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Summary
D EFINITION /P ROCEDURE
E XAMPLE
R EFERENCE
Introduction to Integers
Section 2.1
Positive Integers
Integers used to name whole numbers to the right of the origin
on the number line.
p. 122
Negative Integers
Integers used to name the opposites of whole numbers.
Negatives are found to the left of the origin on the number line.
Integers
Whole numbers and their opposites. The integers are
The origin
3 2 1 0
Negative
integers
1
2
3
Positive
integers
{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
Absolute Value
The distance (on the number line) between the point named
by a signed number and the origin.
The absolute value of x is written x.
7 7
10 10
Addition of Integers
Adding Integers
1. If two integers have the same sign, add their absolute
values. Give the result the sign of the original integers.
2. If two integers have different signs, subtract their absolute
values, the smaller from the larger. Give the result the sign
of the integer with the larger absolute value.
Section 2.2
9 7 16
(9) (7) 16
Section 2.3
16 8 16 (8)
8
8 15 8 (15)
7
9 (7) 9 7
2
Multiplication of Integers
Multiplying Integers
Multiply the absolute values of the two integers.
1. If the integers have different signs, the product is negative.
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2. If the integers have the same sign, the product is positive.
p. 133–134
15 (10) 5
(12) 9 3
Subtraction of Integers
Subtracting Integers
1. Rewrite the subtraction problem as an addition problem by
a. Changing the subtraction symbol to an addition symbol
b. Replacing the integer being subtracted with its opposite
2. Add the resulting integers as before.
p. 124
p. 139
Section 2.4
5(7) 35
(10)(9) 90
8 7 56
(9)(8) 72
(2)2 (2)(2) 4
22 2 2 4
p. 145–146
Continued
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INTEGERS AND INTRODUCTION TO ALGEBRA
D EFINITION /P ROCEDURE
E XAMPLE
Division of Integers
Dividing Integers
Divide the absolute values of the two integers.
1. If the integers have different signs, the quotient is negative.
2. If the integers have the same sign, the quotient is positive.
Section 2.5
32
8
4
18
2
9
Introduction to Algebra: Variables and Expressions
Multiplication
xy
(x)(y) These all mean the product of x and y or x times y.
xy
The product of m and n is mn.
The product of 2 and the sum
of a and b is 2(a b).
Combining Like Terms
To combine like terms:
1. Add or subtract the coefficients (the numbers multiplying
the variables).
2. Attach the common variable.
Evaluate 2x 3y if x 5 and
y 2.
2x 3y
Solution
A value for a variable that makes an equation a true statement.
Section 2.8
3xy is a term
The Addition Property of Equality
If a b, then a c b c
p. 177
p. 179
5x 2x 7x
8a 5a 3a
Section 2.9
2x 3 5 is an equation
p. 186
4 is a solution for the
above equation because
2(4) 3 5
p. 187
The Addition Property of Equality
Equivalent Equations
Equations that have exactly the same solutions.
p. 167
2 5 (3)(2)
10 (6) 4
Introduction to Linear Equations
Equation
A statement that two expressions are equal.
p. 159
Section 2.7
Simplifying Algebraic Expressions
Term
A number or the product of a number and one or more variables.
p. 151
Section 2.6
Evaluating Algebraic Expressions
Evaluating Algebraic Expressions
To evaluate an algebraic expression:
1. Replace each variable or letter with its number value.
2. Do the necessary arithmetic, following the rules for the
order of operations.
R EFERENCE
Section 2.10
2x 3 5 and x 4 are
equivalent equations
p. 192
If 2x 3 7,
then 2x 3 3 7 3
p. 192
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Summary Exercises
This summary exercise set is provided to give you practice with each of the objectives of this chapter. Each exercise is
keyed to the appropriate chapter section. When you are finished, you can check your answers to the odd-numbered exercises against those presented in the back of the text. If you have difficulty with any of these questions, go back and reread
the examples from that section. The answers to the even-numbered exercises appear in the Instructor’s Solutions Manual.
Your instructor will give you guidelines on how to best use these exercises in your instructional setting.
[2.1]
Represent the integers on the number line shown.
1. 6, 18, 3, 2, 15, 9
20
10
0
10
20
Place each of the groups of integers in ascending order.
