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Using Variables
ALGEBRA 1 LESSON 1-1
Write the operation (+, –, , ÷) that corresponds to each phrase.
1. divided by
2. difference
3. more than
4. product
5. minus
6. sum
7. multiplied by 8. quotient
Find each amount.
9. 12 more than 9
10. 8 less than 13
11. 16 divided by 4
12. twice 25
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Solutions
1.
divided by: ÷
2.
difference: –
3.
more than: +
4.
product: 
5.
minus: –
6.
sum: +
7.
multiplied by: 
8.
quotient: ÷
9.
12 more than 9: 9 + 12 = 21
10. 8 less than 13: 13 – 8 = 5
11. 16 divided by 4: 16 ÷ 4 = 4
12. twice 25: 2(25) = 50
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Write an algebraic expression for “the sum of n and 8.”
“Sum” indicates addition.
Add the first number, n, and the second number, 8.
n+8
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Define a variable and write an algebraic expression for “ten
more than twice a number.”
Relate: ten more than twice a number
Define: Let y = the number.
Write:
2 • y + 10
2y + 10
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Write an equation to show the total income from selling
tickets to a school play for $5 each.
Relate: The total income is 5 times the number of tickets sold.
Define: Let t = the number of tickets sold.
Let i = the total income
Write:
i =
5• t
i = 5t
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Write an equation for the data in the table.
Gallons
Miles
4
6
8
10
80
120
160
200
Relate: Miles traveled equals 20 times the number of gallons.
Define: Let m = the number of miles traveled.
Let g = the number of gallons.
Write:
20 • g
m =
m = 20g
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
pages 6–8 Exercises
1. p + 4
11. 9 – n
19.
2. y – 12
12. n
3. 12 – m
13. 5n
4. 15c
14. 13 + 2n
5. n
8
6. 17
k
15. n
7. x – 23
8. v + 3
9–16. Choice of variable
may vary.
9. 2n + 2
10. n – 11
= total length in feet,
n = number of tents,
82
= 60n
20.
= number of slices left,
e = number of slices eaten,
6
16. 11
n
=8–e
17. c = total cost,
n = number of cans,
c = 0.70n
18. p = perimeter,
s = length of a side,
p = 4s
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
21–24. Choices of variables
may vary. Samples are given.
21. w = number of workers,
25. 9 + k – 17
26. 5n + 6.7
34–38. Answers may vary.
Samples are given.
r = number of radios,
27. 37t – 9.85
34. 5 more than q
r = 13w
28. 3b
35. the difference of 3 and t
22. n = number of tapes,
c = cost, c = 8.5n
23. n = number of sales,
t = total earnings,
t = 0.4n
24. n = number of hours,
4.5
29. 15 + 60
w
30. 7 – 3v
31. 5m – t
7
p
q
32.
+
14
3
33. 8 – 9r
p = pay,
p = 8n
1-1
36. one more than the product
of 9 and n
37. the quotient of y and 5
38. the product of 7
times h and b
Using Variables
ALGEBRA 1 LESSON 1-1
39–40. Choices of variables may vary.
Samples are given.
39. n = number of days,
c = change in height (m),
worked would be a multiple
of the number of lawns mowed.
42–44. Choices of variables may vary.
Samples are given.
42. a. Let a = the amount in dollars
c = 0.165n
and n = the number of quarters,
40. t = time in months,
a = 0.25q.
= length in inches,
= 4.1t
41. a. i. yes; 6 = 3 • 2
b. $3.25
43. a. d = drop height (ft),
f = height of first bounce (ft), f = 1 d
2
ii. yes; 6 = 3 + 3
b. 10 ft
b. Answers may vary. Sample: i;
it makes sense that an equation
relating lawns mowed and hours
44. s = height of second bounce (ft),
s =1 d
4
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
45–47. Answers may vary.
Samples are given.
45. You jog at a rate of 5 miles
per hour. How far do you jog
in 2 hours? Let d = distance
in miles and t = time in hours.
46. Anabel is three years older
than her brother Barry. How
old will Anabel be when Barry
is 12? Let a = Anabel’s age in
years and b = Barry’s age
in years.
47. The Merkurs have budgeted
$40 for a baby sitter. What
hourly rate can they afford to
pay if they need the sitter for
5 hours? Let h = the number
of hours and c the cost per hour.
48. D
60. 0.14
49. H
61. 0.93
50. A
62. 1
51. G
63. 0.23
52. D
64. 3
53. F
65. 0.11
54. 0.9
66. any four of 23,
55. 1.04
29, 31, 37, 41,
56. 1.35
43, and 47
57. 1.46
58. 0.63
59. 0.09
1-1
Using Variables
ALGEBRA 1 LESSON 1-1
Write an algebraic expression for each phrase.
1. 7 less than 9
9–7
2. the product of 8 and p
8p
3. 4 more than twice c
2c + 4
Define variables and write an equation to model each situation.
4. The total cost is the number of sandwiches times $3.50.
Let c = the total cost and s = the number of sandwiches; c = 3.5s
5. The perimeter of a regular hexagon is 6 times the length of one side.
Let p = the perimeter and s = the length of a side; p = 6s
1-1
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Find each product.
1. 4 • 4
2. 7 • 7
3. 5 • 5
Perform the indicated operations.
5. 3 + 12 – 7
6. 6 • 1 ÷ 2
7. 4 – 2 + 9
8. 10 – 5 – 4
9. 5 • 5 + 7
10. 30 ÷ 6 • 2
1-2
4. 9 • 9
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Solutions
1. 4 • 4 = 16
2. 7 • 7 = 49
3. 5 • 5 = 25
4. 9 • 9 = 81
5. 3 + 12 – 7 = (3 + 12) – 7 = 15 – 7 = 8
6. 6 • 1 ÷ 2 = (6 • 1) ÷ 2 = 6 ÷ 2 = 3
7. 4 – 2 + 9 = (4 – 2) + 9 = 2 + 9 = 11
8. 10 – 5 – 4 = (10 – 5) – 4 = 5 – 4 = 1
9. 5 • 5 + 7 = (5 • 5) + 7 = 25 + 7 = 32
10. 30 ÷ 6 • 2 = (30 ÷ 6) • 2 = 5 • 2 = 10
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 32 + 62 – 14 • 3.
