2 13u 6
145. x3 x2 2x 2 x2x 1 2x 1
5u 2u 3
146. x3 5x2 5x 25 x2x 5 5x 5
x 1x2 2
147. 2x3 x2 6x 3 x22x 1 32x 1
x 5x2 5
148. 5x3 10x2 3x 6 5x2x 2 3x 2
2x 1x2 3
149. 6 2x 3x3 x 4 23 x x33 x
x 25x2 3
150. x5 2x3 x2 2 x3x2 2 x2 2
3 x2 x3
151. 6x3 2x 3x2 1 2x3x2 1 13x2 1
x2 2x3 1
3x2 12x 1
x 2 2x 1x 2 x 1
152. 8x 5 6x2 12x 3 9 2x24x 3 3 34x 3 3
4x 3 32x2 3
153. a c 38 24. Rewrite the middle term,
10x 6x 4x, since 64 24 and 6 4 10.
3x2 10x 8 3x2 6x 4x 8
3xx 2 4x 2
x 23x 4
154. a c 29 18. Rewrite the middle term,
9x 6x 3x, since 63 18 and 6 3 9.
155. a c 62 12. Rewrite the middle term,
x 4x 3x, since 43 12 and 4 3 1.
2x2 9x 9 2x2 6x 3x 9
6x2 x 2 6x2 4x 3x 2
2xx 3 3x 3
2x3x 2 13x 2
x 32x 3
2x 13x 2
156. a c 615 90. Rewrite the middle term,
x 10x 9x, since 109 90 and
10 9 1.
157. a c 152 30. Rewrite the middle term,
11x 6x 5x, since 65 30 and
6 5 11.
6x2 x 15 6x2 10x 9x 15
15x2 11x 2 15x2 6x 5x 2
2x3x 5 33x 5
3x5x 2 15x 2
2x 33x 5
3x 15x 2
158. a c 121 12. Rewrite the middle term, 13x 12x x, since 121 12 and 12 1 13.
12x2 13x 1 12x2 12x x 1
12xx 1 1x 1 x 112x 1
Appendix A.3
159. 6x2 54 6x2 9
160. 12x2 48 12x2 4
6x 3x 3
161. x3 4x2 x2x 4
164. 16 6x x 2 16 8x 2x x 2
8 x2 x
xx 3x 3
165. 1 4x 4x2 1 2x2
166. 9x2 6x 1 9x2 3x 3x 1
3x 13x 1
167. 2x2 4x 2x3 2xx 2 x2
168. 2y3 7y2 15y y2y2 7y 15
2xx2 x 2
y2y2 10y 3y 15
2xx 1x 2
y2y 3 y 5
169. 9x2 10x 1 9x 1x 1
170. 13x 6 5x2 5x2 13x 6
5x2 10x 3x 6
5x 3x 2
171.
1 2
81 x
1157
12x 2x 2
163. x2 2x 1 x 12
162. x 3 9x xx2 9
Polynomials and Factoring
1 2
648
29x 8 81
x 18
81 x 81
1 2
81
x 18x 648
1
1
1
1
172. 8 x2 96 x 16 96 12x2 x 6 96 4x 33x 2
1
1
81
x 36x 18
173. 3x3 x2 15x 5 x23x 1 53x 1
3x 1x2 5
175. x 4 4x3 x2 4x xx3 4x2 x 4
174. 5 x 5x2 x3 15 x x25 x
5 x1 x2
176. 3u 2u2 6 u3 u3 2u2 3u 6
xx2x 4 x 4
u2u 2 3u 2
xx 4x2 1
u 2u2 3
u 23 u2
3
36
2
177. 14x3 3x2 34x 9 14x3 12
4 x 4x 4
1
1
178. 5x3 x2 x 5 5x3 5x2 5x 25
14x3 12x2 3x 36
15x2x 5 5x 5
14x2x 12 3x 12
1
5x2 5x 5
14x 12x2 3
179. t 12 49 t 12 72
180. x2 12 4x2 x2 1 2xx2 1 2x
t 1 7t 1 7
x2 2x 1x2 2x 1
t 6t 8
x 12x 12
181. x2 82 36x2 x2 82 6x2
x2 8 6xx2 8 6x
x2 6x 8x2 6x 8
x 4x 2x 4x 2
182. 2t 3 16 2t 3 8 2t 2t 2 2t 4
1158
Appendix A
Review of Fundamental Concepts of Algebra
183. 5x3 40 5x3 8
184. 4x2x 1 2x 12 2x 14x 2x 1
5x3 23
5x 2
x2
2x 16x 1
2x 4
185. 53 4x2 83 4x5x 1 3 4x53 4x 85x 1
3 4x15 20x 40x 8
3 4x23 60x
186. 2x 1x 32 3x 12x 3 x 1x 32x 3 3x 1
x 1x 32x 6 3x 3
x 1x 3x 9
x 1x 3x 9
187. 73x 221 x2 3x 21 x3 3x 21 x273x 2 1 x
3x 21 x221x 14 1 x
3x 21 x220x 15
53x 21 x24x 3
188. 7x2x2 12x x2 127 7x2 14x2 x2 1
7x2 13x2 1
189. 3x 22x 14 x 234x 13 x 22x 133x 1 4x 2
x 22x 133x 3 4x 8
x 22x 137x 5
190. 2xx 54 x24x 53 2xx 53x 5 2x
2xx 53x 5
2xx 53x 5
191. 5x6 146x53x 23 33x 223x6 15 3x6 143x 2210x53x 2 3x6 1
3x6 143x 2230x6 20x5 3x6 3
3x6 143x 2233x6 20x5 3
3x23 143x 2233x6 20x5 3
3x2 1x 4 x2 143x 2233x6 20x5 3
3x2 14x 4 x2 143x 2233x6 20x5 3
192.
x2 2
x2
x 14 x2 15 x2 14
x2 1
2
2
x2 14 x2 14
x2
1
2
x2 1
2
Appendix A.3
193. For x2 bx 15 to be factorable, b must equal m n
where mn 15.
Polynomials and Factoring
1159
194. For x2 bx 50 to be factorable, b must equal m n
where mn 50.
Factors of 15
Sum of factors
Factors of 50
Sum of factors
151
15 1 14
150
1 50 51
151
15 1 14
150
1 50 51
35
3 5 2
510
5 10 15
35
3 5 2
510
5 10 15
225
2 25 27
225
2 25 27
The possible b-values are 14, 14, 2, or 2.
The possible b-values are 51, 51, 15, 15, 27, 27.
195. For x2 bx 12 to be factorable, b must equal m n
where mn 12.
196. For x2 bx 24 to be factorable, b must equal m n
where mn 24.
Factors of 12
Sum of factors
121
12 1 11
124
1 24 25
121
12 1 11
124
1 24 25
26
2 6 4
212
2 12 14
26
2 6 4
212
2 12 14
34
3 4 1
38
3 8 11
34
3 4 1
38
3 8 11
46
4 6 10
46
4 6 10
Factors of 24
The possible b-values are 11, 11, 4, 4, 1, 1.
Sum of factors
The possible b-values are 25, 25, 14, 14
11, 11, 10, 10.
197. For 2x2 5x c to be factorable, the factors of 2c must add up to 5.
Possible c-values
2c
Factors of 2c that add up to 5
2
4
14 4 and 1 4 5
3
6
23 6 and 2 3 5
3
6
61 6 and 6 1 5
7
14
72 14 and 7 2 5
12
24
83 24 and 8 3 5
These are a few possible c-values. There are many correct answers.
If c 2:
2x 2 5x 2 2x 1x 2
If c 3:
2x 2 5x 3 2x 3x 1
If c 3:
2x 2 5x 3 2x 1x 3
If c 7:
2x 2 5x 7 2x 7x 1
If c 12: 2x 2 5x 12 2x 3x 4
1160
Appendix A
Review of Fundamental Concepts of Algebra
198. For 3x2 10x c to be factorable, the factors of 3c must add up to 10.
Possible c-values
3c
Factors of 3c that add up to 10
3
9
19 9 and 1 9 10
8
24
122 24 and 12 2 10
8
24
64 24 and 6 4 10
These are a few possible c-values. There are many correct answers.
If c 3:
3x2 10x 3 3x 1x 3
If c 8: 3x2 10x 8 3x 2x 4
If c 8:
3x2 10x 8 3x 4x 2
199. For 3x2 x c to be factorable, the factors of 3c must add up to 1.
Possible c-values
3c
Factors of 3c must add up to 1
2
6
23 6 and 2 3 1
4
12
34 12 and 3 4 1
10
30
56 30 and 5 6 1
These are a few possible c-values. There are many correct answers.
If c 2:
3x 2 x 2 3x 2x 1
If c 4:
3x 2 x 4 3x 4x 1
If c 10: 3x 2 x 10 3x 5x 2
200. For 2x2 9x c to be factorable, the factors of 2c must add up to 9. There are many possibilities.
Possible c-values
2c
Factors of 2c that add up to 9
4
8
18 8 and 1 8 9
7
14
27 14 and 2 7 9
9
18
36 18 and 3 6 9
10
20
45 20 and 4 5 9
11
22
211 22 and 2 11 9
18
36
312 36 and 3 12 9
These are a few possible c-values.
2x2 9x 4 2x 1x 4
2x2 9x 7 2x 7x 1
2x2 9x 9 2x 3x 3
2x2 9x 10 2x 5x 2
2x2 9x 11 2x 11x 1
2x2 9x 18 2x 3x 6
Appendix A.3
201. (a) Profit Revenue Cost
Polynomials and Factoring
202. (a) Profit Revenue Cost
Profit 95x 73x 25,000
P 36x 460 12x
95x 73x 25,000 22x 25,000
36x 460 12x
(b) For x 5000:
24x 460
Profit 225000 25,000
(b) When x 42, P 2442 460 $548.
110,000 25,000 $85,000
203. (a) 5001 r2 500r 12 500r 2 2r 1
500r 2 1000r 500
(b)
r
212%
3%
4%
412%
5%
5001 r2
$525.31
$530.45
$540.80
$546.01
$551.25
(c) As r increases, the amount increases.
204. (a) 12001 r3 12001 3r 3r 2 r 3
1200r 3 3r 2 3r 1
1200r 3 3600r 2 3600r 1200
(b)
r
2%
3%
312%
4%
412%
12001 r3
$1273.45
$1311.27
$1330.46
$1349.84
$1369.40
(c) Amount increases with increasing r.
205. (a) V l w
h 26 2x18 2xx
(b)
x cm
1
2
3
384
616
720
x (cm)
3
5
7
Volume cm3
486
375
84
213 x29 xx
V
cm3
4x1x 131x 9
4xx 13x 9
4x3 88x 2 468x
206. (a) Volume length width height
1245 3x15 2xx
1
2 45
1
2 675x
3x15x 2x2
90x2 45x2 6x3
126x3 135x2 675x
(b) When x 3: V 12633 13532 6753 126
27 135 9 2025
1
2 162
1215 2025 12972
486 cubic centimeters
When x 5:
V
1
3
2 65
1355 6755 126
2
125 135 25 3375
1
2 750
3375 3375 12750
375 cubic centimeters
When x 7:
V
1
3
2 67
1357 6757 126
2
343 135 49 4725
1
2 2058
6615 4725 12168
84 cubic centimeters
1161
1162
Appendix A
Review of Fundamental Concepts of Algebra
208. A 18 2x14 x 252 18x 28x 2x2
207. Area length width
2x 1422
2x2 46x 252
2x22 1422
44x 308
209. (a) Area of shaded region Area of outer rectangle Area of inner rectangle
A 2x2x 6 xx 4
4x2 12x x2 4x
3x2 8x
(b) Area of shaded region Area of outer triangle Area of inner triangle
A 12 9x12x 12 6x8x
54x2 24x2
30x2
210. (a) T R B 1.1x 0.0475x2 0.001x 0.23
(b)
0.0475x2 1.099x 0.23
x mi
hr
30
40
55
T feet
75.95
120.19
204.36
(c) Stopping distance increases at an accelerating rate
as speed increases.
212. x2 4x 3 x 3x 1
211. 3x2 7x 2 3x 1x 2
x
x
x
x
x
1
x
1
1
x
x
x
x
1
x
1
x
x
1
x
x
1
x
x
1
1
x
x
x
1
1
1
x
x
x
1
x
1
x
1
1
1
1
1
1
213. 2x2 7x 3 2x 1x 3
x
214. x2 3x 2 x 2x 1
x
x
x
x
1
1
x
x
x
1
x
1
x
1
x
1
x
x
1
x
1
1
x
x
x
1
x
1
1
x
1
x
1
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
215. A r 22 r 2
216. Area 2r2 r2
217. A 818 4x2
r 22 r 2
4r2 r2
436 x2
r 2 4r 4 r 2
r24 46 x6 x
4r 4
4 r 1
Appendix A.3
1
5
1
218. Area 2x 34 x 3 254
Polynomials and Factoring
219. (a) V R 2h r 2h
58x2 6x 9 5816
hR 2 r 2
58x2 6x 9 16
hR rR r
5 2
8 x
5
8 x
1163
(b) The average radius is R r
2. The thickness of
the tank is R r.