2. 4, 3, 6, 7, 0, 1, 2
3. 7, 8, 8, 1, 2, 3, 3, 0, 7
Find the opposite of each number.
5. 63
4. 17
Evaluate.
6. 9
7. 9
8. 9
9. 9
10. 12 8
11. 812
12. 8 12
13. 18 12
14. 73
15. 95
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[2.2]
Add.
16. 3 (8)
17. 10 (4)
18. 6 (6)
19. 16 (16)
20. 18 0
[2.3]
Subtract.
21. 8 13
22. 7 10
23. 10 (7)
24. 5 (1)
25. 9 (9)
26. 0 (2)
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INTEGERS AND INTRODUCTION TO ALGEBRA
Multiply.
27. (10)(7)
28. (8)(5)
29. (3)(15)
30. (1)(15)
31. (0)(8)
Evaluate each of the expressions.
32. 18 3 5
33. (18 3) 5
34. 5 42
35. (5 4)2
36. 5 32 4
37. 5(32 4)
38. 5(4 2)2
39. 5 4 22
40. (5 4 2)2
41. 3(5 2)2
42. 3 5 22
43. (3 5 2)2
[2.5]
Divide.
44.
80
16
45.
63
7
46.
81
9
47.
0
5
48.
32
8
49.
7
0
51.
6 1
5 (2)
50.
8 6
8 (10)
52.
25 4
5 (2)
[2.6]
Write, using symbols.
53. 5 more than y
54. c decreased by 10
55. The product of 8 and a
56. The quotient when y is divided by 3
© 2007 McGraw-Hill Companies
Perform the indicated operations.
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SUMMARY EXERCISES
57. 5 times the product of m and n
58. The product of a and 5 less than a
59. 3 more than the product of 17 and x
60. The quotient when a plus 2 is divided by
a minus 2
Identify which are expressions and which are not.
61. 4(x 3)
62. 7 8
63. y 5 9
64. 11 2(3x 9)
[2.7]
Evaluate the expressions if x 3, y 6, z 4, and w 2.
65. 3x w
66. 5y 4z
67. x y 3z
68. 5z 2
69. 3x2 2w2
70. 3x3
71. 5(x2 w2)
72.
6z
2w
73.
2x 4z
yz
74.
3x y
wx
75.
x(y2 z2)
(y z)(y z)
76.
y(x w)2
x 2xw w2
[2.8]
2
List the terms of the expressions.
77. 4a3 3a2
78. 5x2 7x 3
Circle like terms.
© 2007 McGraw-Hill Companies
79. 5m 2, 3m, 4m 2, 5m 3, m 2
80. 4ab2, 3b2, 5a, ab2, 7a2, 3ab2, 4a2b
Combine like terms.
81. 5c 7c
82. 2x 5x
83. 4a 2a
84. 6c 3c
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85. 9xy 6xy
86. 5ab2 2ab2
87. 7a 3b 12a 2b
88. 6x 2x 5y 3x
89. 5x3 17x2 2x3 8x2
90. 3a3 5a2 4a 2a3 3a2 a
[2.9]
Tell whether the number shown in parentheses is a solution for the given equation.
91. 7x 2 16
92. 5x 8 3x 2
(2)
93. 7x 2 2x 8
[2.10]
94. 4x 3 2x 11
(2)
95. x 5 3x 2 x 23
(4)
(6)
96.
1
x 2 10
3
(7)
(21)
Solve the equations and check your results.
97. x 5 7
99. 5x 4x 5
98. x 9 3
100. 3x 9 2x
101. 5x 3 4x 2
102. 9x 2 8x 7
103. 7x 5 6x 4
104. 3 4x 1 x 7 2x
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105. 4(2x 3) 7x 5
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Self-Test for Chapter 2
Name
Section
The purpose of this self-test is to help you check your progress so that you can find sections
and concepts that you need to review before the next in-class exam. Allow yourself about
an hour to take this test. At the end of that hour, check your answers against those given in
the back of this text. If you missed a question, notice the section reference that accompanies the answer. Go back to that section and reread the examples until you have mastered
that particular concept.
Represent the integers on the number line shown.
1. 5, 12, 4, 7, 18, 17
Date
ANSWERS
1.
2.
3.
4.
20
10
0
10
20
5.