32 + 62 – 14 • 3 = 32 + 36 – 14 • 3 Simplify the power: 62 = 6 • 6 = 36.
= 32 + 36 – 42
Multiply 14 and 3.
= 68 – 42
Add and subtract in order from left to right.
= 26
Subtract.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Evaluate 5x = 32 ÷ p for x = 2 and p = 3.
5x + 32 ÷ p = 5 • 2 + 32 ÷ 3
Substitute 2 for x and 3 for p.
=5•2+9÷3
Simplify the power.
= 10 + 3
Multiply and divide from left to right.
= 13
Add.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Find the total cost of a pair of jeans that cost $32 and have
an 8% sales tax.
total cost
original price
C
=
p
+
sales tax
r•p
sales tax rate
C=p+r•p
= 32 + 0.08 • 32
Substitute 32 for p. Change 8% to 0.08 and
substitute 0.08 for r.
= 32 + 2.56
Multiply first.
= 34.56
Then add.
The total cost of the jeans is $34.56.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 3(8 + 6) ÷ (42 – 10).
3(8 + 6) ÷ (42 – 10) = 3(8 + 6) ÷ (16 – 10)
Simplify the power.
= 3(14) ÷ 6
Simplify within parentheses.
= 42 ÷ 6
Multiply and divide from left to right.
=7
Divide.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Evaluate each expression for x = 11 and z = 16.
a. (xz)2
b. xz2
(xz)2 = (11 • 16)2
Substitute 11 for x and 16 for z.
= (176)2
Simplify within parentheses. Multiply.
= 30,976
Simplify.
1-2
xz2 = 11 • 162
= 11 • 256
= 2816
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify 4[(2 • 9) + (15 ÷ 3)2].
4[(2 • 9) + (15 ÷ 3)2] = 4[18 + (5)2]
First simplify (2 • 9) and (15 ÷ 3).
= 4[18 + 25]
Simplify the power.
= 4[43]
Add within brackets.
= 172
Multiply.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
A carpenter wants to build three decks in the shape of
regular hexagons. The perimeter p of each deck will be 60 ft. The
perpendicular distance a from the center of each deck to one of the
sides will be 8.7 ft.
pa
Use the formula A = 3 ( 2
pa
A=3( 2
=3(
) to find the total area of all three decks.
)
60 • 8.7
)
2
Substitute 60 for p and 8.7 for a.
522
Simplify the numerator.
=3( 2
= 3(261)
)
Simplify the fraction.
= 783
Multiply.
The total area of all three decks is 783 ft2.
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
pages 12–15 Exercises
23. 185
35. 8 cm3
47. 242
1. 59
12. 169
24. 361
36. 91 in.3
48. 61
2. 22
13. $37.09
25. 57
37. 21 ft3
49. 71.6
3. 7
14. $347.10
26. 9
38. 28 cm3
50. 58 1
4. 113
15. 22
27. 1936
39. 15,000 ft3
51. 5
5. 32
16. 3
28. 3872
40. 2596
6. 29
17. 44
29. 18
41. 15
7. 21
18. 5
30. 186
42. 7
8. 27
19. 16
31. 0
43. 111
9. 124
20. 4
32. 7
44. 1
10. 15
21. 704
33. 81
45. 51
11. 180
22. 7744
34. 36
46. 50.4
1-2
m3
3
6
52. 7
53. 1 7
15
54. 7 5
8
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
55. a. left side = 1
right side = 1
b. left side = 4
right side = 2
c. Answers may vary.
Sample: For a = 2 and
b = 3, left side = 25,
right side = 13
d. No; as seen in part (b),
(a + b)2 = a2 + b2 is
not true for all values
of a and b.
56. 17
57. 9
58. 143
59. 135
67. a. 523.60 cm3
60. 143
b. 381.70 cm3
61. 27
c. about 73%
62. 14 3
68. 38
63. 308
69. 127
64. a. 135 in.2
70. 9
4
b. The frame is
71. 10
2 in. wide, so
72. 10 1
it adds 2 in.
73. a. 23.89 in.3
on each side.
65. $.16
66. 407.72 cm3
1-2
3
b. 2.0 in.3
c. 47.38 in.2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
74. Yes; the rules for
simplifying are
designed to
produce exactly
one result.
75. (10 + 6) ÷ 2 – 3 = 5
76. 14 – (2 + 5) – 3 = 4
77. (32 + 9) ÷ 9 = 2
78. (6 – 4) ÷ 2 = 1
79. a. 22; 22
b. No; part (a) shows
the value is unaffected
for the given numbers.
80. a. 16, 2
b. Yes; part (a) shows
that the placement of
parentheses can
affect the value of
the expression,
when both add. and
subtr. are involved.
81. Answers may vary. Samples:
2(4 – 1) – 5 = 1
5 + 2 – (1 + 4) = 2
(24 – 1) ÷ 5 = 3
1+2+5–4=4
2 • 5 – (1 + 4) = 5
(52 – 1) ÷ 4 = 6
1-2
5+4–1•2=7
25 ÷ (4 • 1) = 8
25 ÷ 4 + 1 = 9
42 – (1 + 5) = 10
(42 – 5) ÷ 1 = 11
1 + 2 + 4 + 5 = 12
2(4 – 1) + 5 = 13
2 • 5 + 1 • 4 = 14
5(4 – 12) = 15
2(1 + 5) + 4 = 16
5 + 4(2 + 1) = 17
4(5 – 1) + 2 = 18
4 • 5 – 12 = 19
42 + 5 – 1 = 20
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
82. a. 38 ft2
b1 + b2
2
b1 + b2
=2 h
2
No; h 2b1 + b2 =/ 2h b1 + b2
2
2
since b2 =/ 0.
b. Yes; 2h
.