6x 7
7x 1
V hR rR r 2
R 2 rR rh
2 average radiusthicknessh
220. kQx kx2 kxQ x
221. False. 4x2 13x 1 12x 3 4x2 3x 1
222. False.
223. True. a2 b2 a ba b
(4x 3) (4x 6) 4x 3 4x 6
369
224. False. A perfect square trinomial can be factored as the
binomial sum squared.
225. Since x mx n x mn, the degree of the product is m n.
226. If the degree of one polynomial is m and the degree of the second polynomial is n
(and n > m), the degree of the sum of the polynomials is n.
227. The unknown polynomial may be found by adding x3 3x2 2x 1 and 5x2 8:
x 3 3x2 2x 1 5x2 8 x 3 3x2 5x2 2x 1 8
x 3 8x2 2x 7
228. x y2 x2 y 2
Let x 3 and y 4.
3 42 72 49
32 42 9 16 25
Not Equal
If either x or y is zero, then x y2 would equal x2 y2.
229. x2n y 2n x n2 y n2
x n y nx n y n
This is not completely factored unless n 1.
For n 2: x 2 y 2x 2 y 2 x 2 y 2x yx y
For n 3: x3 y3x3 y3 x yx 2 xy y 2x yx 2 xy y 2
For n 4: x 4 y 4x 4 y 4 x 4 y 4x2 y 2x yx y
230. x 3n y 3n x n 3 ( y n3 x n y nx 2n x n y n y 2n
Depending on the value of n, this may factor further.
232. Answers will vary. A possible answer: A polynomial
is in factored form when written as a product of polynomials of lesser degree than the given polynomial.
231. x 3n y 2n x n3 y n2 x 3n y 2n is completely
factored. For integer values of n greater than 4, the
factorizations become more complicated.
233. Answers will vary. Some examples:
x 2 3; x 2 x 1; x 2 16
1164
Appendix A
Appendix A.4
Review of Fundamental Concepts of Algebra
Rational Expressions
■
You should be able to find the domain of a rational expression.
■
You should know that a rational expression is the quotient of two polynomials.
■
You should be able to simplify rational expressions by reducing them to lowest terms. This may involve factoring both
the numerator and the denominator.
■
You should be able to add, subtract, multiply, and divide rational expressions.
■
You should be able to simplify complex fractions.
■
You should be able to simplify expressions with negative or fraction exponents.
Vocabulary Check
1. domain
2. rational expression
3. complex
4. smaller
5. equivalent
6. difference quotient
1. The domain of the polynomial 3x2 4x 7 is the set of
all real numbers.
2. The domain of the polynomial 2x2 5x 2 is the set of
all real numbers.
3. The domain of the polynomial 4x3 3, x ≥ 0 is the set
of non-negative real numbers, since the polynomial is
restricted to that set.
4. The domain of the polynomial 6x2 9, x > 0 is the set
of all positive real numbers because the polynomial is
restricted to that set.
5. The domain of 1x 2 is the set of all real numbers x
such that x 2.
6. The domain of x 12x 1 is the set of all real
numbers such that x 12.
7. The domain of x 1 is the set of all real numbers x
such that x ≥ 1.
8. The domain of 6 x is the set of all real numbers x
such that x ≤ 6.
9.
5
53x
53x
,
2x 2x3x
6x2
x0
10.
The missing factor is 3x, x 0.
3 3x 1
4 4x 1
11.
15x2 5x3x 3x
,
10x
5x2
2
x0
The missing factor is x 1,
where x 1.
12.
6y2 3
3
18y 2
2
5
3
60y
6y 10y 10y3
13.
3xy
x3y
3y
, x0
xy x xy 1 y 1
14.
2x2y
2x2y
2x2
xy y yx 1 x 1
15.
4y 8y2 4y2y 1
10y 5
52y 1
16.
9x2 9x 9xx 1
2x 2
2x 1
17.
x5
x5
10 2x 2x 5
18.
4y
1
, y
5
2
12 4x 43 x
4, x 3
x3
x3
19.
1
, x5
2
9x
, x 1
2
y2 16 y 4 y 4
y4
y4
y 4, y 4
20.
x2 25 x 5x 5
5x
1x 5
x 5, x 5
Appendix A.4
21.
x3 5x2 6x xx 2x 3 xx 3
,
x2 4
x 2x 2
x2
23.
y2 7y 12 y 3 y 4 y 4
,
y2 3y 18 y 6 y 3 y 6
25.
2 x 2x2 x3 2 x x22 x
x2 4
x 2x 2
z2
y3
22.
24.
26.
x2 8x 20
x 10x 2 x 2
,
x2 11x 10 x 10x 1 x 1
x 10
x2 7x 6
x 6x 1
11x 10 x 10x 1
x3
x2 9
x2 9
2
2
x 9x 9 x x 1 9x 1
2 x1 x2
x 2x 2
x2 9
x2 9x 1
x 2x2 1
x 2x 2
1
, x ±3
x1
x2 1
, x2
x2
z3 8
z 2z2 2z 4
z2
2z 4
z2 2z 4
28.
y3 2y2 3y
y y 3 y 1
y3 1
y 1 y2 y 1
29.
1165
x2
27.
x 2
Rational Expressions
x
0
1
2
3
4
5
6
x2 2x 3
x3
1
2
3
Undef.
5
6
7
x1
1
2
3
4
5
6
7
30.
The expressions are equivalent except at x 3.
y y 3
, y 1
y2 y 1
x
0
1
2
3
4
5
6
x3
x2 x 6
1
2
1
3
1
4
Undef.
1
6
1
7
1
8
1
x2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
The expressions are equivalent except at x 3.
31.
5x3
5x3
3
4 2x 2
32.
2x3
There are no common factors so this expression cannot be
simplified. In this case factors of terms were incorrectly
cancelled.
33.
r2
r2 , r0
2 2r
4r2
4
x2
xx2 25
x3 25x
2x 15
x 5x 3
The expression cannot be simplified.
34. Area of shaded portion:
x5
2
2
x 5 2
4
Area of total figure: 2x 3x 5
x 52
x 5
x5
4
4
Ratio:
2x 3x 5 2x 3 42x 3
35.
5
x1
x1
1
25x 2 5x 2,
x1
36.
x 13
x33 x
xx 3
x 13
3
5
x x 31
xx 3
5
x 13
x 13
, x3
5x2
5x2
1166
37.
Appendix A
r
r1
Review of Fundamental Concepts of Algebra
r 2 1 rr 1r 1 r 1
, r 1, r 0
r2
r 2r 1
r
38.
4y 16
5y 15
2y 6 4 y 4
4y
5 y 3
t 3t 2t 3
39.
t2 t 6
t2 6t 9
40.
x 2 xy 2y 2
x3 x2y
41.
x2 36 x3 6x2 x2 36
2
x
x x
x
43.
x
x 2 3xy 2y 2 t3
t 2
x2 x
x3 6x2
x 6x 6 xx 1
x2x 6
x
x 6x 1
, x6
x2
5
5
6x 3
x3
x 3
x3
42.
x2 14x 49 3x 21 x 7x 7
x2 49
x7
x 7x 7
1
, x ±7
3
44.
2x 1 1 x 2x 1 1 x
x
x3
x3
x3
x3
46.
3
3
5x 1
5
x1
x1
x1
6x 3 5
x3
3 5x 1
x1
6x 18 5
x3
3 5x 5
x1
6x 13
x3
8 5x
x1
5
3
5
2
3
x2 2x x2 x2
x2
48.
2x
5
2x
51
x 5 5 x x 5 15 x
49.
50.
8
8
, y 3, 4
5
5
x 2yx y
x
xy
x 2yx y xx y)2, x 2y
x 2x y
5
x
5x x5
x1 x1 x1 x1
45. 6 47.
t3
t2 4 t 32t 2t 2 t 3t 2,
2 y 3
1 y 4
5
2x 5
2x
x5 x5
x5
1
x
1
x
x2 x 2 x2 5x 6 x 2x 1 x 2x 3
x 3 xx 1
x 3 x2 x
x 1x 2x 3 x 1x 2x 3
x2 3
x2 3
x 1x 2x 3
x 1x 2x 3
2
10
2
10
x2 x 2 x2 2x 8 x 2x 1 x 4x 2
2x 4
10x 1
x 2x 1x 4 x 2x 1x 4
12x 18
62x 3
2x 8 10x 10
x 2x 1x 4 x 2x 1x 4 x 2x 1x 4
x7
3x 7
Appendix A.4
Rational Expressions
1167
2
1
x2 1
1
2x
1
51. 2
x
x 1 x3 x
xx2 1
xx2 1 xx2 1
x2 1 2x 1 x2 2x xx 2
xx2 1
xx2 1
xx2 1
52.
53.
2x
x2
,
x2 1 x2 1
x0
2
1
2
2
1
2
x 1 x 1 x2 1 x 1 x 1 x 1x 1
2x 1
2x 1
1
x 1x 1 x 1x 1 x 1x 1
2x 2 2x 2 1
4x 1
x 1x 1
x 1x 1
x 4 3x 8 x 4 3x 8
x2
x2
x2
x 4 3x 8 2x 12 2x 6
x2
x2
x2
The error was incorrect subtraction in the numerator.
54.
6x
x2
8
x6 x
x 22
8
2
2
2
2
xx 2
x
x x 2 x x 2 x x 2 x2x 2
6x x2 x2 4x 4 8 10x 12 25x 6
2
2
x2x 2
x x 2
x x 2
The error was an incorrect expansion of x 22 in the numerator.
x4
x4
x 4
1
1
2
2
56.
x
4
x
16
x 16
4
x
4x
4x
4x
2x 1 2x 22
55.
x 2
x 1 2
x2
2
1
,
2
1
x2
2
2
x 1
x
x4
1
x2 16
x4
1
x 4x 4 x 4 ,
2
3
2
3
xx 1, x 1, 0
x 21 x x 21 x
x
4x
4x
4x
2
2
x x 1 x
58.
x 1
x x2
59.
x2
x 1 x
57.
x
x 1 x 1
x
2x 2x 1
, x>0
2x
2x
1
x
x
x 12
x 1x 1
x
x 1x 1
x
x1
, x0
x1
x 0, 4
1168
Appendix A
Review of Fundamental Concepts of Algebra
t2
t2
t 2 1
t 2 1
2
t 1
t 1
60.
t2
t2
2
t 2 1
t 2 1
t 2 t 2 1
1
2 2
t 2t 2 1
t t 1
61. x5 2x2 x2x7 2 x7 2
x2
x8 5
x3
62. x 5 5x3 x3x8 5 63. x2x2 15 x 2 14 x 2 15 x 2 x 2 1
1
x 2 15
64. 2xx 53 4x2x 54 2xx 54 x 5 2x
2xx 54x 5 2x
2xx 5
x 54
65. 2x2x 112 5x 112 x 112 2x2x 11 5
2x3 2x 2 5
x 112
66. 4x 32x 132 2x2x 112 2x 1124x 32x 12 2x
67.
68.
3x13 x23 3x13 x23
3x23
3x23
1 x3
2x1 x4
x2 12
x23
x23 x2 12
4x32x 12 2x
2x 112
3x1 x0 3x 1
, x0
3x0
3
x3
2x1 x212
1 x212
x4
x3
2x1 x2121 x212
x3 2x1 x2
2
12
2
12
1 x 1 x 1 x212
4
x
x4
69.
x 1 h 1x x 1 h 1x h
h
x3 2x 2x3
1 x212
x2 2
xx2 2
3
2 12
x 1 x x 1 x212
1
x3 2x
1
x4 1 x212 x4
4
xx h
xx h
x 1 h
2
70.
h
1
x2
x 1 h
2
1
x2
h
x2x h2
x2x h2
x h2
hx2x h2
x x h
hxx h
h
hxx h
x2 x2 2xh h2
hx2x h2
h2x h
hx2x h2
1
,
xx h
h0
x2
2x h
, h0
x2x h2
Appendix A.4
71.
x h1 4 x 1 4 x h1 4 x 1 4
h
h
x 4 x h 4
hx 4x h 4
h
hx 4x h 4
xh
72.
73.