2. Place the following group of integers in ascending order: 4, 3, 6, 5, 0, 2, 2
7.
Evaluate.
3. 7
6.
4. 7
8.
9.
5. 18 7
6. 187
10.
11.
7. 24 5
12.
Find the opposite of each integer.
8. 40
9. 19
13.
14.
Add.
10. 8 (5)
15.
11. 6 (9)
16.
17.
12. (9) (12)
18.
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Subtract.
19.
13. 9 15
14. 9 15
15. 5 (4)
16. 7 (7)
Multiply.
17. (8)(5)
18. (9)(7)
19. (6)(4)
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ANSWERS
20.
Evaluate each expression.
21.
20.
75
3
21.
36 9
9
22.
(15)(3)
9
23.
9
0
22.
23.
24.
25.
24. 23 4 5
25. 4 52 35
26.
26. 4(2 4)2
27. 16 (4) (5)
27.
28.
28. If x 2, y 1, and z 3, evaluate the expression
29.
Write, using symbols.
30.
29. 5 less than a
9x2y
.
3z
30. The product of 6 and m
31.
31. 4 times the sum of m and n
32.
32. The quotient when the sum of a and b is divided by 3
33.
Identify which are expressions and which are not.
34.
33. 5x 6 4
34. 4 (6 x)
35.
36.
Combine like terms.
37.
35. 8a 7a
38.
37. Subtract 9a2 from the sum of 12a2 and 5a2.
39.
38. Number Problem
36. 10x 8y 9x 3y
Tom is 8 years younger than twice Moira’s age. Write an
expression for Tom’s age. Let x represent Moira’s age.
40.
The length of a rectangle is 4 more than twice the width. Write an
expression for the length of the rectangle.
42.
43.
Tell whether the number shown in parentheses is a solution for the given equation.
40. 7x 3 25
(5)
41. 8x 3 5x 9
44.
Solve the equations and check your results.
42. x 7 4
44. 9x 2 8x 5
210
43. 7x 12 6x
(4)
© 2007 McGraw-Hill Companies
39. Geometry
41.
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Cumulative Review
for Chapters 1 and 2
Name
Section
Date
ANSWERS
The following exercises are presented to help you review concepts from earlier chapters
that you may have forgotten. This section is meant as review material and not as a comprehensive exam. The answers are presented in the back of the text. If you have difficulty
with any of these exercises, be certain to at least read through the summary related to
that section.
1.
2.
3.
1. Give the place value of 7 in 3,738,500.
4.
2. Give the word name for 302,525.
5.
3. Write two million, four hundred thirty thousand as a numeral.
6.
7.
In exercises 4 to 6, name the property of addition that is illustrated.
4. 5 12 12 5
5. 9 0 9
6. (7 3) 8 7 (3 8)
8.
9.
10.
11.
In exercises 7 and 8, perform the indicated operations.
7.
593
275
98
8. Find the sum of 58, 673, 5,325, and 17,295.
In exercises 9 and 10, round the numbers to the indicated place value.
© 2007 McGraw-Hill Companies
9. 5,873 to the nearest hundred
10. 953,150 to the nearest ten thousand
In exercise 11, estimate the sum by rounding to the nearest hundred.
11.
943
3,281
778
2,112
570
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ANSWERS
12.
13.
Evaluate.
14.
12. 5 14
15.
13. 514
14. What is the opposite of 3?
16.
17.
18.
In exercises 15 to 24, evaluate each expression.
15. 12 (6)
16. 7 7
17. 5 (7)
18. 9 (4) (7)
19. 8 (8)
20. 5 (5)
21. (8)(12)
22. (6)(15)
23. 14 (7)
24. 25 0
19.
20.
21.
22.
23.
24.
25.
26.
In exercises 25 to 28, evaluate the expressions if a 5, b 3, c 4, and d 2.
27.
25. 6ad
26. 3b2
27. 3(c 2d)
28.
28.
29.
2a 7d
ab
30.
In exercises 29 and 30, combine like terms.
31.
29. 6x 14 3x 5
30. 4x 8y 2x 7y
32.
31. x 5 17
212
32. 3x 5 2x 3
© 2007 McGraw-Hill Companies
In exercises 31 and 32, solve each equation and check your solution.