95. 95%
105. 60, 150, 240
96. 145%
106. 26, 39, 52
97. 6%
98. 43
99. 18.9
Yes; h 2(b1 + b2) = h 2(b1 + b2) =
2
2
2 h b1 + b2
.
2
100. 1.25
101. 60.3
102–106. Answers may vary.
Samples are given.
83. D
87. B
91. t – 21
84. F
88. F
92. y
102. 8, 24, 56
85. B
89. c + 2
93. 50%
103. 55, 100, 250
86. H
90. 36m
94. 34%
104. 44, 66, 121
5
1-2
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify each expression.
1. 50 – 4 • 3 + 6
44
2. 3(6 + 22) – 5
25
3. 2[(1 + 5)2 – (18 ÷ 3)]
60
Evaluate each expression.
4. 4x + 3y for x = 2 and y = 4
20
5. 2 • p2 + 3s for p = 3 and s = 11
51
6. xy2 + z for x = 3, y = 6 and z = 4
112
1-2
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
(For help, go to skills handbook page 725.)
Write each decimal as a fraction and each fraction as a decimal.
1. 0.5
5.
2
5
2. 0.05
6.
3
8
3. 3.25
7.
1-3
2
3
4. 0.325
8. 3 5
9
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Solutions
1. 0.5 = 5 = 5 • 1 = 1
10
5•2
2
2. 0.05 = 5 = 5 • 1 = 1
100 5 • 20
20
3. 3.25 = 3 25 = 3 25 • 1 = 3 1 or 13
25 • 4
4
100
4
4. 0.325 = 325 = 25 • 13 = 13
1000
5.
25 • 40
40
2 = 2 ÷ 5 = 0.4
5
3 = 3 ÷ 8 = 0.375
8
7. 2 = 2 ÷ 3 = 0.6
3
8. 3 5 = 3 + (5 ÷ 9) = 3.5
9
6.
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each number belongs.
a. –13
b. 3.28
integers
rational numbers
rational numbers
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Which set of numbers is most reasonable for displaying
outdoor temperatures?
integers
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Determine whether the statement is true or false. If it is false,
give a counterexample.
All negative numbers are integers.
A negative number can be a fraction, such as –
The statement is false.
1-3
2
. This is not an integer.
3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Write – 3 , – 7 , and – 5 , in order from least to greatest.
4
12
8
– 3 = –0.75
Write each fraction as a decimal.
–0.75 < –0.625 < –0.583
Order the decimals from least to greatest.
4
– 7 = –0.583
12
– 5 = –0.625
8
From least to greatest, the fractions are – 3 , – 5 , and – 7 .
4
1-3
8
12
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Find each absolute value.
a. |–2.5|
b. |7|
–2.5 is 2.5 units from 0
on a number line.
7 is 7 units from 0
on a number line.
|–2.5| = 2.5
|7| = 7
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
pages 20–23 Exercises
9. rational numbers
18. rational numbers
1. integers, rational numbers 10. irrational numbers 19. true
2. rational numbers
11. Answers may vary. 20. False; answers may vary.
3. rational numbers
Sample: –17
Sample: – 2
3
4. natural numbers,
whole numbers, integers,
rational numbers
12. Answers may vary. 21. False; answers may vary.
5. rational numbers
13. Answers may vary. 22. true
6. integers, rational numbers
7. whole numbers, integers,
rational numbers
8. rational numbers
Sample: 53
Sample: 0.3
14. rational numbers
Sample: 6
23. False; answers may vary.
Sample: –6 < | –6 |
15. whole numbers
24. >
16. integers
25. <
17. whole numbers
26. >
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
27. =
39. 0
28. 2.001, 2.01, 2.1
40. 1295
29. –9 3 , –9 2 , –9 7
41. 4
4
3
30. – 5 , – 1 , 2
6
2 3
12
31. –1.01, –1.001, –1.0009
32. 0.63, 0.636, 7
11
33. 22 , 0.8888, 8
25
9
34. 4
35. 9
46. Answers may vary.
Sample: – 4
1
5
42. Answers may vary.
Sample: 1
5
47. natural numbers,
whole numbers,
integers,
rational numbers
43. Answers may vary.
Sample: 5
1
44. Answers may vary.
Sample: 213
48. natural numbers,
whole numbers,
integers,
rational numbers
10
36. 9
45. Answers may vary.
49. rational numbers
37. 0.5
Sample: 1034
50. rational numbers
14
1000
38. 3
5
1-3
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
51. =
63. a. irrational
52. >
b. No;
has no final digit.
68. sometimes
53. <
64. 0
54. >
65. Neg.; 0 is the point between
R and S since the coordinates
of Q and T are opposites,
so if R is to the left of 0,
then R is neg.
55. =
56. >
57. 6
58. 9
59. a
60. 28
61. 48
67. Answers may vary. Sample:
25, |25| + | –25| = 50
66. Q; 0 is the point to the right
of S because the coordinates
of R and T are opposites,
therefore the point Q is the
farthest from 0, so it has the
greatest absolute value.
62. 5
1-3
69. sometimes
70. sometimes
71. always
72. Yes; all can be expressed as
ratios of themselves to 1.
73. –6
74. 17
75. 1
76. 10
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
g
g
cm
cm
77. a. 2.75 cm3 , 19.3 cm3 ,
10.5 g 3 , 3.5 g 3
80. G
92. 25
81. D
93. 4
b. aluminum, diamond,
82. G
silver, gold
83. C
78. Answers may vary.
84. I
94. 195
95–98. Choices of variables may vary.
95. n = number of tickets, 6.25n
Samples are given.
85. C
a. –2.2
86. 33
96. i = cost of item, i + 3.98
b. –2.81
87. 48
c. –2 1
8
d. Yes; find the average
of the two given numbers.
