Chapter 1
INTRODUCTION TO REAL
NUMBERS AND ALGEBRAIC
EXPRESSIONS
1.1 Real Numbers
Exercises
2. The whole numbers consist of 0 and the
natural numbers.
10. $1000+
12. 1
62
+lb.
14. $1500.45−
22.
24.
34. 1.5
36. 15 15−=
38. 11
44
−=
40. 2.6 2.6−=
42. 666
⎛⎞
⎟
⎜
−=− =−
⎟
⎜⎟
⎜
52. True, because 6− is equal to 6.−
54. 4 is to the right of 7,− so 47.>−
56. 1.6− is to the right of 2,− so 1.6 2.−>−
62. 8.6 8.6−= and 7.4 is to the left of 8.6, so
7.4 8.6 .<−
64.
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
8
70. 5
6
72. Impossible; absolute value is always
positive or 0.
74. 7.8− is to the right of 8.2,−so
7.8 8.2.
−>−
82. a. The lever was invented earlier.
b.
Glass, the lever, the compass, and the
windmill were invented between 2000
B.C. and A.D. 1000.
Mindstretchers
1. Answers may vary.
2. a. Yes. 4 is the number.
1.2 Addition of
Real Numbers
Exercises
2. The additive inverse property states that
for any real number a, there is exactly one
8. The associative property of addition states
that regrouping numbers that are added
gives us the same sum.
10.
22. Associative property of addition
24. Commutative property of addition
26.
()
10 6
+−
Section 1.2 Addition of Real Numbers
9
34.
()
()
210
10 2
8
+−
=− −
=−
36. 10 10 0−+ =
42.
(
)
00.30.3+− =−
44.
()
7.2 2.8
7.2 2.8
4.4
−+
=− −
=−
50. 57
612
10 7
12 12
3
12
1
4
−+
=− +
=−
=−
54. 11
61
43
25 4
43
75 16
⎛⎞
⎟
⎜
+− ⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
=+−
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
()
36 45
9
=+−
=−
58.
()
()()
()
45 ( 27) 0 18
45 27 18
45 45
+− + +−
=+− +−
=+−
()()()
()
7641019
793
86
=+−+−+−
=+−
=−
66.
()()
()( )( )
()
10 6.17 10.005 4.519
6.17 10 10.005 4.519
6.17 24.524
18.354
− + +− +−
=+−+− +−
=+−
=−
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
10
68. Additive inverse property
74.
()( )
()( )
()
4.81 0.63 10.002 1.05
4.81 1.05 0.63 10.002
5.86 10.632
4.772
+− +− +
= + +− +−
=+−
=−
78. (5) 15
15 5
10
10 C
−+
=−
=
°
82.
()
()()
()()
$1.82 $0.49 $0.04 $0.77
$1.26 $0.94
$1.82 $0.49 $0.77 $0.94
$0.04 $1.26
++− +
+− +
=+++
+− +−
Mindstretchers
2. a. xyz+=
b. z; the sum of z and any element results in
the same element. That is, ,xz x+=
,yz y+= and .zz z+=
1.3 Subtraction of
Real Numbers
4.
()
69
69
15
−−
=− + −
=−
6.
(
)
()()
49 2
49 2
−−
=− +−
Section 1.3 Subtraction of Real Numbers
11
12.
()
30 1
30 1
29
−−−
=− +
=−
18.
()
25 25
25 25
50
−−
=− + −
=−
24.
()
10.1 11.84
10.1 11.84
1.74
−
=+−
=−
30.
(
)
9.4 2.5
9.4 2.5
11.9
−−
=+
=
34.
1
12 4
1
12 4
⎛⎞
⎟
⎜
−−−
⎟
⎜⎟
⎜
⎝⎠
=− +
36. 11
61
24
−− ⎟
⎜⎟
⎜
⎝⎠
11
61
24
13 5
=+
=+
89
17
=+
=
40.
()
12 16 5
−+−
12 8
4
=− +
=−
44.
()
7412
7412
−−
=+− −
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
12
52.
()
28 17
28 17
11
−−−
=− +
=−
58. 61.8 ( 71)
61.8 71
9.2
−−−
=− +
=
The boiling point is 9.2 C
D higher.
60. Alabama
114 ( 27) 141−− =
66. a.
()
238 196
238 196
42
−−−
=− +
=−
42 C below the boiling point
D
2. a.