1
, h0
x 4x h 4
x hx 1
x
h
xx h 1
x 2 x
2
h1
x hx 1
x h 1x 1 x h 1x 1 h
h
1
1
, h0
x h 1x 1 h x h 1x 1
x2 x hx h x2 xh x
x h 1x 1
x 2 x
2
h
1
x 2 x
74.
x 2 x
z 3 z
3
z 3 z
3
z 3 z
z 3 z
x 2 x
2x 2 x z 3 z
3z 3 z 2
2x 2 x 3
3z 3 z 1
x 2 x
h
x h 2 x 2
x h 1 x 1
h
x h 1 x 1
x h 1 x 1
hx h 1 x 1 h
hx h 1 x 1 1
x h 1 x 1
h0
,
x h 2 x 2
xh2x2
hx h 2 x 2 h
hx h 2 x 2 x h 2 x 2 Shaded area
xx2
x2
Total area
x2x 1
2x 1
1
z 3 z
x h 1 x 1
77. Probability 1
x h 1 x 1
h
xx h 1
76.
x 4x h 4
x 4x h 4
x h 1 x 1 x h 1x 1 x h 1x 1
75.
Rational Expressions
2
1
x h 2 x 2
x
2 22x 1
,
h0
1169
1170
Appendix A
78. Probability 79. (a)
1
2
4
x x 2x x 4
Shaded area
Total area
1
4
x 4 x 2 x 2
2
x
4x 22x 4
x
4
x 4x 2 1 x
8x 22
x
4
2x 22
x
x 4x 2 1 4
x
1
x 4x 2 1 4
x
8x 22
8x 2
x 4x 2x 4 x 42
1
minute
16
(b) x
(c)
Review of Fundamental Concepts of Algebra
80.
t
t
5t 3t
8t
3 5
15
15
161 16x minutes
60 15
minutes
16
4
24[48(400) 16,000]
48
81. (a) r 0.0909 9.09%
48400
16,000 12
24(NM P)
N
24NM P
(b) r NM
N
P
12
r
288NM P
12
12P NM N12P NM
28848400 16,000
0.0909 9.09%
481216,000 48400
24NM P
2460 525 28,000
231,500 28,000
N
60
5
82. (a) r NM
60 525
31,500
P
28,000 28,000 12
12
12
23500
5
2700
1400
4.57%
28,000 2625 30,625 30,625
(b) r 24NMN P
12P NM
12
24NM P
N
12
288NM P
12P NM N12P NM
28831,500 28,000
2883500
1,008,000
28860 525 28,000
4.57%
6012 28,000 60 525 60336,000 31,500 60367,500 22,050,000
Appendix A.4
83. T 10
(a)
4tt
2
2
16t 75
4t 10
Rational Expressions
1171
t
0
2
4
6
8
10
12
14
16
18
20
22
T
75
55.9
48.3
45
43.3
42.3°
41.7
41.3
41.1
40.9
40.7
40.6
(b) T is approaching 40.
84. (a)
Year
2002
2003
2004
2005
2006
2007
Banking
21.9
27.0
31.4
35.6
40.2
45.6
Paying Bills
13.9
17.8
21.5
25.0
28.1
31.0
(b) The estimates and actual values are quite close for banking. For paying bills, they are close for 2002–2004,
but the model generally tends to overestimate the number.
4.39t 5.5
0.002t 2 0.01t 1.0
4.39t 5.5
(c)
0.728t 2 23.81t 0.3
0.002t 2 0.01t 1.0
0.049t 2 0.61t 1.0
(d)
0.049t 2 0.61t 1.0
0.728t 2 23.81t 0.3
0.21511t 3 2.6779t 2 4.39t 0.2695t2 3.355t 5.5
0.001456t 4 0.04762t 3 0.0006t 2 72.8t 3 0.2381t 2 0.003t 0.728t 2 23.81t 0.3
0.21511t3 2.4084t 2 7.745t 5.5
0.001456t 4 72.75238t 3 0.4905t2 23.807t 0.3
Year
2002
2003
2004
2005
2006
2007
Paying BillsBanking
0.63
0.66
0.69
0.70
0.70
0.68
The ratio is approximately 23, and it appears to peak in 2005–2006.
85. False. In order for the simplified expression to be equivalent to the original expression, the domain of
the simplified expression needs to be restricted. If n is even, x ± 1. If n is odd, x 1.
86. False. The two expressions are equivalent for all values of x such that x 1.
87. Completely factor the numerator and the denominator. A rational expression is in simplest form if there
are no common factors in the numerator and the denominator other than ± 1.
1172
Appendix A
Appendix A.5
Review of Fundamental Concepts of Algebra
Solving Equations
■
You should know how to solve linear equations.
ax b 0
■
An identity is an equation whose solution consists of every real number in its domain.
■
To solve an equation you can:
(a) Add or subtract the same quantity from both sides.
(b) Multiply or divide both sides by the same nonzero quantity.
■
To solve an equation that can be simplified to a linear equation:
(a) Remove all symbols of grouping and all fractions.
(b) Combine like terms.
(c) Solve by algebra.
(d) Check the answer.
■
A “solution” that does not satisfy the original equation is called an extraneous solution.
■
You should be able to solve a quadratic equation by factoring, if possible.
■
You should be able to solve a quadratic equation of the form u2 d by extracting square roots.
■
You should be able to solve a quadratic equation by completing the square.
■
You should know and be able to use the Quadratic Formula: For ax2 bx c 0, a 0,
x
b ± b2 4ac
.
2a
■
You should be able to solve polynomials of higher degree by factoring.
■
For equations involving radicals or fractional powers, raise both sides to the same power.
■
For equations with fractions, multiply both sides by the least common denominator to clear the fractions.
■
For equations involving absolute value, remember that the expression inside the absolute value can be positive or negative.
Vocabulary Check
1. equation
2. solve
3. identities; conditional
4. ax b 0
5. extraneous
6. quadratic equation
7. factoring; extracting square roots; completing the square; Quadratic Formula
1. 2x 1 2x 2 is an identity by the Distributive
Property. It is true for all real values of x.
2. 3x 2 5x 4 is conditional. There are real
values of x for which the equation is not true (for
example, x 0).
3. 6x 3 5 2x 10 is conditional. There are
real values of x for which the equation is not true.
4. 3x 2 5 3x 1 is an identity by simplification.
It is true for all real values of x.
3x 2 5 3x 6 5 3x 1
5. 4x 1 2x 4x 4 2x 2x 4 2x 2
This is an identity by simplification. It is true for all real
values of x.
6. 7x 3 4x 37 x is an identity by
simplification. It is true for all real values of x.
7x 3 4x 7x 21 4x
21 3x 37 x
Appendix A.5
7. x 42 11 x2 8x 16 11 x2 8x 5
2
1
4x
is conditional. There are real values
x1 x1
of x for which the equation is not true.
9. 3 11.
x 11 15
x 11 11 15 11
10.
5 3
24 is conditional. There are real values of x for
x
x
which the equation is not true (for example, x 1).
7 x 19
12.
7 2x 25
13.
7 x x 19 x
x4
1173
8. x2 23x 2 x2 6x 4 is an identity by
simplification. It is true for all real values of x.
Thus, 8x 5 x 4 11 is an identity by
simplification. It is true for all real values of x.
x2
Solving Equations
7 7 2x 25 7
7 19 x
2x 18
7 19 19 x 19
18
2x
2
2
12 x
x 9
14.
7x 2 23
8x 5 3x 20
15.
7x 3 3 3x 3x 17 3 3x
8x 3x 5 3x 3x 20
7x 2 2 23 2
7x 3 3x 17
16.
7x 21
5x 5 20
4x 20
7x 21
7
7
5x 5 5 20 5
x 5
5x 25
x3
5x 25
5
5
x5
17. 2x 5 7 3x 2
3x 3 51 x 1
18.
2x 10 7 3x 6
19. x 32x 3 8 5x
3x 9 5 5x 1
x 6x 9 8 5x
2x 3 3x 6
3x 9 4 5x
5x 9 8 5x
2x 3x 3 3x 3x 6
3x 9 5x 9 4 5x 5x 9
5x 5x 9 8 5x 5x
8x 5
9 8
x 3 6
x 3 3 6 3
x
x 9
58
No solution
x9
20. 9x 10 5x 22x 5
1
5x 1
x
4
2
2
21.
9x 10 5x 4x 10
4
9x 10 9x 10
The solution is the set of all real
numbers.
4
5x4 21 4x 21
5x4 412 4x 412
5x 2 4x 2
x 4
x
x
3x
3
5 2
10
22.
10
5x 2x 103 103x 2x 5x 30 3x
6x 30
x 5
1174
Appendix A
Review of Fundamental Concepts of Algebra
3
1
z 5 z 24 0
2
4
23.
3x 1
x 2 10
2
4
24.
32 z 5 4z 24 40
1
4
4
2 4 x 2 410
4
32z 5 414z 24 40
4
3x
1
2 44 x 2 410
3x
1
6z 5 z 24 0
6x x 2 40
6z 30 z 24 0
7x 2 40
5z 6
7x 42
6
z
5
x6
25. 0.25x 0.7510 x 3
26. 0.60x 0.40100 x 50
0.25x 7.5 0.75x 3
0.60x 40 0.40x 50
0.50x 7.5 3
0.20x 10
0.50x 4.5
x 50
x9
27. x 8 2x 2 x
28. 8x 2 32x 1 2x 5
x 8 2x 4 x
8x 16 6x 3 2x 10
x8x4
2x 13 2x 10
8 4
13 10
Contradiction; no solution
100 4x 5x 6
6
3
4
29.
12
31.
Contradiction; no solution
17 y 32 y
100
y
y
30.
100 3 4x 125x 4 6 126
y
17 y
32 y
y
100 y
y
y
4100 4x 35x 6 72
17 y 32 y 100y
400 16x 15x 18 72
49 2y 100y
31x 310
49 98y
x 10
1
y
2
5x 4 2
5x 4 3
32.
10x 3 1
5x 6
2
35x 4 25x 4
210x 3 15x 6
15x 12 10x 8
20x 6 5x 6
5x 20
15x 0
x4
x0
Appendix A.5
33. 10 5
13
4
x
x
34.
10x 13 4x 5
x
x
15
6
4 3
x
x
x3
3z 6 2z 4 2
z0
9 7x
1
2
0
x
x5
Multiply both sides by xx 5.
37.
x
4
20
x4 x4
1x 5 2x 0
x4
20
x4
3x 5 0
120
3x 5
x
38.
30
5
3
Contradiction; no solution
7
8x
4
2x 1
2x 1
Multiply both sides by 2x 12x 1.
72x 1 8x2x 1 42x 12x 1
14x 7 16x2 8x 16x2 4
6x 11
x
39.
11
6
2
1
2
x 4x 2 x 4 x 2
Multiply both sides by x 4x 2.
2 1x 2 2x 4
2 x 2 2x 8
2 3x 10
12 3x
4x
A check reveals that x 4 is an extraneous solution—it makes the denominator zero. There is no real solution.
4
6
15
x 1 3x 1 3x 1
40.
x 13x 1
Multiply both sides by x 13x 1.
4
6
15
x 13x 1
x 13x 1
x1
3x 1
3x 1
43x 1 6x 1 15x 1
12 x 4 6x 6 15x 15
18x 2 15x 15
3x 13
x
13
3
2
z 2
z2
9
x
7
36.
1175
2
z2
3z 2 2 9
7
x
6x 18
32
35.
15 6
7
x
x
10x 13 4x 5
Solving Equations
1176
Appendix A
Review of Fundamental Concepts of Algebra
41.
1
1
10
x 3 x 3 x2 9
1
1
10
x 3 x 3 x 3x 3
Multiply both sides by x 3x 3.
1x 3 1x 3 10
2x 10
x5
42.
3
4
1
2
x2 x3 x x6
3
4
1
x 2 x 3 x 3x 2
Multiply both sides by x 3x 2.
x 3 3x 2 4
x 3 3x 6 4
4x 3 4
4x 7
x
43.
7
4
3
4
1
x2 3x
x
x3
3
4
1
xx 3
x
x3
Multiply both sides by xx 3.
3 4x 3 x
3 4x 12 x
3x 9
x3
A check reveals that x 3 is an extraneous solution since it makes the denominator zero, so there is no solution.
44.
6
2
3x 5
x
x3
xx 3
Multiply both sides by xx 3.
Check:
6x 3 2x 3x 5
2 6x 18 2x 3x 15
x 3
x 22 5 x 32
x2 4x 4 5 x2 6x 9
4x 9 6x 9
2x 0
x0
2
6
0 30
Division by zero is undefined. Thus, x 3 is not a
solution, and the original equation has no solution.
4x 18 3x 15
45.