88. 315
97. n = number of hours,
d = distance traveled,
d = 7n
79. A
89. 7
90. 0
91. 7 1
2
1-3
98. c = total cost,
n = number of books,
c = 3.5n
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each given number belongs.
1. –2.7
2.
rational numbers
11
3. 16
irrational numbers
Use <, =, or > to compare.
4. 3
4
>
5.
5
8
–3 < – 5
4
6. Find |– 7 |.
12
7
12
1-3
8
natural numbers,
whole numbers
integers,
rational numbers
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
(For help, go to skills handbook page 726.)
Find each sum.
1. 4 + 2
2. 10 + 7
3. 9 + 5
4. 27 + 32
5. 0.4 + 0.9
6. 5.2 + 0
7. 4.1 + 6.8
8. 7.6 + 9.5
9. 1 + 3
10. 4 + 7
11. 1 + 3
12.
9
9
2
4
1-4
5
5
3 + 1
8
4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Solutions
1. 4 + 2 = 6
2. 10 + 7 = 17
3. 9 + 5 = 14
4. 27 + 32 = 59
5. 0.4 + 0.9 = 1.3
6. 5.2 + 0 = 5.2
7. 4.1 + 6.8 = 10.9
8. 7.6 + 9.5 = 17.1
9. 1 + 3 = 1 + 3 = 4
5
5
5
5
10. 4 + 7 = 4 + 7 = 11 = 1 2
9
9
9
9
9
11. 1 + 3 = 2 + 3 = 2 + 3 =
2
4
4
4
4
12. 3 + 1 = 3 + 2 = 3 + 2 =
8
4
8
8
8
5 = 11
4
4
5
8
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Use a number line to simplify each expression.
b. –3 + 5
a. 3 + (–5)
Start at 3.
Start at –3.
Move left 5 units.
Move right 5 units.
3 + (–5) = –2
–3 + 5 = 2
c. –3 + (–5)
Start at –3.
Move left 5 units.
–3 + (–5) = –8
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Simplify each expression.
a. 12 + (–23) = –11
b. –6.4 + (–8.6) = –15.0
The difference of the absolute
values is 11.
Since both addends are negative,
add their absolute values.
The negative addend has the
greater absolute value, so the
sum is negative.
The sum is negative.
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
The water level in a lake rose 6 inches and then fell 11 inches.
Write an addition statement to find the total change in water level.
6 + (–11) = –5
The water level fell 5 inches.
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Evaluate 3.6 + (–t) for t = –1.7.
3.6 + (–t) = 3.6 + [–(–1.7)]
Substitute –1.7 for t.
= 3.6 + [1.7]
–(–1.7) means the opposite of –1.7, which is 1.7.
= 5.3
Simplify.
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
A scuba diver who is 88 ft below sea level begins to ascend
to the surface.
a. Write an expression to represent the diver’s depth below sea level after
rising any number of feet.
Relate:
88 ft below sea level plus feet diver rises
Define:
Let r = the number of feet the diver rises.
Write:
–88 + r
–88
+
r
b. Find the new depth of the scuba diver after rising 37 ft.
–88 + r = –88 + 37
Substitute 37 for r.
= – 51
Simplify.
The scuba diver is 51 ft below sea level.
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Add
–6 + 7
= 2.3 + 11.1
=
1
13.4
3.2
8
–6
2.3
8.6
5
11
–3
–6
2.3
8.6
5
11
–3
8.6 + (–5.4)
5+3
+
7
11.1
–5.4
3
–2
–1
+
7
11.1
–5.4
3
–2
–1
11 + (–2)
–3 + (–1)
9
–4
Add corresponding elements.
Simplify.
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
pages 27–30 Exercises
1
6
24. – 13
14
1. 6 + (–3); 3
12. –42
2. –1 + (–2); –3
13. 2.2
3. –5 + 7; 2
14. –0.65
4. 3 + (–4); –1
15. –7.49
25. –47 + 12 = –35, 35 ft
below the surface
5. 15
16. 1.33
26. 8 + (–5) = 3, 3 yd gain
6. –11
17. 14
27. –6 + 13 = 7, 7°F
7. –19
8. 12.14
9. –4
10. 5
11. –8
23. 5
15
18. – 8
9
19. 6 3
16
20. –6 1
8
28. 8.7
29. –1.7
30. 1.7
31. –8.7
21. 0
32. 12.6
22. – 13
18
1-4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
33. –5.6
38.
34. 5.6
35. –12.6
39.
36–37. Choices of variable may vary.
36. c = change in temp., –8 + c
–1
–21
40.
37. c = change in amount
of money, 74 + c
41.
7
43. –13
b. $45
44. 6.6
1-4
50. –1.72
– 1
a. $92
c. $27
49. –20.83
22
42. –2.7
46. 4
48. –18.53
1.8
2
24
35
0
25
–12
b. –11°F
45. 11 19
47. –3 22
–18.2
11.6
19.1
a. –1°F
c. 11°F
1.4
23.2
51. – 17
60
52. –5 11
120
53. 0.8
54. 4 1
3
55. –8.8
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
56. 13.8 million people
65. 5
72. –0.6
57. 6.3 million people
66. –1
73. 8.7
58. Weaving; add the
numbers in each column.
67. 1
74. 0.1
68. The sum of –227 and 319;
the sum of –227 and 319
is positive, while the sum
of 227 and –319 is negative.
75. –1.9
59. a. 100 = 50
442
221
b. 0.23
c. about 23%
62. 1
69. Answers may vary. Sample:
Although 20 and –20 are
opposite numbers, there is
no such thing as opposite
temperatures.
63. –5
70. –0.3
64. 7
71. –13.7
60. 0
61. –2
1-4
76. +2
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
77. Answers may vary. Sample: 80. (continued)
b. 13 5
2 0
1
–1 3 0.5
18 4
6
78. The matrices are not the
same size, so they can’t
be added.