In both problems, expressions are rewritten
so that the given operation is replaced by the
opposite (inverse) operation.
b. When the difference is rewritten as a sum,
143
−−
Section 1.4 Multiplication of Real Numbers
13
1.4 Multiplication of
Real Numbers
Exercises
2. The product of two real numbers with
different signs is negative.
)
)
)
10. Commutative property of multiplication
12. Associative property of multiplication
14. Multiplicative identity property
16. Multiplication property of zero
)
3
26. 12 112 22
257 5 7 35
⎛⎞⎛ ⎞⎛⎞⎛ ⎞
⎟⎟ ⎟⎟
⎜⎜ ⎜⎜
−= −=−
⎟⎟ ⎟⎟
⎜⎜ ⎜⎜
⎟⎟ ⎟⎟
⎜⎜ ⎜⎜
⎝⎠⎝ ⎠⎝⎠⎝ ⎠
34.
()( )
()
8.5 0 2.6
02.6
0
−
=
=
36.
(
)
(
)
(
)
(
)
()( )( )( )
()
61 23 4
63 1 2 4
18 8
−− −
=−−−
=−
()
1.0298 4.8
4.94304
4.94
=−
=−
≈−
44.
(
)
()
42 5 3
4103
14 3
−+ − −
=− + − −
=− −
1
4
1
12 20
8
=− +
=
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
14
11
=
56.
()
()
5410
56
56
1
−− −
⎡⎤
⎣⎦
=−−−
⎡⎤
⎣⎦
=−
=−
58. a.
()
62 2
12 2
10
−+
=− +
=−
60.
1
23 2
54
⎛⎞
⎟
⎜−=
⎟
⎜⎟
⎜
⎝⎠ 2
3
54
−
⋅3
10
=−
62.
()()
()
3.51 0.23 6.4
0.8073 6.4
5.16672
5.17
−− −
=−
=−
≈−
64. Commutative property of multiplication
70.
()
$5000 3 $15, 000−=−
72.
()
150 3 450−=−
450 ft below sea level
()
450ft−
74. a.
() ( ) ()
17 5 1 2 2 0
85 2 0
83
+−+
=−+
=
to multiply
()
43,− we start at 0 and move 3
units to the left 4 times. So
(
)
43is12.−−
2. Writing the sum forward and backward, we
see that the sum of the vertical pairs is 1001.−
(
)
(
)
(
)
(
)
(
)
()() ()()()
1 2 3 . . . 998 999 1000
1000 999 998 . . . 3 2 1
− +− +− + +− +− +−
− +− +− + +− +− +−
Using Gauss’s reasoning, we can conclude
that 1000( 1001)− is twice the correct answer.
So, the sum is
(
)
()
1000 1001
2500 1001
−=−
Section 1.5 Division of Real Numbers
15
1.5 Division of
Real Numbers
Exercises
8. To divide two real numbers, multiply the
dividend by the reciprocal of the divisor.
12.
(
)
12 1 12−÷−=
14. 16 4 4−÷=−
16.
()
0100÷− =
18.
()
300 10 30−÷−=
)
)
28. 35 7 , or 0.875
40 8
−=
−
30.
71
12 6
⎛⎞
⎟
⎜
÷− ⎟
⎜⎟
⎜
⎝⎠
30
=−
1
42. 1.8 18 3
0.6 6
−==
−
44.
(
)
6.4516 3.54 1.82−÷−≈
46.
()
0.3102 0.129 2.40÷− ≈−
48.
(
)
(
)
36 3 2
−÷−−
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
16
54.
()()
15 3 2 4
−−÷−−
56.
()()
()
32 8 5 6
430
34
÷− +− ⋅
=− + −
=−
62.
()()
0.7882 2.36 0.33−÷≈−
64. a.
1
8
66.
()
46
46 10 5
222
−− +
===−
−−−
68.
()
1.1 1.5 8.9 1.2
++− +
72. $48 billion $3÷ billion 16=
74.
4
20 15
20
−÷
=−
3
15
41
or 1 lb
=− −
78. $174,000 $183,000
10
$9000
10
$900
=−
=−
The value of the house depreciated by
$900 per year
$900per year .−
3, 1, 2
−
2. 18
2 and 18; 9
2
36
2 and 36; 18
−=−
−
−
−− =
Section 1.6 Algebraic Expressions, Translations, and Exponents
17
1.6 Algebraic Expressions,
Translations, and
Exponents
Exercises
2. A constant is a known quantity whose
value does not change.