2
33 5
6
3 3 3 33 3
46.
x 12 2x 2 x 1x 2
x2 2x 1 2x 4 x2 x 2
5x 1
1
x5
Appendix A.5
47.
x 22 x2 4x 1
Solving Equations
1177
2x 12 4x2 x 1
48.
x2 4x 4 x2 4x 4
4x2 4x 1 4x2 4x 4
44
14
The equation is an identity; every real number is a solution.
49. 2x2 3 8x
This is a contradiction. Thus, the equation has no solution.
50. x2 16x
General form: 2x2 8x 3 0
51.
x 32 3
x2 6x 9 3
General form: x2 16x 0
General form: x2 6x 6 0
13 3x 72 0
1
53. 53x2 10 18x
13 3x2 14x 49 0
3x2 10 90x
52.
13 3x2 42x 147 0
55.
4x2 2x 1 0
General form:
14x2 2x 1 10
3x2 42x 134 0
General form: 4x2 2x 1 0
6x2 3x 0
3x 0
x0
x
12
3x 1 0 ⇒ x 59. x2 10x 25 0
x 9x 1 0
x 52 0
x90 ⇒ x9
x50
x4
60.
x 2
or
4x2 12x 9 0
2x 3 0 ⇒ x 23
or 1 2x 0
x
1
2
x2 8x 12
2x2 19x 33 0
x2 4x 12 0
2x 3x 11 0
x 6x 2 0
x60
2x 3 0 ⇒ x 23
65.
x2 4x 12
63.
x 6
x 11 0 ⇒ x 11
64.
or x 2 0
2x 32x 3 0
2x2 19x 33
62.
3 x1 2x 0
or
x40
1
3
x 5
3 5x 2x2 0
x3
x2 2x 8 0
x 4x 2 0
3x 1 0 ⇒ x 31
x2 10x 9 0
3x0
57.
3x 13x 1 0
or 2x 1 0
or
9x2 1 0
56.
x10 ⇒ x1
61.
x2 2x 5x2 1
General form: 3x2 90x 10 0
3x2x 1 0
58.
xx 2 5x2 1
54.
3 2
4x
8x 20 0
66.
1 2
8x
or x 2 0
or
x2
x 16 0
x2 8x 12 0
434x2 8x 20 40
x2 8x 128 0
1x2 8x 12 10
3x2 32x 80 0
x 16x 8 0
3x 20x 4 0
x 16 0 ⇒ x 16
x2 8x 12 0
x 6x 2 0
x60 ⇒ x6
x20 ⇒ x2
3x 20 0
or x 4 0
x 20
3
or
x 4
x 8 0 ⇒ x 8
1178
Appendix A
Review of Fundamental Concepts of Algebra
x a2 b2 0
69. x2 49
x a2 0
x a bx a b 0
x ±7
xa0
x a bx a b 0
67. x2 2ax a2 0
68.
x a
x a b 0 ⇒ x a b
x a b 0 ⇒ x a b
70. x2 169
71. x2 11
72. x2 32
x ± 11
x ± 169 ± 13
73. 3x2 81
x ± 32 ± 42
75. x 122 16
74. 9x2 36
x2 4
x2 27
x 12 ± 4
x ± 4 ± 2
x ± 33
x 12 ± 4
x 16
77. x 22 14
76. x 132 25
78. x 52 30
x 2 ± 14
x 13 ± 25
x 5 ± 30
x 5 ± 30
x 2 ± 14
x 13 ± 5
or x 8
x 13 ± 5 8, 18
80. 4x 72 44
79. 2x 12 18
4x 7 ± 44
2x 1 ± 18
2x 1 ± 32
x
4x 7 ± 211
1 ± 32
2
x
81. x 72 x 32
82. x 52 x 42
x 5 ± x 4
x 7 ± x 3
x7x3
7 3
7 11
7 ± 211
±
4
4
2
x 5 x 4 or
or x 7 x 3
54
2x 4
or
x 5 x 4
or x 5 x 4
2x 9
x2
x 92
The only solution to the equation is x 2.
The only solution to the equation is x 92.
83. x2 4x 32 0
84.
85. x 2 12x 25 0
x2 2x 3 0
x2 4x 32
x2 2x 3
x 2 12x 25
x2 4x 22 32 22
x2 2x 12 3 12
x 2 12x 62 25 6 2
x 22 36
x 12 4
x 62 11
x 2 ±6
x 1 ± 4
x1±2
x 2 ± 6
x4
or
x 8
x3
or x 1
x 6 ± 11
x 6 ± 11
Appendix A.5
86. x2 8x 14 0
87.
9x2 18x 3
x2 8x 14
x2
8x 42
x2 2x 14 16
x 4 2
2
x 4 ± 2
x 12 14
4
x2 x 3
9
1
3
32
2
x1 ± 3
8 4x x2 0
90. x 2 x 1 0
x2 4x 8 0
x2 x 1 0
x2 4x 8 0
x2
x2 x 4x 8
x 22 12
2
x 3 2
x 3 2
2
x
6
14 4
9
9
18
9
2
2
± 2
3
x
3
91.
2
± 2
3
2x 2 5x 8 0
2x 2 5x 8
1
1
1 4
4
x 21
x2 4x 22 8 22
2
2
x1±
89.
4
2
x2 x 3
3
2
3
x1 ±
3
4
5
x2 x 4
2
2
x 45
2
5
5
x2 x 2
4
No real solution
x 2 ± 12
x 2 ± 23
x
4
93. 2x2 x 1 0
b ± b2 4ac
2a
2
89
5
±
4
4
x
4x2 4x 99 0
54
89
16
x
92.
1179
9x2 12x 14
88.
1
x2 2x 12 12
3
x 4 ± 2
Solving Equations
5 89
±
4
4
5 ± 89
4
94. 2x2 x 1 0
b ± b2 4ac
2a
x2 x 99
4
2
2
99 1
4
4
1 ± 12 421
22
1 ± 12 421
22
x 2 2
100
4
1 ± 3 1
, 1
4
2
1 ± 1 8
4
x 2 2
1±3
1
1, 4
2
x2 x 1
1
1
x
25
1
± 25
2
x
11 9
1
± 5 ,
2
2
2
x
x
1180
Appendix A
Review of Fundamental Concepts of Algebra
95. 16x2 8x 3 0
x
96. 25x2 20x 3 0
b ± b2 4ac
2a
8 ±
x
4163
216
82
8 ± 16 1 3
,
32
4 4
x
b ± b2 4ac
2a
10 ± 10
21
b ± b2 4ac
2a
x2 2x 2 0
x
122
24
10 ± 100 88
2
x
20 ± 20 4253
225
2
20 ± 400 300
50
20 ± 10 3 1
,
50
5 5
99. x2 14x 44 0
98. x2 10x 22 0
2 2x x2 0
97.
2 ± 22 412
21
2 ± 23
1 ± 3
2
6x 4 x2
100.
x2 6x 4 0
b ± b2 4ac
2a
14 ±
b ± b2 4ac
2a
x
14 4144)
2
21
14 ± 25
7 ± 5
2
10 ± 23
5 ± 3
2
b ± b2 4ac
2a
6 ± 62 414
21
6 ± 36 16
2
6 ± 213
2
3 ± 13
101. x2 8x 4 0
x
b ± b2 4ac
2a
8 ± 82 414
21
8 ± 45
2
4 ± 25
16x2 22 40x
104.
8x2 20x 11 0
x
b ± b2 4ac
2a
102. 4x2 4x 4 0
x2 x 1 0
x
b ± b2 4ac
2a
12x 9x2 3
103.
9x2 12x 3 0
x
b ± b2 4ac
2a
1 ± 12 411
21
12 ± 122 493
29
1 ± 1 4
2
12 ± 67 2 7
±
18
3
3
1 5
±
2
2
105. 9x2 24x 16 0
x
b ± b2 4ac
2a
106. 36x2 24x 7 0
x
b ± b2 4ac
2a
20 ± 202 4811
28
24 ± 242 4916
29
24 ± 242 4367
236
20 ± 400 352
16
24 ± 0
18
24 ± 576 1008
72
24 ± 14411
72
5 3
±
4
4
4
3
1 11
±
3
6
Appendix A.5
108. 16x2 40x 5 0
4x2 4x 7
107.
4x2 4x 7 0
x
x
b ± b2 4ac
2a
447
24
4 ±
42
4 ± 82
1
± 2
8
2
110. 3x x2 1 0
b ± b2 4ac
2a
8t 5 2t2
t
3 13
3 ± 13
±
2
2
2
b ±
4ac
2a
12 ± 122 4125
21
12 ± 211
6 ± 11
2
57 x 14
28 ± 282 4494
249
28 ± 0 2
98
7
h
8 ± 82 425
22
6
8 ± 26
2 ±
4
2
14z 36 0
b ± b2 4ac
z
2a
2
b ± b2 4ac
2a
80 ± 802 42561
225
80 ± 6400 6100
50
8 103
±
5
50
8 3
±
5
5
1 2 3
x x2
2
8
115.
4x 2 3x 16
4x 2 3x 16 0
x
b ± b 2 4ac
2a
14 ± 142 4136
21
3 ± 32 4416
24
14 ± 52
2
3 ± 265
8
7 ± 13
116.
z2 12z 36 2z
z2
b ± b2 4ac
2a
112. 25h2 80h 61 0
b ± b2 4ac
2a
z 62 2z
114.
y2 12y 25 0
y
4165
40 ± 1600 320
32
5 5
±
4
2
3 ± 32 411
21
b2
x
2t2 8t 5 0
y 52 2y
113.
40 ± 40
216
111.
x2 3x 1 0
x
49x2 28x 4 0
2
40 ± 165
32
3 265
±
8
8
8x
25 2
x 20x 196 8x
49
25 2
x 28x 196 0
49
25x2 1372x 9604 0
x
1181
28x 49x2 4
109.
b ± b2 4ac
2a
Solving Equations
b ± b2 4ac 1372 ± 13722 4259604 1372 ± 921,984 686 ± 1966
2a
225
25
50
1182
Appendix A
Review of Fundamental Concepts of Algebra
117. 5.1x2 1.7x 3.2 0
x
1.7 ± 1.72 45.13.2
25.1
x 0.976, 0.643
118. 2x2 2.50x 0.42 0
x
b ± b2 4ac
2a
2.50 ± 2.502 420.42
22
2.50 ± 9.61
4
1.400, 0.150
119. 0.067x2 0.852x 1.277 0
x
0.852 ± 0.8522 40.0671.277
20.067
x 14.071, 1.355
120. 0.005x2 0.101x 0.193 0
x
b ± b2 4ac
2a
0.101 ± 0.1012 40.0050.193
20.005
0.101 ± 0.006341
0.01
2.137, 18.063
121. 422x2 506x 347 0
x
506 ± 5062 4422347
2422
x 1.687, 0.488
123. 12.67x2 31.55x 8.09 0
x
31.55 ± 31.552 412.678.09
212.67
x 2.200, 0.290
125. x2 2x 1 0
Complete the square.
x 2x 1
122. 1100x2 326x 715 0
x
b ± b2 4ac
2a
326 ± 3262 41100715
21100
326 ± 3,252,276
0.672, 0.968
2200
124. 3.22x2 0.08x 28.651 0
x
b ± b2 4ac
2a
0.08 ± 0.082 43.2228.651
23.22
0.08 ± 369.031
2.995, 2.971
6.44
126. 11x2 33x 0
11
2
x2
x2 2x 12 1 12
x 12 2
Factor.
3x 0
xx 3 0
x0
or x 3 0
x 3
x 1 ± 2
x 1 ± 2
127. x 32 81
Extract square roots.
x 3 ±9
x39
x6
x 7 0
2
or x 3 9
or
128. x2 14x 49 0
x 12
x70
x7
Extract square roots.
Appendix A.5
129.
x2 x 11
4 0
x2 x Complete the square.
130.
x2 3x 34 0
Solving Equations
Complete the square.
x2 3x 2 34 94
3 2
11
4
1
x2 x 12 11
4 2 2
x 32 2 3
2
x 12 2 124
x 32 ± 3
x 12 ± 12
4
x 32 ± 3
x 12 ± 3
131. x 12 x2
Extract square roots.
132.
Factor.
ax bax b 0
x2 x 12
x ± x 1
ax b 0 ⇒ x For x x 1:
01
a2 x 2 b2 0
ax b 0 ⇒ x No solution
For x x 1:
b
a
b
a
2x 1
x 12
133. 3x 4 2x2 7
0
x
2x2
Quadratic Formula
134. 4x2 2x 4 2x 8
3x 11
4x2
3 ± 32 4211
22
135.
40
4x2 1 0
x 1x 1 0
3 ± 97
4
x10
or x 1 0
x 1
3 97
±
4
4
4x4 18x2 0
2x22x2 9 0
2x2 0 ⇒ x 0
32
2x2 9 0 ⇒ x ±
2
Factor.