79. No; time and temperature
are different quantities and
can’t be added.
80. a.
8
10
4
3
2
1
5
2
0
1
1
1
5
8
2
2
2
1
1
0
0
1
1
1
2
82. a. 4
6
2
0
2
2
2
c. 4 employees
d. 10 employees
e. Answers may vary. Sample:
Multiply the entries in each
column by the appropriate
hourly wage, then by 8, and
then add all entries to find
the total wages.
f. $3230
81. $7
1-4
b. –4
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
83.
11
4
21
2
84.
81
2
20
85. w
86.
87.
88.
89.
10
–c
2
58a
21
– 2b
9
x
12
–1
90. – x
1
2
96. Pos.; if m is neg.,
–m is pos. and the
sum of two pos.
is pos.
12
91. t
1
32
–27
61
0
2
6
92. –3m + 1
4
93. m
9
97. Zero; sum of neg.
and pos. is the
94. Pos.; if m is neg.,
difference of the
abs. values.
–m is pos. and the
|n| = |m| so |n| – |m| = 0.
sum of two pos.
is pos.
98. zero; n + (–m) = n + (–n) = 0
95. Neg.; if n is pos.,
103. C 107. < 111. 9
–n is neg. and the 99. B
sum of two neg.
100. F 104. H 108. > 112. 2.2
is neg.
101. D 105. < 109. > 113. 18
102. F
1-4
106. =
110. = 114. 21
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Simplify.
2. – 57 + (– 11)
–68
1. 15 + (– 10)
5
4. – 1 1 + (– 3
5
–4
7
15
4
)
15
5.
3
–2
–5
0
6. Evaluate 25 + 4x for x = – 5.
5
1-4
+
3. – 42 + 19
–23
–2
8
–4
–6
1
6
–9
–6
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
(For help, go to Lessons 1-3 and 1-4)
Find the opposite of each number.
1. 6
2. –7
3. 3.79
4. – 7
6. 9.5 + (–3.5)
7. 13 + (–8)
8.
19
Simplify.
5. 3 + (–2)
1-5
1
2
+ (– 6 )
3
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Solutions
1. opposite of 6: –6
2. opposite of –7: 7
3. opposite of 3.79: –3.79
4. opposite of – 7 : 7
5. 3 + (–2) = 1
6. 9.5 + (–3.5) = 6
19
7. 13 + (–8) = 5
8.
2 +
3
(– 16 )
= 4 + (– 1 ) = 4 + (–1) = 3 = 1
6
6
6
1-5
6
2
19
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Subtract –3 – 2 using a number line.
Start at –3.
Move left 2 units
–3 – 2 = –5
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Subtract 4 – (–2) using tiles.
Start with 4 positive tiles.
Add zero pairs until there are 2 negative tiles.
Remove 2 negative tiles.
There are 6 positive tiles left.
4 – (–2) = 6
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify –11.6 – (–14).
–11.6 – (–14) = –11.6 + 14
= 2.4
The opposite of –14 is 14.
Add.
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify |–13 – (–21)|.
|–13 – (–21)| = | –13 + 21|
The opposite of –21 is 21.
=|8|
Add within absolute value symbols.
=8
Find the absolute value.
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Evaluate x – (–y) for x = –3 and y = –6.
x – (–y) = –3 – [–(–6)]
Substitute –3 for x and –6 for y.
= –3 – 6
The opposite of –6 is 6.
= –9
Subtract.
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
The temperature in Montreal, Canada, at 6:00 P.M. was –8°C.
Find the temperature at 10:00 P.M. if it fell 7°C.
Find the temperature at 10:00 P.M. by subtracting 7°C from the
temperature at 6:00 P.M.
–8 – 7 = –8 + (–7)
= –15
Add the opposite.
Simplify.
The temperature at 10:00 P.M. was –15°C.
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
pages 34–36 Exercises
9. –4
20. – 19
31. 1
1. –1
10. 11
21. 3
32. 1
2. –2
11. –10
22. 8
33. 3
12. –4
23. 6
34. 0
13. –2.1
24. 2
35. –7
4. –5
14. –9.2
25. 13
36. –13
5. 1
15. 13.7
26. 2
37. $50.64
16. 3.5
27. 6
38. –9.5
17. – 1
28. 3
39. –1.5
18. –1 1
29. –10
40. –2
19. 11
30. 7
41. 5.5
3. –6
6. –11
7. 14
60
6
10
8. 3
12
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
42. 1.5
52. false; –1 – (–7) = 6, –1 + (–7) = –8
43. –1
53. false; 2 – (–1) = 3, 3 < 2 or –1
44. 16
54. true
45. –16
55. a. 1992:
5.5
1.4
4.2
1.6
46. 11
47. 1
48. –2
49. 12.5
1997:
8.2
3.2
3.8
5.2
4.9
3.9
1.3
5.1
7.9
1.8
4.1
4.9
4.9
1.7
1.3
2.9
1.3 –0.3
0
–0.4 –1.4 –2.2
1.4
0.3
0
0.2 –0.3 –2.2
c. Answers may vary. Sample: Invest in soccer;
it is the only sport that has not lost any participants.
50. 650 ft
51. Answers may vary. Sample:
5
–3
6.8
1.0
5.6
1.8
b.
12
7
–
4
5
6
10
=
1
–8
6
–3
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
56.
–2
3
2
–9
57. [– 1
0
58.
5
12
4
1 –
–3]
–1
59. a. Yes, answers may vary. Sample:
n and –n always have the same
absolute value.
b. No, answers may vary. Sample:
|1| + | –1| =/ | 1 + (–1)|, 2 =/ 0
60. – 9
20
61. 37
60
62. –4x + 9
73. –9
63. –12t – m
74. –0.7
64. 29r – 35
16
65. –15w – 8
9
75. –4.1
76.
66. |x| – |y|, |x – y|
and x – y, |x + y|
4 0
0 16
77.