10. 1
12. 2
14. 3
16. the sum r and 2
18. the difference between y and 10
20. the product of negative 4 and x
38. 10y+
40. 12n−
42. 7x−
44. 2z+
46. 1
56. xy
xy
−
58.
(
)
(
)
(
)
2
44416−=−−=
60.
(
)
()()( )( )
2
2
43
44 3 3
16 9
−−
=− − −
=− ⋅
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
18
74. 2
55xyyz xyz−⋅⋅⋅=−
3n
86.
()
4x+−
88. 86,ab+++ or 14 ab++
96. 3
eee e⋅⋅=
98.
()
ab cd e++ students
100.
()
21 2xy xy+= + runs
Mindstretchers
1. Answers may vary.
2. Yes; 2 and 4. We have 4
216=
1.7 Evaluating Algebraic
24
=
6. 22
44(3)
4(9)
b−=−
=−
24
=
12.
(
)
(
)
()
4443
41
4
ab−= −
=
=
14.
(
)
2
232
34
1
bc−+ =−+−
=− +
=
16.
(
)
(
)
()
32
32
2224
8216
ca−=−−
=− −
Section 1.7 Evaluating Algebraic Expressions and Formulas
19
22.
() ( ) ( ) ( )
() ( )
2
2
23
2 2 3 0.5 3 1.5
2 2 3 0.5 9 1.5
41.591.5
5.5 9 1.5
3.5 1.5
2
xwyz−−+
=−−−−+
=−−−+
=+ −+
=−+
=− +
=−
26.
(
)
2
2
9
zy
wx
−−
()()
(
)
2
21.5 3
−−−
=−
12.8 1.8
22.8 0.8
32.8 0.2
42.81.2
−=−
−=−
−=
−=
32. 3
5x−
300
34.
5
51
5
10 2
5
=
=
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
20
38. 222aa−− +
2
40.
()
()
532
9
5832
9
CF
=−
=−−
44. 22Plw=+
13
23 22
24
P⎛⎞ ⎛⎞
⎟⎟
⎜⎜
=+
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
46.
4
abcd
A+++
=
50.
150lb
W
CA
=⋅
52.
()
()
2
2
2
2
6
6 2.5 cm
6 6.25 cm
Ae=
=
=
()
()
300
4
386
4
−=
−=−
60. 23
43
ca
ba
++
=
−−−
Section 1.8 Simplifying Algebraic Expressions
21
64.
(
)
2Plw=+
66.
2
s
R
A
=
74. a. 22
15
fm=
b.
()
22 60 88ft/sec
15
f==
87 lb
=
Mindstretchers
1.
34 1 34 7 1 F T
)
)
)
a bababa baba b aba b ab++++=++≥+
−
2. 1
2
n
C+
=
3. Answers may vary.
1.8 Simplifying Algebraic
Expressions
Exercises
10. 18−
12. 1
14. 1−
4
20. 1; 10−
22. Terms: 10 and ; likerr
24. Terms: 30 and 2 ; unlikeA−
32.
(
)
(
)
312 3 12
312
xx x
xx
−⋅=⋅+−⋅
=−
34.
(
)
(
)
0.4 5 0.4 0.4 5
0.4 2
pp
p
−+=−⋅+−⋅
=− −
Chapter 1 Introduction to Real Numbers and Algebraic Expressions
22
38.
()
515
6
yy y
y
−− =−−
=−
46. 2
35xx− The terms are not like terms.
This cannot be simplified.
48.
(
)
22 22 22
22
212
3
mn mn mn
mn
+=+
=
56.
()()()
25 12 15
25
xx
x
−+=−⋅ +−⋅
=− −
58.
(
)
59 459 4
413
xx xx
x
−− + = −−−
=−
60.
(
)
(
)
52 1 4 10 5 4 4
14 4 5
bcbbcb
bc
−+ + = −+ +
=+−
66.
(
)
[
]
[]
52 7 4 52 74
523 7
zz zz
z
⎡
⎤
+−+ + =+−−+
⎣⎦
=+ −
9
n
=− −
74.
()()
563215663
11 3
aaaa
a
−+ += −+ +
=−
76. 11 11
11
cccc c
⎛⎞
⎟
⎜
++ + = ++ +⎟
⎜⎟
82.
(
)
43100 43003
300
yyyy
y
+−=+−
=+
(
)
300 roomsy+
Mindstretchers
1. a.
Parentheses indicate the operation of
Section 1.8 Simplifying Algebraic Expressions
23
3. a. Combining like terms means
combining like measures – feet with
feet and inches with inches.