136.
x1
20x3 125x 0
5x4x2 25 0
5x2x 52x 5 0
5x 0 ⇒ x 0
2x 5 0 ⇒ x 2x 5 0 ⇒ x 137.
x4 81 0
x2 9x 3x 3 0
x2 9 0 ⇒ No real solution
x 3 0 ⇒ x 3
2
x30 ⇒ x3
2
5
2
5
2
1183
1184
Appendix A
Review of Fundamental Concepts of Algebra
x6 64 0
138.
x3 8x3 8 0
x 2x2 2x 4x 2x2 2x 4 0
x20 ⇒ x2
x2 2x 4 0 ⇒ No real solution (by the Quadratic Formula)
x 2 0 ⇒ x 2
x2
139.
2x 4 0 ⇒ No real solution (by the Quadratic Formula)
x3 216 0
x3 63 0
x 6x2 6x 36 0
x 6 0 ⇒ x 6
x2 6x 36 0 ⇒ No real solution (by the Quadratic Formula)
140.
27x 3 512 0
3x 89x2 24x 64 0
3x 8 0 ⇒ x 83
9x2 24x 64 0 ⇒ No real solution (by the Quadratic Formula)
141. 5x3 30x2 45x 0
142. 9x4 24x3 16x2 0
5xx2 6x 9 0
x29x2 24x 16 0
5xx 32 0
x23x 42 0
5x 0 ⇒ x 0
x2 0 ⇒ x 0
x 3 0 ⇒ x 3
143.
x3 3x2 x 3 0
3x 4 0 ⇒ x 43
144.
x3 2x2 3x 6 0
x2x 3 x 3 0
x2x 2 3x 2 0
x 3x2 1 0
x 2x2 3 0
x 3x 1x 1 0
x30 ⇒ x3
x 1 0 ⇒ x 1
x10 ⇒ x1
145.
x4 x3 x 1 0
x3x 1 x 1 0
x 1x3 1 0
x 1x 1x2 x 1 0
x10 ⇒ x1
x 1 0 ⇒ x 1
x2
x 1 0 ⇒ No real solution (by the Quadratic Formula)
x 2 0 ⇒ x 2
x2 3 0 ⇒ No real solution
Appendix A.5
146.
Solving Equations
1185
x4 2x3 8x 16 0
x3x 2 8x 2 0
x3 8x 2 0
x 2x2 2x 4x 2 0
x20 ⇒ x2
x2
2x 4 0 ⇒ No real solution (by the Quadratic Formula)
x 2 0 ⇒ x 2
147.
x4 4x2 3 0
x4 5x2 36 0
148.
x2 3x2 1 0
x2 9x2 4 0
x 3 x 3 x 1x 1 0
x2 9x 2x 2 0
x 3 0 ⇒ x 3
x2 9 0 ⇒ No real solution
x 3 0 ⇒ x 3
x 2 0 ⇒ x 2
x 1 0 ⇒ x 1
x20 ⇒ x2
x10 ⇒ x1
149.
4x4 65x2 16 0
36t4 29t2 7 0
150.
4x2 1x2 16 0
36t 2 7t 2 1 0
2x 12x 1x 4x 4 0
6t 7 6t 7 t 2 1 0
2x 1 0 ⇒ x 21
2x 1 0 ⇒ x 6t 7 0 ⇒ t 1
2
1x 4 0 ⇒ x 4
6t 7 0 ⇒ t 1x 4 0 ⇒ x 4
151.
x3 8x3 1 0
x 2x2 2x 4x 1x2 x 1 0
x 2 0 ⇒ x 2
x 2x 4 0 ⇒ No real solution (by the Quadratic Formula)
2
x10 ⇒ x1
152.
x 1 0 ⇒ No real solution (by the Quadratic Formula)
x6 3x3 2 0
x3 2x3 1 0
3 2 x2 3 2x x 3 2 2x 1x2 x 1 0
3 2 0 ⇒ x 3 2
x 3 2x x2 3 2 0 ⇒ No real solution (by the Quadratic Formula)
2
x 1 0 ⇒ x 1
x x 1 0 ⇒ No real solution (by the Quadratic Formula)
2
6
7
6
t 1 0 ⇒ No real solution
2
x6 7x3 8 0
x2
7
1186
Appendix A
Review of Fundamental Concepts of Algebra
153. 2x 10 0
2x 10
2x 100
x 50
156. 5 x 3 0
5 x 3
5x9
x 4
154. 4x 3 0
155. x 10 4 0
4x 3
x 10 4
16x 9
x 10 16
9
16
x 26
x
3 2x 5 3 0
157. 3 3x 1 5 0
158. 3
3
2x 5 3
3
3x 1 5
2x 5 27
3x 1 125
2x 32
3x 124
x 16
159. 26 11x 4 x
x
160. x 31 9x 5
4 x 26 11x
31 9x 5 x
16 8x x2 26 11x
31 9x 25 10x x2
x2 3x 10 0
0 x2 x 6
x 5x 2 0
0 x 3x 2
x 5 0 ⇒ x 5
0x3 ⇒ x3
x20 ⇒ x2
0 x 2 ⇒ x 2
161. x 1 3x 1
162. x 5 x 5
x 1 3x 1
x5x5
2x 0
5 5
x0
163. x 532 8
x 53 82
3
x 5 64
x549
No solution
164. x 332 8
165. x 323 8
x 33 82
x 32 83
x 33 64
x 3 ± 83
3 64
x3
x 3 ± 512
x 3 4 1
166. x 223 9
x 22 93
x 2 ± 729
x 2 ± 27
29, 25
124
3
x 3 ± 162
167. x2 532 27
x2 53 272
3 272
x2 5 x2 5 9
x2 14
x ± 14
Appendix A.5
Solving Equations
168. x2 x 2232 27
x2 x 22 2723
x2 x 22 9
x2 x 31 0
x
1 ± 12 4131 1 ± 125 1 ± 55
21
2
2
169. 3xx 112 2x 132 0
170.
4x2x 113 6xx 143 0
x 1123x 2x 1 0
2x2xx 113 3x 143 0
x 1125x 2 0
2xx 1132x 3x 1 0
2xx 1135x 3 0
x 112 0 ⇒ x 1 0 ⇒ x 1
2x 0 ⇒ x 0
5x 2 0 ⇒ x 25, extraneous
x10 ⇒ x1
3
5x 3 0 ⇒ x 5
x
171.
3
1
x
2
2xx 2x
172.
x 2x2
3
1
4 5
x
x
3 6
4
5
x
6x 6x 6x
x
3
6
24 10x x2
2x2 6 x
x2 10x 24 0
2x2 x 6 0
x 12x 2 0
2x 3x 2 0
x 12 0 ⇒ x 12
3
2x 3 0 ⇒ x 2
x20 ⇒ x2
x20 ⇒ x2
1
1
3
x
x1
173.
174.
1
1
xx 1 xx 1
xx 13
x
x1
x 1 x 3xx 1
1
3x2
3x
0
3x2
3x 1
a 3, b 3, c 1
x
3 ± 3 431 3 ± 21
23
6
2
4
3
1
x1 x2
4x 2 3x 1 x 1x 2
4x 8 3x 3 x2 3x 2
x2 2x 3 0
x 1x 3 0
x10 ⇒ x1
x 3 0 ⇒ x 3
1187
1188
Appendix A
Review of Fundamental Concepts of Algebra
20 x
x
x
175.
4x 1 176.
20 x x2
3
x
x4x x1 x
0 x2 x 20
3
x
4x2 x 3
0 x 5x 4
4x2 x 3 0
x 5 0 ⇒ x 5
4x 3x 1 0
x40 ⇒ x4
4x 3 0 ⇒ x 3
4
x 1 0 ⇒ x 1
x
1
3
x2 4 x 2
177.
x 2x 2
x1 x1
0
3
x2
178.
x
1
x 2x 2
3x 2x 2
x2 4
x2
3x 2
x 2x 1) 3x 1 0
x x 2 3x2 12
3x2
x1
x1
3x 2
0
3
x2
x2 3x 2 3x 3 0
2x 10 0
a 3, b 2, c 10
x
x2 1 0
2 ± 22 4310
23
x 1x 1 0
x 1 0 ⇒ x 1
2 ± 124 2 ± 231 1 ± 31
6
6
3
179.
x10 ⇒ x1
2x 1 5
180.
3x 2 7
3x 2 7 ⇒ x 53
2x 1 5 ⇒ x 3
3x 2 7
2x 1 5 ⇒ x 2
3x 2 7 ⇒ x 3
181. x x2 x 3
First equation:
x x2 x 3
x2 3 0
x ± 3
Second equation:
x x2 x 3
x2 2x 3 0
x 1x 3 0
x10 ⇒ x1
x 3 0 ⇒ x 3
Only x 3 and x 3 are solutions to the original equation. x 3 and x 1 are extraneous.
Appendix A.5
Solving Equations
1189
182. x2 6x 3x 18
Second equation:
First equation:
x2 6x 3x 18
x2 6x 3x 18
x2 3x 18 0
0 x2 9x 18
x 3x 6 0
0 x 3x 6
x30 ⇒ x3
0 x 3 ⇒ x 3
x 6 0 ⇒ x 6
0 x 6 ⇒ x 6
The solutions to the original equation are x ± 3 and x 6.
183. x 1 x2 5
Second equation:
First equation:
x 1 x2 5
x 1 x2 5
x2 x 6 0
x2 x 4 0
x 3x 2 0
x30 ⇒ x3
x 2 0 ⇒ x 2
Only x 3 and x x 1 x2 5
x
1 ± 17
2
1 17
1 17
are solutions to the original equation. x 2 and x are extraneous.
2
2
184. x 10 x2 10x
First equation:
Second equation:
x 10 x 10 x2 10x
x2
10x
0 x2 11x 10
0 x2 9x 10
0 x 1x 10
0 x 10x 1
0x1 ⇒ x1
0 x 10 ⇒ x 10
0 x 10 ⇒ x 10
0 x 1 ⇒ x 1
The solutions to the original equation are x 10 and x 1. x 1 is extraneous.
185. (a) Female: y 0.432x 10.44
For y 16:
16 0.432x 10.44
26.44 0.432x
26.44
x
0.432
x 61.2 inches
—CONTINUED—
(b) Male: y 0.449x 12.15
For y 19:
19 0.449x 12.15
31.15 0.449x
69.4 x
Yes, it is likely that both bones came from the same
person because the estimated height of a male with
a 19-inch thigh bone is 69.4 inches.
1190
Appendix A
Review of Fundamental Concepts of Algebra
185. —CONTINUED—
(c)
Height x
Female Femur
Length
Male Femur
Length
60
15.48
14.79
70
19.80
19.28
80
24.12
23.77
90
28.44
28.26
100
32.76
32.75
110
37.08
37.24
The lengths of the male and female femurs are approximately equal when the lengths are 100 inches.
186. 10,000 0.32m 2500
y 0.25t 8
187.
1 0.25t 8
7500 0.32m
0.25t 7
7500
m
0.32
t 28 hours
m 23,437.5 miles
189.
188. (a)
S x2 4xh
84 x2 4x2
w
0 x2 8x 84
w + 14
0 x 14x 6
(b) ww 14 1632
x 14
(c) w2 14w 1632 0
Since x must be positive, we have x 6 inches. The
dimensions of the box are 6 inches 6 inches 2 inches.
w 48w 34 0
w 48
or w 34
Since w must be greater than zero, we have w 34
feet and the length is w 14 48 feet.
190. x2 x2 52
Pythagorean Theorem
2x2 25
x2 x
25
2
25
2
5
x
5
2
or x 6
x
52
3.54 centimeters
2
Each leg in the right triangle is approximately 3.54 centimeters.
Appendix A.5
191. Model:
Labels:
Solving Equations
1191
height2 half of side2 side2
height 10 inches, side s, half of side Equation: 102 2s 2
s
2
s
10
s2
1s
2
s2
100 s2
4
3 2
s 100
4
s2 s
400
3
4003 203 3 11.55 inches
Each side of the equilateral triangle is approximately 11.55 inches long.
dN 3 hoursr 50 mph
192.
dE 3 hoursr mph
dN dE2 24402
2
2440 miles
3(r + 50)
9r 502 9r 2 24402
18r 2 900r 5,931,100 0
r
900 ± 9002 4185,931,100 900 ± 60118,847
218
36
3r
Using the positive value for r, we have one plane moving northbound at r 50 600 miles per hour
and one plane moving eastbound at r 550 miles per hour.
193. (a) P 200 million when:
(b) For P 230:
182.45 3.189t
200
1.00 0.026t
182.45 3.189t
230
1.00 0.026t
182.45 3.189t 2001.00 0.026t
182.45 3.189t 2301.00 0.026t
182.45 3.189t 200 5.2t
182.45 3.189t 230 5.98t
2.011t 17.55
t 8.7
So the total voting-age population reached
200 million during 1998.