1.2
–2.5
5.2
78.
70. H
–3 2
71. D
6
67. C
68. F
69. C
72. 4
1-5
1
2
–3
1
1
–4
3
3
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
79–81. Choices of variables may vary.
79. t = total cost,
p = pounds of pears,
t = 1.19p
80. p = bouquet’s price,
m = money left,
m = 20 – p
81. c = check ($),
c
s = your share, s = 6
1-5
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify each expression.
1. 7 – 12
–5
2. 3 – (–8)
11
3. –7 – 5
–12
Evaluate each expression for x = 3 and y = 4.
4. x – y
–1
5. | – y – x|
6.
–2
y
x
4
7
–5
9
1-5
–5
0
–
x
–5
8
y
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
(For help, go to Lessons 1-4 and 1-5.)
Simplify each expression.
1. –2 + (–2) + (–2) + (–2)
2. –5 + (–5) + (–5) + (–5) + (–5)
3. –6 – 6 – 6 – 6
4. –12 – 12 – 12 – 12 – 12 – 12
Write the next three numbers in each pattern.
5. 2, 4, 6, • , • , •
6. 6, 4, 2,• , • , •
7. 12, 9, 6, • , • , •
8. –18, –12, –6, • , • , •
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Solutions
1.
2.
3.
4.
–2 + (–2) + (–2) + (–2) = 4(–2) = –8
–5 + (–5) + (–5) + (–5) + (–5) = 5(–5) = –25
–6 – 6 – 6 – 6 = 4(–6) = –24
–12 – 12 – 12 – 12 – 12 – 12 = 6(–12) = –72
5. 4 – 2 = 2, and 6 – 4 = 2, so the pattern is to add 2: 6 + 2 = 8,
8 + 2 = 10, and 10 + 2 = 12. 2, 4, 6, 8, 10, 12
6. 4 – 6 = –2, and 2 – 4 = –2, so the pattern is to subtract 2:
2 – 2 = 0, 0 – 2 = –2, and –2 – 2 = –4. 6, 4, 2, 0, –2, –4
7. 9 – 12 = –3, and 6 – 9 = –3, so the pattern is to subtract 3: 6 – 3 = 3,
3 – 3 = 0, and 0 – 3 = –3. 12, 9, 6, 3, 0, –3
8. –12 – (–18) = 6, and –6 – (–12) = 6, so the pattern is to add 6:
–6 + 6 = 0, 0 + 6 = 6, and 6 + 6 = 12. –18, –12, –6, 0, 6, 12
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
3
a. –3(–11)
–3(–11) = 33
b. –6( 4 )
–6( 3 ) = – 18 The product of a positive
4
4 number and a negative
number is negative.
The product of
two negative
numbers is
positive.
1
= –4 2
Write – 18 as a
4
mixed number.
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Evaluate 5rs for r = –18 and s = –5.
5rs = 5(–18)(–5)
Substitute –18 for r and –5 for s.
= –90(–5)
5(–18) results in a negative number, –90.
= 450
–90(–5) results in a positive number, 450.
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Use the expression –5.5(
a
1000
) to calculate the change in
temperature for an increase in altitude a of 7200 ft.
a
–5.5( 1000
7200
) = –5.5 ( 1000 )
Substitute 7200 for a.
= –5.5(7.2)
Divide within parentheses.
= –39.6°F
Multiply.
The change in temperature is –39.6°F.
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Use the order of operations to simplify each expression.
a. –0.24 = –(0.2 • 0.2 • 0.2 • 0.2)
= –0.0016
b. (–0.2)4 = (–0.2)(–0.2)(–0.2)(–0.2)
= 0.0016
Write as repeated multiplication.
Simplify.
Write as repeated multiplication.
Simplify.
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
a.
70 ÷ (–5) = –14
b.
The quotient of a positive number
and a negative number is negative.
1-6
–54 ÷ (–9) = 6
The quotient of a negative
number and a negative number
is positive.
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
x
Evaluate – y – 4z2 for x = 4, y = –2, and z = –4.
–4
– x – 4z2 = –2 – 4(–4)2
y
–4
Substitute 4 for x, –2 for y, and –4 for z.
= –2 – 4(16)
Simplify the power.
= 2 – 64
Divide and multiply.
= –62
Subtract.
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Evaluate p for p = 3 and r = – 3 .
r
p
=p÷r
r
2
4
Rewrite the equation.
3
(– 34 )
Substitute 2 for p and – 4 for r.
3
4
Multiply by – 3 , the reciprocal of – 4 .
= 2 ÷
= 2 (– 3
= –2
)
3
3
4
Simplify.
1-6
3
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
pages 41–44 Exercises
23. –432
45. 12 4
33. 8
5
1. –15
12. 81
24. 1
34. –25
46. –2
2. –15
13. –12
25. –64
35. 81
47. –8
3. 15
14. 12
26. –87
36. –81
48. –1
4. 15
15. –15
27. 4
37. –192
49. –7
5. –34.4
16. 5
28. 24
38. –5
50. –28 1
2
6. –2 1
2
17. –8
29. 36
39. 675
51. 0
7. –120
18. –13
30. 12
40. –2
52. 4 2
8. –105
19. 48
9. –80
20. 1
10. 80
21. 12
31. a.
b.
c.
d.
11. –78
22. –8
32. –1
1-6
–24°F
–75°F
–51°F
–31.5°F
41. –4
42. 5 1
2
43. 6
44. –11
3
53. 3 1
2
54. 1 1
3
55. –15
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
56. 1
68. –60
57. – 6
69. 30
58. –2
70. a. i. 2
18
25
59. 18
60. – 1
2
61. – 8
15
62. –125
63. 0.75
64. –27
65. 22.32
66. 44
67. –27
71. a. When a and b are both neg.
or both pos.
b. When a is neg. and b is pos.,
or when a is pos. and b is neg.
ii. –6
72. –1 5
12
73. – 5
6
74. 1 7
8
75. – 1
5
iii. 24
iv. –120
b. pos.
c. neg.
d. No; answers may vary.
