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1
Summary
DEFINITION /P ROCEDURE
EXAMPLE
From Arithmetic to Algebra
Addition
x y means the sum of x and y or x plus y. Some other words
indicating addition are “more than” and “increased by.”
Subtraction
x y means the difference of x and y or x minus y. Some other
words indicating subtraction are “less than” and “decreased by.”
Multiplication
xy
(x)(y) These all mean the product of x and y or x times y.
xy
Division
x
means x divided by y or the quotient when x is divided by y.
y
REFERENCE
Section 1.1
The sum of x and 5 is x 5.
7 more than a is a 7.
b increased by 3 is b 3.
p. 54
The difference of x and 3
is x 3.
5 less than p is p 5.
a decreased by 4 is a 4.
p. 54
The product of m and n is mn.
The product of 2 and the sum
of a and b is 2(a b).
p. 55
n
n divided by 5 is .
5
The sum of a and b, divided
ab
by 3, is
.
3
p. 57
The Properties of Signed Numbers
Section 1.2
The Commutative Properties
If a and b are any numbers,
1. a b b a
2. a b b a
3883
2552
p. 63
3 (7 12) (3 7) 12
2 (5 12) (2 5) 12
p. 63
6 (8 15) 6 8 6 15
p. 65
The Associative Properties
If a, b, and c are any numbers,
1. a (b c) (a b) c
2. a (b c) (a b) c
© 2001 McGraw-Hill Companies
The Distributive Property
If a, b, and c are any numbers, a(b c) a b a c
Adding and Subtracting Signed Numbers
Adding Signed Numbers
1. If two numbers have the same sign, add their absolute values.
Give the sum the sign of the original numbers.
2. If two numbers have different signs, subtract their absolute
values, the smaller from the larger. Give the sum the sign of
the number with the larger absolute value.
Section 1.3
9 7 16
(9) (7) 16
p. 72
15 (10) 5
(12) 9 3
p. 73
Continued
131
132
CHAPTER 1
THE LANGUAGE OF ALGEBRA
DEFINITION /P ROCEDURE
EXAMPLE
REFERENCE
Adding and Subtracting Signed Numbers
Subtracting Signed Numbers
1. Rewrite the subtraction problem as an addition problem by
a. Changing the subtraction symbol to an addition symbol
b. Replacing the number being subtracted with its opposite
2. Add the resulting signed numbers as before.
Section 1.3
16 8 16 (8)
8
8 15 8 (15)
7
9 (7) 9 7
2
Multiplying and Dividing Signed Numbers
Multiplying Signed Numbers
Multiply the absolute values of the two numbers.
1. If the numbers have different signs, the product is negative.
2. If the numbers have the same sign, the product is positive.
Dividing Signed Numbers
Divide the absolute values of the two numbers.
1. If the numbers have different signs, the quotient is negative.
2. If the numbers have the same sign, the quotient is positive.
p. 77
Section 1.4
5(7) 35
(10)(9) 90
8 7 56
(9)(8) 72
32
4
75
5
20
5
18
9
p. 89
p. 90
8
15
4
2
Evaluating Algebraic Expressions
p. 92
Section 1.5
Algebraic Expressions
An expression that contains numbers and letters
(called variables).
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.
p. 103
Evaluate 2x 3y if x 5 and
y 2.
2x 3y
2 5 (3)(2)
10 6 4
Adding and Subtracting Terms
p. 103
Section 1.6
Term
p. 115
A number or the product of a number and one or more variables.
1. Add or subtract the coefficients (the numbers multiplying
the variables).
2. Attach the common variable.
52
8a 5a 3a
85
Multiplying and Dividing Terms
Property 1 of Exponents
a m a n a mn
p. 117
Section 1.7
27 23 273 210
p. 123
37
37 3 34
33
p.125
Property 2 of Exponents
am
am n
an
© 2001 McGraw-Hill Companies
5x 2x 7x
Combining Like Terms
To combine like terms:
Summary Exercises
This exercise set is provided to give you practice with each of the objectives of the chapter. Each exercise is keyed to the
appropriate chapter section. The answers are provided in the Instructor’s Manual. Your instructor will give you guidelines
on how to best use these exercises.
[1.1] Write, using symbols.
1. 5 more than y
2. c decreased by 10
3. The product of 8 and a
4. The quotient when y is divided by 3
5. 5 times the product of m and n
6. The product of a and 5 less than a
7. 3 more than the product of 17 and x
8. The quotient when a plus 2 is divided by
a minus 2
Identify which are expressions and which are not.
9. 4(x 3)
10. 7 8
11. y 5 9
12. 11 2(3x 9)
[1.2] Identify the property that is illustrated by each of the following statements.
13. 5 (7 12) (5 7) 12
14. 2(8 3) 2 8 2 3
15. 4 (5 3) (4 5) 3
16. 4 7 7 4
Verify that each of the following statements is true by evaluating each side of the equation separately and comparing the
results.