194. When C 2.5 we have:
2.5 0.2x 1
so
2.791t 47.55
t 17
The model predicts that the total voting-age population
will reach 230 million during 2007. This value is reasonable but the model is reaching its limit since it soon
begins to rise very fast due to its asymptotic behavior.
195. 1x 37.55 40 0.01x 1
0.01x 1 2.45
6.25 0.2x 1
0.01x 1 6.0025
5.25 0.2x
1 0.01x 5.0025
x 26.25 26,250 passengers
1 0.01x 500.25
Rounding x to the nearest whole unit yields x 500 units.
1192
Appendix A
Review of Fundamental Concepts of Algebra
197. False. x3 x 10 ⇒ 3x x2 10
196. When p $750, we have:
This is a quadratic equation. The equation cannot be
written in the form ax b 0.
750 800 0.01x 1
50 0.01x 1
2500 0.01x 1
0.01x 2499
x 249,900
So the demand is 249,900 units when the price is $750.
198. False. The product must equal zero for the Zero-Factor
property to be used.
200. False. x 0 has only one solution to check, 0.
199. False—See Example 14 on page A55.
201. Equivalent equations are derived from the substitution
principle and simplification techniques. They have the
same solution(s).
2x 3 8 and 2x 5 are equivalent equations.
202. To transform an equation into an equivalent equation,
you should first remove symbols of grouping, combine
like terms, and reduce fractions. Then, as needed, you
may add (or subtract) the same quantity to (from) both
sides of the equation, multiply (divide) both sides of
the equation by the same nonzero quantity, or interchange the two sides of the equation.
203. The student should have subtracted 15x from both sides
so that the equation is equal to zero. By factoring out an
x, there are two solutions, x 0 and x 6.
204. 3x 42 x 4 2 0
(b) 3x2 8x 16 x 4 2 0
(a) Let u x 4
3u2 u 2 0
3x2 24x 48 x 4 2 0
3u 2u 1 0
3x2 25x 50 0
3u 2 0
u
2
3
x4
2
3
x 10
3
or
u 1
x 4 1
or
3x 10x 5 0
u10
x 5
205. 3 and 6
3x 10 0
x
10
3
or
x50
x 5
(c) The method of part (a) reduces the number of algebraic
steps.
206. x 4x 11 0
One possible equation is:
x 4x 11 0
x 3x 6 0
x2 15x 44 0
x 3x 6 0
x2 3x 18 0
Any non-zero multiple of this equation would also have
these solutions.
Appendix A.5
Solving Equations
1
208. x 6 ⇒ 6x 1 ⇒ 6x 1 is a factor.
207. 8 and 14
One possible equation is:
x 25 ⇒ 5x 2 ⇒ 5x 2 is a factor.
x 8x 14 0
6x 15x 2 0
22x 112 0
30x2 7x 2 0
x2
1193
Any non-zero multiple of this equation would also have
these solutions.
209. 1 2 and 1 2
210. x 3 5, x 3 5, so:
One possible equation is:
x 1 2 x 1 2 0
x 1 2x 1 2 0
2
x 12 2 0
x 3 5 x 3 5 0
x 3 5 x 3 5 0
x 2 6x 4 0
x 2 2x 1 2 0
x 2 2x 1 0
Any non-zero multiple of this equation would also have
these solutions.
9 a b 9
211. 9 9 a b
9ab9
OR
a b 18
a 18 b
9 a b 9
9 a b 9
212. Isolate the absolute value by subtracting x from both
sides of the equation. The expression inside the absolute
value signs can be positive or negative, so two separate
equations must be solved. Each solution must be
checked since extraneous solutions may be included.
a b
ab
Thus, a 18 b or a b. From the original equation
we know that b ≥ 9.
Some possibilities are:
b 9, a 9
b 10, a 8 or a 10
b 11, a 7 or a 11
b 12, a 6 or a 12
b 13, a 5 or a 13
b 14, a 4 or a 14
213. (a) ax2 bx 0
(b) ax2 ax 0
xax b 0
axx 1 0
x0
ax b 0 ⇒ x ax 0 ⇒ x 0
b
a
x10 ⇒ x1
1194
Appendix A
Review of Fundamental Concepts of Algebra
Appendix A.6
■
Linear Inequalities in One Variable
You should know the properties of inequalities.
(a) Transitive: a < b and b < c implies a < c.
(b) Addition: a < b and c < d implies a c < b d.
(c) Adding or Subtracting a Constant: a ± c < b ± c if a < b.
(d) Multiplying or Dividing a Constant: For a < b,
■
1. If c > 0, then ac < bc and
a b
< .
c
c
2. If c < 0, then ac > bc and
a b
> .
c
c
You should be able to solve absolute value inequalities.
(b) x > a if and only if x < a
(a) x < a if and only if a < x < a.
or x > a.
Vocabulary Check
1. solution set
2. graph
3. negative
4. equivalent
5. double
6. union
1. Interval: 1, 5
2. Interval: 2, 10
3. Interval: 11, (a) Inequality: 1 ≤ x ≤ 5
(a) Inequality: 2 < x ≤ 10
(a) Inequality: x > 11
(b) The interval is bounded.
(b) The interval is bounded.
(b) The interval is unbounded.
5. Interval: , 2
4. Interval: 5, (a) Inequality:
5 ≤ x <
or x
6. Interval: , 7
(a) Inequality: x < 2
≥ 5
(a) Inequality:
< x ≤ 7 or x ≤ 7
(b) The interval is unbounded.
(b) The interval is unbounded.
(b) The interval is unbounded.
9. 3 < x ≤ 4
8. x ≥ 5
7. x < 3
Matches (b).
Matches (f).
10. 0 ≤ x ≤
Matches (d).
9
2
Matches (c).
11. x < 3 ⇒ 3 < x < 3
12. x > 4 ⇒ x > 4 or x < 4
Matches (e).
Matches (a).
13. 5x 12 > 0
(b) x 3
(a) x 3
?
53 12 > 0
?
53 12 > 0
3 > 0
27 > 0
Yes, x 3 is a
solution.
No, x 3 is not a
solution.
(d) x 32
(c) x 52
?
552 12 > 0
1
2
> 0
Yes, x 52 is a solution.
?
532 12 > 0
92 > 0
No, x 32 is not a
solution.
Appendix A.6
Linear Inequalities in One Variable
1195
14. 2x 1 < 3
(c) x 4
(b) x 14
(a) x 0
?
20 1 < 3
?
2 14 1 < 3
1 < 3
No, x 0 is not
a solution.
1
2
No, x a solution.
?
2 32 1 < 3
7 < 3
2 < 3
< 3
14
is not
(d) x 32
?
24 1 < 3
Yes, x 4 is a
solution.
No, x 32 is not
a solution.
x2
< 2
4
15. 0 <
(a) x 4
(b) x 10
? 42 ?
0 <
< 2
4
Yes, x 4 is a
solution.
16. 1 <
No, x 10 is not a
solution.
(d) x ? 02 ?
0 <
< 2
4
0 < 1
< 2
2
No, x 0 is not a
solution.
7
2
? 72 2 ?
0 <
< 2
4
0 <
3
< 2
8
Yes, x 27 is a solution.
3x
≤ 1
2
(a) x 0
1 <
(c) x 1
(b) x 5
? 30 ?
1 <
≤ 1
2
3
≤ 1
2
No, x 0 is not a
solution.
? 10 2 ?
0 <
< 2
4
0 < 2 < 2
1
< 2
2
0 <
(c) x 0
(d) x 5
? 35 ?
1 <
≤ 1
2
? 31 ?
1 <
≤ 1
2
1 < 4 ≤ 1
1 < 1 ≤ 1
No, x 5 is not a
solution.
Yes, x 1 is a solution.
? 35 ?
1 <
≤ 1
2
?
1 < 1 ≤ 1
No, x 5 is not a
solution.
17. x 10 ≥ 3
(a) x 13
(b) x 1
?
13 10 ≥ 3
3 ≥ 3
Yes, x 13 is a
solution.
(c) x 14
?
1 10 ≥ 3
9 10 ≥ 3
4 ≥ 3
1 ≥ 3
11 ≥ 3
Yes, x 1 is a
solution.
(d) x 9
?
14 10 ≥ 3
Yes, x 14 is a
solution.
?
No, x 9 is not a
solution.
18. 2x 3 < 15
(a) x 6
(b) x 0
?
26 3 < 15
15 < 15
No, x 6 is not
a solution.
(c) x 12
?
20 3 < 15
3 < 15
Yes, x 0 is a
solution.
(d) x 7
?
212 3 < 15
27 3 < 15
21 < 15
11 < 15
No, x 12 is not
a solution.
?
Yes, x 7 is a
solution.
1196
19.
Appendix A
Review of Fundamental Concepts of Algebra
20. 10x < 40
4x < 12
1
4 4x
1
4 12
<
2x > 3
21.
122x
x < 4
x < 3
−5
−4
−3
1
3
2
x <
x
−6
< 2 3
−2
3
2
x
1
2
3
4
5
x
−2
22. 6x > 15
x <
or x <
52
−3
−2
11
12
x ≥
3
4
28. 6x 4 ≤ 2 8x
6
7
7x ≥ 2
1
2
x ≥
2
7
2
7
0
1
x
−2
2
−1
0
1
2
30. 4x 1 < 2x 3
4 2x < 9 3x
4x 4 < 2x 3
x < 5
2x < 1
x ≥ 3
x < 12
x
x
−2
5
27. 2x 1 ≥ 1 5x
29. 4 2x < 33 x
2x ≤ 6
−3
4
x
−1
−4
3
1
2
x
−5
x
14
2x ≥ 1
x > 2
2
13
26. 3x 1 ≥ 2 x
2x < 4
1
3
−1
25. 2x 7 < 3 4x
0
2
x
10
x
−4
1
x ≤ 5
x ≥ 12
− 52
0
24. x 7 ≤ 12
23. x 5 ≥ 7
15
6
−1
−1
3
4
5
6
7
− 21
x
−2
3
31. 4x 6 ≤ x 7
32.
3 27 x > x 2
−1
0
1
33. 128x 1 ≥ 3x 52
14x ≤ 1
21 2x > 7x 14
4x 12 ≥ 3x 52
x ≥ 4
5x > 35
x ≥ 2
x < 7
x
2
3
4
5
x
6
0
1
2
3
4
x
5
34.
9x 1 < 34 16x 2
6
1
6
x
1
36. 15.6 1.3x < 5.2
1.3x < 20.8
x ≥ 4
x > 16
x
−6
1
6
0
9
3.6x ≥ 14.4
12x < 2
−1
8
35. 3.6x 11 ≥ 3.4
36x 4 < 48x 6
x >
7
2
3
−5
−4
−3
−2
x
14
15
16
17
18
Appendix A.6
38. 8 ≤ 3x 5 < 13
37. 1 < 2x 3 < 9
2 < 2x < 6
8 ≤ 3x 5 < 13
1 < x < 3
3 ≤ 3x < 18
0
1
2
39. 14 <
2x 3
< 4
3
19 < 2x < 15
1 92 < x <
3
x
− 6 − 5 − 4 − 3 − 2 −1
1197
12 < 2x 3 < 12
6 < x ≤ 1
x
−1
Linear Inequalities in One Variable
0
15
2
15
2
−9
2
1
x
− 6 −4 − 2
x3
< 5
2
0 ≤
40.
0 ≤ x 3 < 10
3 ≤ x < 7
−3
x
0
2
4
6
−
3
4
4
6
8
x
< 1
3
3 < 6 x < 3
1
3
> x > 4
4
8
2
42. 1 < 2 9 < x < 3
3
1
< x < 4
4
7
−4 −2
3
1
> x1 >
4
4
41.
0
3 < x < 9
x
−1
4
1
3
5
7
9
11
4
6
x
−1
3.2 ≤ 0.4x 1 ≤ 4.4
43.
44. 4.5 >
4.2 ≤ 0.4x ≤ 5.4
13.5
11
12
1.5x 6
> 10.5
2
45. x < 6
6 < x < 6
x
3 > 1.5x > 15
13
−6 −4 −2
0
2
2 > x > 10
x
10
1
9 > 1.5x 6 > 21
10.5 ≤ x ≤ 13.5
10.5
0
14
There is no solution.
47.
46. x > 4
x < 4 or x > 4
x
− 6 − 4 −2
0
2
4
x
> 1
2
48.
x
> 3
5
x
x
< 3 or
> 3
5
5
x
x
< 1 or
> 1
2
2
6
x < 2
x < 15
x > 2
x > 15
x
− 3 − 2 −1
49. x 5 < 1
No solution. The absolute value of a number cannot be
less than a negative number.