Sample: The sign of the
product is not affected by
the number of pos. factors,
only by the number of
neg. factors.
1-6
76. –6
77. –4 1
2
78. – 1
10
79. 1
5
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
80–84. Answers may vary.
Samples are given.
80. ac + b
81. b – a + c
82. ab – c
83. –ab + c
84. –bc + a
90.
b. Pos.; the expression
will involve an even
number of neg. factors.
–22
18
8
91.
–15
c. Neg.; the expression
will involve an odd
number of neg. factors.
92.
87. a. 4, –8, 16, –32
9, –27, 81, –243
88. 0.1 is the multiplicative
85. 31¢
inverse of 10 because
86. Yes; whatever the
0.1(10) = 1. (–10 is the
signs of a and b, | ab |,
opposite of 10.)
| a |, and | b | are pos.,
89. The opposite of a nonzero
and | ab | = | a | • | b |.
number n is –n while the
multiplicative inverse is 1 .
n
1-6
2
3
10
–12
–6
21
–9
4.7
–1.3
–0.02 6.4
0.79
0
93. [–12 –2 2 ]
3
94.
95.
–8
6.2
–10.6
0
1 –1 3
4 16
2
0
9
4
12
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
96. a. 139 ft; 91 ft
102. 1
64
b. Less than 4 s;
103. – 1
64
for t = 4, h = 155 – 16t 2
104. –8
= –101, so
h = 0 for some value
105. 45 , or 11 1
4
4
less than 4 (about 3.1).
106. – 9
16
97. a. a = 5000 – 25t
107. – 5
b. 312.5 ft
48
108. 1
c. 4687.5 ft
2
109. A
98. –23°C
110. F
99. – 1
8
100. 1
16
101. – 1
32
114. –2
126. 196
115. –20
127. $34.93
116. 0.8
117. 9.8
118. 2 1
4
119. 1
72
120. 4.95
121. 56
122. 4.59
111. C
123. 3
112. H
124. 23
113. B
125. 121
4
1-6
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify.
1. –8(–7)
2. –6(–7 + 10) – 4
– 22
56
Evaluate each expression for m = –3, n = 4, and p = –1.
3. 8m + p
n
–7
4. (mp)3
5. mnp
27
12
1
2
6. Evaluate 2a ÷ 4b – c for a = –2, b = – 1 , and c = – .
3
1
32
1-6
The Distributive Property
ALGEBRA 1 LESSON 1-7
(For help, go to Lessons 1-2 and 1-6.)
Use the order of operations to simplify each expression.
1. 3(4 + 7)
2. –2(5 + 6)
4. –0.5(8 – 6)
5.
1
t(10 – 4)
2
1-7
3. –1(–9 + 8)
6. m(–3 – 1)
The Distributive Property
ALGEBRA 1 LESSON 1-7
Solutions
1. 3(4 + 7) = 3(11) = 33
2. –2(5 + 6) = –2(11) = –22
3. –1(–9 + 8) = –1(–1) = 1
4. –0.5(8 – 6) = –0.5(2) = –1
5.
1
1
1
1
t(10 – 4) = t(6) = (6)t = ( • 6) t = 3t
2
2
2
2
6. m(–3 – 1) = m(–4) = –4m
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Use the Distributive Property to simplify 26(98).
26(98) = 26(100 – 2)
Rewrite 98 as 100 – 2.
= 26(100) – 26(2)
Use the Distributive Property.
= 2600 – 52
Simplify.
= 2548
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Find the total cost of 4 CDs that cost $12.99 each.
4(12.99) = 4(13 – 0.01)
Rewrite 12.99 as 13 – 0.01.
= 4(13) – 4(0.01)
Use the Distributive Property.
= 52 – 0.04
Simplify.
= 51.96
The total cost of 4 CDs is $51.96.
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify 3(4m – 7).
3(4m – 7) = 3(4m) – 3(7)
= 12m – 21
Use the Distributive Property.
Simplify.
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify –(5q – 6).
–(5q – 6) = –1(5q – 6)
Rewrite the expression using –1.
= –1(5q) – 1(–6)
Use the Distributive Property.
= –5q + 6
Simplify.
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify –2w2 + w2.
–2w2 + w2 = (–2 + 1)w2
= –w2
Use the Distributive Property.
Simplify.
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Write an expression for the product of –6 and the quantity 7
minus m.
Relate: –6 times the quantity 7 minus m
Write:
–6
•
(7 – m)
–6(7 – m)
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
pages 50–52 Exercises
1. 2412
12. $209.51
2. 663
13. $98.97
3. 5489
14. $2.76
4. 2448
15. 7t – 28
5. 686
16. –2n + 12
6. 2997
17. 3m + 12
7. 20,582
18. b – 4
5
8. 24,480
19. –2x – 6
9. $3.96
20. 4y + 6
10. $11.82
21. 1.5q + 8
11. $29.55
22. 18n – 42
23. 5 – 15 r
35. –3t
24. –4.5b + 13.5
36. 20k2
25. 2w + 4
37. 7x
26. –36 + 16n
38. 24w
27. –x – 3
39. 5v2
28. –x + 3
40. 6m
29. –3 – x
41. –17q
30. –3 + x
42. –45x
31. –6k – 5
43. 3(m – 7)
32. –7x + 2
44. –4(4 + w)
33. –2 + 7x
45. 2(b + 9)
34. –4 + z
46. –11(n – 8)
2
1-7
16
The Distributive Property
ALGEBRA 1 LESSON 1-7
47. 2(3c + 9)
59. 18.6 + 15m
70. 8x + 38
48. (3 + r )(r – 7)
60. 5 d – 42
71. No; 2a • 2b = 4ab.
49. 44,982
61. –8.4 – 300g + 512h
50. 2392
62. –12k – 5m + 62
51. 14.021
63. 6p – 14q + 30w
52. 84.012
64. 36x + 10.8y – 72.9
72. The student did not mult.
the second number in
parentheses, 10, by 4 to
get the correct answer,
12x + 40.