17. 8(5 4) 8 5 8 4
18. 2(3 7) 2 3 2 7
19. (7 9) 4 7 (9 4)
20. (2 3) 6 2 (3 6)
21. (8 2) 5 8(2 5)
22. (3 7) 2 3 (7 2)
© 2001 McGraw-Hill Companies
Use the distributive law to remove parentheses.
23. 3(7 4)
24. 4(2 6)
25. 4(w v)
26. 6(x y)
27. 3(5a 2)
28. 2(4x2 3x)
[1.3] Add.
29. 3 (8)
30. 10 (4)
33. 18 0
34.
3
11
8
8
31. 6 (6)
32. 16 (16)
35. 5.7 (9.7)
36. 18 7 (3)
133
134
CHAPTER 1
THE LANGUAGE OF ALGEBRA
Subtract.
37. 8 13
38. 7 10
39. 10 (7)
40. 5 (1)
41. 9 (9)
42. 0 (2)
43. 5
17
4
4
44. 7.9 (8.1)
Find the median for each of the following sets.
45. 2, 4, 9, 10, 15
46. 7, 3, 2, 4, 5
47. 3, 8, 4, 1, 6
48. 6, 3, 2, 5, 1
49. 2, 4, 1, 8, 6, 7
50. 3, 1, 5, 3, 4, 1
Determine the range for each of the following sets.
52. 4, 5, 6, 4, 2, 1
53. 5, 2, 1, 3, 8
54. 7, 3, 5, 3, 4
55. (10)(7)
56. (8)(5)
57. (3)(15)
58. (1)(15)
59. (0)(8)
60.
51. 3, 5, 1, 8, 9
[1.4] Multiply.
32
2
3
61. (4)
8
3
62.
4(1)
5
Divide.
63.
80
16
64.
63
7
65.
81
9
66.
0
5
67.
32
8
68.
7
0
70.
6 1
5 (2)
71.
25 4
5 (2)
Perform the indicated operations.
69.
8 6
8 (10)
72. 18 3 5
73. (18 3) 5
74. 5 42
75. (5 4)2
76. 5 32 4
77. 5(32 4)
78. 5(4 2)2
79. 5 4 22
80. (5 4 2)2
81. 3(5 2)2
82. 3 5 22
83. (3 5 2)2
© 2001 McGraw-Hill Companies
Evaluate each of the following expressions.
SUMMARY EXERCISES
[1.5] Evaluate the expressions if x 3, y 6, z 4, and w 2.
84. 3x w
85. 5y 4z
86. x y 3z
87. 5z 2
88. 3x2 2w2
89. 3x3
90. 5(x2 w2)
91.
6z
2w
92.
2x 4z
yz
3x y
wx
94.
x(y2 z2)
(y z)(y z)
95.
y(x w)2
x2 2xw w2
93.
[1.6] List the terms of the expressions.
96. 4a3 3a2
97. 5x2 7x 3
Circle like terms.
98. 5m 2, 3m, 4m 2, 5m 3, m 2
99. 4ab2, 3b2, 5a, ab2, 7a2, 3ab2, 4a2b
Combine like terms.
100. 5c 7c
101. 2x 5x
102. 4a 2a
103. 6c 3c
104. 9xy 6xy
105. 5ab2 2ab2
106. 7a 3b 12a 2b
107. 6x 2x 5y 3x
108. 5x 17x 2x 8x
3
2
3
2
109. 3a3 5a2 4a 2a3 3a2 a
110. Subtract 4a3 from the sum of 2a3 and 12a3.
111. Subtract the sum of 3x2 and 5x 2 from 15x 2.
© 2001 McGraw-Hill Companies
[1.7] Divide.
112.
x10
x3
113.
a5
a4
114.
x2 x3
x4
115.
m2 m3 m4
m5
116.
18p7
9p5
117.
24x17
8x13
118.
30m7n5
6m2n3
119.
108x9y4
9xy4
120.
48p5q3
6p3q
121.
52a5b3c5
13a4c
122. (4x3)(5x4)
123. (3x)2(4xy)
124. (8x2y3)(3x3y2)
125. (2x3y3)(5xy)
126. (6x4)(2x 2y)
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CHAPTER 1
THE LANGUAGE OF ALGEBRA
Write the algebraic expression that answers the question.
[1.1–1.7]
127. Carpentry. If x feet (ft) are cut off the end of a board that is 23 ft long, how much is left?
128. Money. Joan has 25 nickels and dimes in her pocket. If x of these are dimes, how many of the
coins are nickels?
129. Age. Sam is 5 years older than Angela. If Angela is x years old now, how old is Sam?
130. Money. Margaret has $5 more than twice as much money as Gerry. Write an expression for the amount of money
that Margaret has.
131. Geometry. The length of a rectangle is 4 meters (m) more than the width. Write an expression for the length of the
rectangle.
132. Number problem. A number is 7 less than 6 times the number n. Write an expression for the number.
133. Carpentry. A 25-ft plank is cut into two pieces. Write expressions for the length of each piece.
134. Money. Bernie has x dimes and q quarters in his pocket. Write an expression for the amount of money that Bernie
© 2001 McGraw-Hill Companies
has in his pocket.