0
1
2
52. x 8 ≥ 0
6 ≤ x 20 ≤ 6
x 8 ≥ 0 or x 8 ≥ 0
14 ≤ x ≤ 26
x 8 ≥ 0
x ≥ 8
x ≥ 8
26
14
x
10
15
20
− 10
25
0
10
20
50. There is no solution because the absolute value of a
number cannot be less than a negative number.
51. x 20 ≤ 6
x
− 20
3
x ≤ 8
30
All real numbers x
x
−3 −2 −1
0
1
2
3
1198
Appendix A
Review of Fundamental Concepts of Algebra
53. 3 4x ≥ 9
54. 1 2x < 5
or 3 4x ≥ 9
3 4x ≤ 9
5 < 1 2x < 5
4x ≥ 6
4x ≤ 12
x ≥ 3
x ≤
6 < 2x < 4
32
3 > x > 2
2 < x < 3
−3
2
x
−2 −1
55.
0
1
2
3
x
4
−2
−1
56. 1 x3
≥ 4
2
x 3 ≤ 8
x3 ≥ 8
x ≤ 5
2 < x ≥ 11
5
3
2x
< 1
3
1 < 1 2x
< 0
3
3 > x > 0
x
0
2
2x
< 1
3
11
− 15 − 10 − 5
1
x3
≥ 4
2
x3
≤ 4 or
2
0
10
0 < x < 3
15
x
−1
57.
9 2x 2 < 1
9 2x < 1
0
1
2
3
4
58. x 14 3 > 17
1 < 9 2x < 1
x 14 > 14
x 14 < 14 or x 14 > 14
10 < 2x < 8
x < 28
x > 0
5 > x > 4
x
− 35 − 28 − 21 −14 −7
0
7
4 < x < 5
x
3
4
5
6
4 5x ≤ 3
x 10 ≥ 29
60. 3 4 5x ≤ 9
59. 2 x 10 ≥ 9
x 10 ≤ 29
x ≤
−
x 10 ≥
or
29
2
29
2
x ≥
3 ≤ 4 5x ≤ 3
11
2
7 ≤ 5x ≤ 1
− 11
2
−12
−8
−4
62. 3x 1 ≤ 5
61. 6x > 12
≥ x ≥
1
5
1
5
≤ x ≤
7
5
5 2x ≤ 2
10
10
−10
2
52x ≥ 4
x ≤ 2
10
7
5
1
63. 5 2x ≥ 1
3x ≤ 6
6x > 2
7
5
x
0
9
2
x
−16
1
5
10
− 10
10
− 10
10
− 10
−10
− 10
Appendix A.6
64. 3x 1 < x 7
Linear Inequalities in One Variable
65. x 8 ≤ 14
2x 9 < 13 or 2x 9 > 13
6 ≤ x ≤ 22
2x < 4
x < 2
66. 2x 9 > 13
14 ≤ x 8 ≤ 14
3x 3 < x 7
1199
10
2x < 22
2x > 4
x < 11
x > 2
10
10
−10
−10
24
− 15
10
9
−10
−10
−10
x 7 ≥ 132
67. 2 x 7 ≥ 13
68.
x 7 ≤ 13
2
x7 ≥
or
x ≤ 27
2
1
2
x 1 ≤ 3
x 1 ≤ 6
6 ≤ x 1 ≤ 6
13
2
7 ≤ x ≤ 5
x ≥ 21
10
10
−15
− 10
1
10
− 10
− 10
69. y 2x 3
2
70. y 3 x 1
3
(a) 2 3y ≥ 1
−4
8
y ≤ 5
(a)
2
3x
2x 3 ≥ 1
32x ≥ 4
1 ≤ 5
2
3x
−5
2 3x ≥ 2
2
3x
−2
1 ≥ 0
2
3x
32x ≤ 3
≥ 1
3
x ≥ 2
3
2
71. y 12x 2
72. y 3x 8
6
0 ≤
12 x
2 ≤ 3
2 ≤
12 x
≤ 1
−6
1 ≤ 3x 8 ≤ 3
6
9 ≤ 3x ≤ 5
−2
3 ≥ x ≥
5
3
y ≥ 0
12 x 2 ≥ 0
12 x ≥ 2
x ≤ 4
9
(a) 1 ≤ y ≤ 3
0 ≤ y ≤ 3
4 ≥ x ≥ 2
(b)
≤ 4
7
y ≥ 0
(b)
2x 3 ≤ 0
(a)
−5
x ≤ 6
(b) 2 3y ≤ 0
2 3x ≤
6
(b)
5
3
≤ x ≤ 3
y ≤ 0
3x 8 ≤ 0
3x ≤ 8
x ≥
8
3
−9
9
−3
1200
Appendix A
Review of Fundamental Concepts of Algebra
73. y x 3
74. y 8
1
2
x 3 ≤ 2
−5
x 1
1
2
10
2 ≤ x 3 ≤ 2
−2
or
or
x 1
1
2
x3 ≥ 4
1
2x
x ≥ 7
≥ 1
1 ≤ 1 or
76. x 10
5, x ≥ 10
x ≤
3x ≥ 0
x ≥ 3
3, 3 ≥ x
, 6x ≥ 15
82. x 8 > 4
81. x 10 < 8
6x 15 ≥ 0
All real numbers within
8 units of 10.
All real numbers more
than 4 units from 8
x ≥ 52
7
2
52, 83. The midpoint of the interval 3, 3
is 0. The interval represents all real
numbers x no more than 3 units
from 0.
84. The graph shows all real numbers
more than 3 units from 0.
86. The graph shows all real numbers
no more than 4 units from 1.
87. All real numbers within 10 units
of 12
x 1 ≤ 4
89. All real numbers more than 4 units from 3
85. The graph shows all real numbers
at least 3 units from 7.
x 7 ≥ 3
x 0 > 3
x > 3
x 0 ≤ 3
x ≤ 3
x 3 > 4
x 3 > 4
, 3
4 6x 15
80. 7
2
≥ 0
78. 3 x
10, 2x ≥ 7
1 ≥ 1
x ≥ 0
77. x 3 ≥ 0
x 10 ≥ 0
x ≥ 5
1
2x
1
2x
≤ 2
x ≤ 4
79. 7 2x ≥ 0
−1
(b) y ≥ 1
1
2x
75. x 5 ≥ 0
1
10 ≤ x ≤ 6
x 3 ≥ 4
x ≤ 1
−5
1
5 ≤ 2 x ≤ 3
y ≥ 4
x 3 ≤ 4
≤ 4
4 ≤ 12 x 1 ≤ 4
1 ≤ x ≤ 5
(b)
3
(a) y ≤ 4
y ≤ 2
(a)
x 1
x 12 < 10
88. All real numbers at least 5 units
from 8
x 8 ≥ 5
90. All real numbers no more than 7 units from 6
x 6 ≤ 7
Appendix A.6
91. Let x the number of checks written in a month.
Type A account charges: 6.00 0.25x
Type B account charges: 4.50 0.50x
6.00 0.25x < 4.50 0.50x
1.50 < 0.25x
Linear Inequalities in One Variable
1201
92. 3000 0.03x ≤ 0.1x
3000 < 0.07x
42,857 < x
You must make more than 42,857 copies to justify buying
the copier.
6 < x
If you write more than six checks a month, then the
charges for the type A account are less than the charges
for the type B account.
93. 10001 r 2 > 1062.50
1 2r > 1.0625
2r > 0.0625
r > 0.03125
r > 3.125%
94. 825 < 7501 r 2
825 < 7501 2r
825 < 750 1500r
75 < 1500r
0.05 < r
The rate must be more than 5%.
96. 24.55x > 15.4x 150,000
95. 115.95R > C
115.95x > 95x 750
9.15 > 150,000
120.95x > 750
x > 16,393.44262
120.95x > 35.7995
Because the number of units x must be an integer, the
product will return a profit when at least 16,394 units
are sold.
120.95x ≥ 36 units
97. Let x daily sales level (in dozens) of doughnuts.
Revenue: R 2.95x
Cost:
C 150 1.45x
Profit:
PRC
2.95x 150 1.45x
98. The goal is to lose 164 128 36 pounds. At 112 pounds
per week, it will take 24 weeks.
36 112 36 23
12 2
24
1.50x 150
50 ≤ P ≤ 200
50 ≤ 1.50x 150 ≤ 200
200 ≤ 1.50x ≤ 350
13313 ≤ x ≤ 23313
In whole dozens, 134 ≤ x ≤ 234.
99. (a) y 0.067x 5.638
(b) From the graph we see that y ≥ 3 when x ≥ 129.
Algebraically we have:
5
3 ≤ 0.067x 5.638
8.638 ≤ 0.067x
75
150
0
x ≥ 129
IQ scores are not a good predictor of GPAs. Other factors
include study habits, class attendance, and attitude.
1202
Appendix A
Review of Fundamental Concepts of Algebra
100. (a) and (b)
300
x
165
184
150
210
196
240
y
170
185
200
255
205
295
1.3x 36
179
203
159
237
219
276
x
202
170
185
190
230
160
y
190
175
195
185
250
155
1.3x 36
227
185
205
211
263
172
(c) One estimate is x ≥ 181 pounds.
(d) 1.3x 36 ≥ 200
1.3x ≥ 236
x ≥ 181.5385 181.54 pounds
101. S 1.05t 31.0, 0 ≤ t ≤ 12
(a)
32 ≤ 1.05t 31 ≤ 42
140
140
260
(e) An athlete’s weight is not a particularly good
indicator of the athlete’s maximum bench press
weight. Other factors, such as muscle tone and
exercise habits, influence maximum bench press
weight.
102. (a) 70 ≤ 1.64t 67.2 ≤ 80
2.8 ≤ 1.64t ≤ 12.8
1.7 ≤ t ≤ 7.8
1 ≤ 1.05t ≤ 11
0.95 ≤ t ≤ 10.48
Rounding to the nearest year, 1 ≤ t ≤ 10. The
average salary was at least $32,000 but not more
than $42,000 between 1991 and 2000.
(b) 1.05t 31 > 48
The number of eggs produced was between 70 and
80 billion from late 1991 until late 1997.
(b) E 1.64t 67.2 ≥ 95 when t ≥ 16.95.
So the annual egg production will exceed 95 billion
in 2007.
1.05t > 17
t > 16
According to the model, the average salary will
exceed $48,000 in 2006.
103. s 10.4 ≤
1
16
1
16
104. 24.2 0.25 ≤ s ≤ 24.2 0.25
≤ s 10.4 ≤
1
16
0.0625 ≤ s 10.4 ≤ 0.0625
10.3375 ≤ s ≤ 10.4625
Since A s2, we have
23.95 ≤ s ≤ 24.45
The interval containing the possible side lengths s in
centimeters of the square is 23.95, 24.45, so the interval containing the possible areas in square centimeters is
23.95 2, 24.45 2, or 573.6025, 597.8025.
10.33752 ≤ area ≤ 10.46252
106.864 ≤ area ≤ 109.464.
105. x 15 ≤
1
10
1
10
≤ x 15 ≤
1
1
106. 1 oz 16
lb, so 21 oz 32
lb
1
10
14.9 ≤ x ≤ 15.1 gallons
1
10 $1.89
$0.19
You might have been undercharged or overcharged
by $0.19.
1
14.99 32
0.4684375, so you could have been underor overcharged as much as $0.47.
Appendix A.6
107.
1 <
t 15.6
1.9
< 1
t 15.6
< 1
1.9
1.9 < t 15.6 < 1.9
1203
h 68.5
≤ 1
2.7
1 ≤
17.5
t
12 13 14 15 16 17 18 19
h 68.5
≤ 1
2.7
2.7 ≤ h 68.5 ≤ 2.7
65.8 inches ≤ h ≤ 71.2 inches
13.7 < t < 17.5
Two-thirds of the workers could perform the task in the
time interval between 13.7 minutes and 17.5 minutes.
h 50 ≤ 30
109.
108.
13.7
Linear Inequalities in One Variable
65.8
71.2
h
65 66 67 68 69 70 71 72
110. (a) Estimate from the graph: when the plate thickness is
2 millimeters, the frequency is approximately 330
vibrations per second.
30 ≤ h 50 ≤ 30
20 ≤ h ≤ 80
The minimum relative humidity is 20 and the maximum
is 80.
(b) Estimate from the graph: when the frequency is 600,
the plate thickness is approximately 3.6 millimeters.
(c) Estimate from the graph: when the frequency is
between 200 and 400 vibrations per second, the plate
thickness is between 1.2 and 2.4 millimeter.
(d) Estimate from the graph: when the plate thickness
is less than 3 millimeters, the frequency is less than
500 vibrations per second.
111. False. If c is negative, then
ac ≥ bc.
112. False. If 10 ≤ x ≤ 8, then
10 ≥ x and x ≥ 8.
113. x a ≥ 2
Matches (b).
x a ≤ 2
x ≤ a 2 or
xa ≥ 2
x ≥ a2
114. ax b ≤ c ⇒ c must be greater than or equal to zero.
c ≤ ax b ≤ c
b c ≤ ax ≤ b c
Let a 1, then b c 0 and b c 10. This is true when b c 5.
One set of values is a 1, b 5,c 5.
(Note: This solution is not unique. Any positive multiple of these values will also work, such as
a 2, b c 10 or a 3, b c 15.
1204
Appendix A
Appendix A.7
Review of Fundamental Concepts of Algebra
Errors and the Algebra of Calculus
■
You should be able to recognize and avoid the common algebraic errors involving parentheses, fractions, exponents,
radicals, and cancellation.
■
You should be able to “unsimplify” algebraic expressions by the following methods.
(a) Unusual Factoring
(b) Rewriting with Negative Exponents
(c) Writing a Fraction as a Sum of Terms
(d) Inserting Factors or Terms
Vocabulary Check
1. numerator
1. 2x 3y 4 2x 3y 4
3.
2. reciprocal
2. 5z 3x 2 5z 3x 2
Change all signs when distributing the minus sign.
The 3 is distributed to both terms.
2x 3y 4 2x 3y 4
5z 3x 2 5z 3x 6
4
4
16x 2x 1 14x 1
4.
Change all signs when distributing the minus sign.
The expression on the right should be negative.
4
4
4
16x 2x 1 16x 2x 1 14x 1
1x
x1
5 xx
xx 5
5. 5z6z 30z
6. x yz xyxz
z occurs twice as a factor.
yz is one term, not two.
5z6z xyz xyz
7. a
30z2
xy ayax
8. 4x2 4x2
The exponent applies to the coefficient also.
The fraction as a whole is multiplied by a, not the
numerator and denominator separately.
a
4x2 16x2
xy 1a xy axy
9. x 9 x 3
Do not apply the radical to the terms.
10. 25 x2 5 x
Do not apply radicals term-by-term.
x 9 does not simplify.
11.
1x
x1
5 xx xx 5
2x2 1 2x 1
5x
5
Divide out common factors not common terms.
2x 1
cannot be simplified.
5x
2
25 x2 5 x5 x
12.
6x y
does not simplify.
6x y
Reduce common factors of the numerator and
denominator, not common factors of terms.
Appendix A.7
13.
1
1
a1 b1
ab
1
14.
ab
ab
ab b a
1
y
x y1 x 1
1
1
x y1 x 1y
y
y
y xy 1
16. x2x 12 2x2 x2
15. x2 5x12 xx 512
17.
Factor within grouping symbols before applying the
exponent to each factor.
Factor within grouping symbols before applying the
exponent to each factor.
x2 5x12 xx 512 x12x 512
x2x 12 x4x2 4x 1
3 4 3
x
y
x
4
y
3y 4x
xy
x
yyx
1205
The negative exponent is on a term of the denominator,
not a factor.
To get rid of negative exponents:
1
1
a1 b1 a1 b1
Errors and the Algebra of Calculus
18.
To add fractions, they must have a common denominator.
1
1
y
2y
2
Be careful when using a slash to denote division.
12y 21 y 2
y
19.
3x 2 1
3x 2
5
5
20.
The required factor is 3x 2.
7x2
7
x2
10
10
The required factor is x2.
2
1
2
1
15
1
21. 3x2 3x 5 3x2 3x 3 32x2 x 15
3
1
3
2
1
22. 4x 2 4x 4 43x 2
The required factor is 3x 2.
The required factor is 2x2 x 15.
1
23. x2x3 14 x3 143x2
3
24. x1 2x23 4x
1
1 2x23 4x1 2x23
4
4
411 2x 4x
1
The required factor is 3.
2 3
1
The required factor is 4.
25.
4x 6
22x 3
2 2x 3
1
1
x2 3x 73 2x2 3x 73 2x 3
x2 3x 73 x2 3x 73 1
1
The required factor is 2.
26.
x1
1
x 2x 32 2
2
2x 1
x 2 2x 32
12x
The required factor is 12.
2
27.
1
2x 2
2x 32
5
6x
3
3
5
3x 3
2 x 2 2 2
x
2x
2
2x
2x
2x
2x1 6x 5 3x The required factor is
3
2
1
.
2x2
1206
28.
Appendix A
Review of Fundamental Concepts of Algebra
x 12
x 1x 12
y 52 y 52
169
x 1169
29.
9x2 16y2
9
25
49
25
x 13
y 52
169x 1
The required factor is x 1.
x2 16
1
49
3x2 9y 2
4
16
13 3x2 19 9y 2 x2 y 2
13 4 19 16 43 169
31.
4
16
The required factors are 3 and 9 .
32.
1
x2
1
4916
1
259
x2
y2
259 4916
x2
y2
9
8
9x2 8y 2
y2
x2
49 78
4
7
4
7
34. 32x 1x12 4x32 x1232x 1 4x
2x
2x
35. 1 3x43 4x1 3x13 1 3x131 3x1 4x
1 3x131 7x
The required factor is 1 7x.
5x322x 10x522x 2x
2x
1
1 10x32x 20x52x 2x
1
1 10x2 20x3
2x
The required factor is 1 10x2 20x3.
37.
1
10 2x
3
152 162x 132 30
2x 1 322x 11 305 2x 132
1
30
2x 13232x 1 5
1
30
2x 1326x 2
1
30
2x 13223x 1
1
15
2x 1323x 1
The required factor is 3x 1.
38.
3
3
12
21
t 173 t 143 t 143t 133 t 143
7
4
28
28
3t 143
4t 1 7
28
3t 143
4t 3
28
The required factor is 4t 3.
The required factor is 1 5x.
The required factor is 10x 3.
1
49
and 16.
33. x13 5x43 x131 5x33 x131 5x
x1210x 3
5x32 10x52 y2
1
x2
12x2 3y2
y2
12
3
y2
x2
112 23
1
2
1
2
x126x 3 4x
1
25
9
The required factors are 1 and 2.
The required factors are 4 and 7.
36.
y2
1
The required factors are
30.
Appendix A.7
Errors and the Algebra of Calculus
1207
39.
3x2
3x22x 13
2x 13
40.
x1
x 1x16 x12
x6 x12
41.
4
4
7x
3
43x1 4x4 7x2x13
3x x4 2x
42.
x
8
8
1
xx 21 x2 9x3
x 2 x2
39x3
3
43.
16 5x x2 16 5x x2 16
5x
x
x
x
x
x
44.
x3 5x2 4 x3 5x2
4
4
2 2 2x5 2
x2
x
x
x
x
45.
4x3 7x2 1
4x3
7x2
1
13 13 13
13
x
x
x
x
46.
2x5 3x3 5x 1
2x5
3x3
5x
1
32 32 32 32
32
x
x
x
x
x
4x313 7x213 47.
50.
51.
7x
53
2x72 3x32 1
x13
3 5x2 x4
3
5x2
x4
x
x
x
x
49.
4x83
2x532 3x332 5x132 x32
1
x13
3
x
48.
x3
5x4
x
5x2
x3 5x4
2 2 2
3x
3x
3x
3
3
5x212 x412
3
5x32 x72
x12
2x2 332xx 13 3x 12x2 32 x 2 33x 124xx 1 3x2 3
x 132
x 16
4x2 4x 3x2 9
x2 33x 14
7x2 4x 9
x2 33x 14
x53x2 142x x2 135x4 x 4x2 146x2 5x2 1
x52
x10
x 46x2 5x2 5
x10x2 14
11x2 5
x6x2 14
6x 1327x2 2 9x3 2x36x 126
6x 126x 127x2 2 189x3 2x
3
2
6x 1 6x 16
162x3 12x 27x2 2 162x3 36x
6x 14
27x2 24x 2
6x 14
5
1
x12 x32
1208
52.
53.
Appendix A
Review of Fundamental Concepts of Algebra
4x2 9122 2x 312 4x2 9128x 24x2 9124x2 9 2x2x 3
4x2 9122
4x2 9
24x2 9 4x2 6x
4x2 932
29 6x
4x2 932
62x 3
4x2 932
x 234x 323 x 313x 214 x 214x 323x 2 x 3
x 2342
x 264
x2x3
x 214x 323x 264
1
x 323x 274
54. 2x 112 x 22x 112 2x 112 55.
56.
x 2
2x 112
2x 1
x 2
2x 112 2x 112
2x 1 x 2
2x 112
x3
2x 112
23x 113 2x 1133x 1233 3x 12323x 1 2x 1
3x 123
3x 123
6x 2 2x 1
3x 1233x 123
4x 3
3x 143
x 1122x 3x2122 6x 2x 3x212 x 12x 3x2121 3x 2x 3x212
x 12
x 12
2x 3x212x 11 3x 2x 3x2
x 12
x 3x2 1 3x 2x 3x2
2x 3x212x 12
1 4x
2x 3x212 x 12
Appendix A.7
57.
58.
Errors and the Algebra of Calculus
1209
1
1
1
1
1
x2 4122x x2 412 x2 412 22x
x2 412 2
x2
1
x
x2 41
x
x2 4
1
1
2x2x 5 2x2 6
2x 2 6
2x 5
x2 62x 5
59. x2 512
323x 2
4x2 10x 2x2 12
6x2 10x 12
2
2
x 62x 5
x 62x 5
23x2 5x 6
x2 62x 5
3 3x 232
12
12x
2
9
5122x x2 5123x 212 xx2 5123x 232
2
9
2
x2 5123x 212 xx2 5123x 232
2
2
1
x2 5123x 2129x2 51 2x3x 21
2
1
x2 5123x 2129x2 45 6x2 4x
2
213x 2
60. 3x 2123x 6121 x 63 32
3x 21215x2 4x 45
2x2 512
3 33x 212x 612 32x 6 3x 2
32
3
3
x 6123x 23223x 2 x 652
2
61. t (a)
x2 4
2
3x 6126x 4 x 652
23x 232
4 x2 4
6
1
x
t
0.5
1.70
(b) She should swim to a point about 2 mile down the coast to minimize the time required
to reach the finish line.
1.0
1.72
1 2
1
3
1
xx 412 x 4x2 8x 2012 xx 2 412 x 4x 2 8x 2012
2
6
6
6
1.5
1.78
1
3xx2 412 x 4x2 8x 2012
6
2.0
1.89
2.5
2.02
1
3x
x4
6 x2 412 x2 8x 2012
3.0
2.18
3.5
2.36
3xx2 8x 20 x 4x2 4
6x2 4 x2 8x 20
4.0
2.57
(c)
1210
Appendix A
62. (a) y1 x2
13x
2
Review of Fundamental Concepts of Algebra
1232x x2 1132x
2xx2 123
2xx2 123
x2
2x
123
x
2
1
12
0
1
2
5
2
y1
8.7
2.9
1.1
0
2.9
8.7
12.5
x2 3x2 1
3
3
y2
8.7
2.9
1.1
0
2.9
8.7
12.5
3 x
x2
(b)
2
1
4x2 3
3
2x4x2 3
3x2 123
y2
63. True.
x1 y2 1
1
2
x
y
64. False. Cannot move term-by-term from denominator
to numerator.
x
xy2
y2
x2
1
1
x2y
1
1
2
1
y
1
yx
y x2
2
2
x
y
xy
66. False. x2 9 does not factor into x 3x 3.
65. True.
1
x 4
1
x 4
x 4
x 4
x 4
x2 9
x 3x 3
x 3
x 3
x 16
67. x n x 3n x 3n
x 3
x 3
x 3x 3x 3
x9
68. xn2n x2nn x2n x2n 2x2n
2
2
2
There is no error.
Add exponents when multiplying powers with like bases.
xn
x 3n x 4n
69. x2n y2n x n y n2
70.
When squaring binomials, there is also a middle term.
x n y n2 x2n 2x ny n y 2n
x2n x3n
x2n3n
x5n
3n
3n
3n
2
2
x x
x x
x x2
There is no error.
71. The two answers are equivalent and can be obtained by factoring.
1
10 2x
1
152 16 2x 132 60
2x 13262x 1 10
1
60
2x 13212x 4
4
60
2x 1323x 1
1
15
2x 1323x 1
(a)
2
3 x2x
2
2
332 15
2x 352 15
2x 332 5x 2x 3
2
15
2x 3323x 3
2
15
2x 3323x 1
25 2x 332x 1
(b)
2
3 x4
2
2
x32 15
4 x52 15
4 x325x 4 x
2
15
4 x324x 4 152 4 x324x 1 158 4 x32x 1
2