8
53. 11.82
65.
54. 15.992
66.
55. 30.1 + 7x
56. 40 d – 50 dh
7
7
67.
57. 12.8 – n
68.
58. 135b + 128
69.
2 1 (5 1 – k)
4 2
6 7 (8 + 4 p)
100
3
11 (b – 13 )
30
20
17
z – 34
4 1 (x – 11 )
3
12
1-7
73. Answers may vary.
Sample: 2(x + 5) = 2x + 10
74. 4.78d
75. –76p2 – 20p – 9
76. –6.1t 2 + 13.7t
77. 1.5m – 12.5v
The Distributive Property
ALGEBRA 1 LESSON 1-7
78. 71 n + n3
86. 10 pennies, 7 nickels,
42
79. – 107 k + 3 h
20
120
4 quarters, 1 dime
80. 7m2 – mz + 4
87. 4(1.02) + 3(0.99) + 3(0.52)
81. 19 – 7t + 6y
= t ; $8.61
82. 1.4b – 5b2 + 5c
88. 27 + 3t
83. –4xyz + 6xy
89. –6r + 37
84. terms: –7t, 6v, 7, –19y, 90. –3m – 9
coefficients: –7, 6, –19, 91. 2a + 2ab + abc
constant: 7
85. a. (84 + 10)50
b. 4700 ft2
92. 14b + 42b2
ft2
93. 5y + 13z
94. 3x – 10
3
1-7
95. a. 2; 2
b. –2; –2
c. Yes; the two
expressions
are equal for
the values of
a, b, and c in
parts (a) and (b).
96. a. 30; –8
b. 12, –6
c. No; parts (a) and
(b) show the
expressions are
not equal.
The Distributive Property
ALGEBRA 1 LESSON 1-7
97. D
109. 1
121. –7.7
98. G
110. – 5
122. 23
99. A
111.
123. a. 4 + m
100. F
112. –3
101. D
113. –17
102. H
114. –2.386
103. C
115. 12.127
104. I
116. 45.3
105. –68
117. 45.7
106. –4
118. 2.57
107. 4
119. 0.92
108. 7
120. –9.6
6
4
–1
5
3
b. 7; 5; 8
1-7
The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify each expression.
3. – 3(2y – 7)
– 6y + 21
1. 11(299)
3289
2. 4(x + 8)
4x + 32
4. –(6 + p)
5. 1.3a + 2b – 4c + 3.1b – 4a
–6–p
–2.7a + 5.1b – 4c
6. Write an expression for the product of 4 and the quantity b minus 3 .
7
4
3
b
–
(
)
7
5
1-7
5
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
(For help, go to Lessons 1-4 and 1-6.)
Simplify each expression.
1. 8 + (9 + 2)
2. 3 • (–2 • 5)
3. 7 + 16 + 3
4. –4(7)(–5)
5. –6 + 9 + (–4)
6. 0.25 • 3 • 4
7. 3 + x – 2
8. 2t – 8 + 3t
9. –5m + 2m – 4m
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Solutions
1. 8 + (9 + 2) = 8 + (2 + 9) = (8 + 2) + 9 = 10 + 9 = 19
2. 3 • (–2 • 5) = 3 • (–10) = –30
3. 7 + 16 + 3 = 7 + 3 + 16 = 10 + 16 = 26
4. –4(7)(–5) = –4(–5)(7) = 20(7) = 140
5. –6 + 9 + (–4) = –6 + (–4) + 9 = –10 + 9 = –1
6. 0.25 • 3 • 4 = 0.25 • 4 • 3 = 1 • 3 = 3
7. 3 + x – 2 = 3 + (–2) + x = 1 + x
8. 2t – 8 + 3t = 2t + 3t – 8 = (2 + 3)t – 8 = 5t – 8
9. –5m + 2m – 4m = (–5 + 2 – 4)m = –7m
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Name the property each equation illustrates.
a. 3 • a = a • 3
Commutative Property of Multiplication,
because the order of the factors changes
b. p • 0 = 0
Multiplication Property of Zero, because a
factor multiplied by zero is zero
c. 6 + (–6) = 0
Inverse Property of Addition, because the sum of a
number and its inverse is zero
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Suppose you buy a shirt for $14.85, a pair of pants for
$21.95, and a pair of shoes for $25.15. Find the total amount you
spent.
14.85 + 21.95 + 25.15 = 14.85 + 25.15 + 21.95
Commutative Property of Addition
= (14.85 + 25.15) + 21.95 Associative Property of Addition
= 40.00 + 21.95
Add within parentheses first.
= 61.95
Simplify.
The total amount spent was $61.95.
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Simplify 3x – 4(x – 8). Justify each step.
3x – 4(x – 8) = 3x – 4x + 32
Distributive Property
= (3 – 4)x + 32
Distributive Property
= –1x + 32
Subtraction
= –x + 32
Identity Property of Multiplication
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
pages 56–57 Exercises
1. Ident. Prop. of Add.;
0, the identity for
addition, is added.
5. Inv. Prop. of Add.;
a number and its
inverse are added.
6. Comm. Prop. of Mult.;
2. Comm. Prop. of Add.;
the order of the factors
changes.
the order of the terms
changes.
7. Dist. Prop.; a number
3. Ident. Prop. of Mult.;
1, the identity for
multiplication,
is multiplied.
outside parentheses
is distributed to the
two terms inside
the parentheses.
4. Assoc. Prop. of Add.; 8. Assoc. Prop. of Mult.;
the grouping of the
the grouping of the
factors changes.
terms changes.
1-8
9. Inv. Prop. of Mult.;
a number and its
mult. inverse are
multiplied.
10. 100
11. 7400
12. 14.95
13. 4200
14. –5
15. 13
16. $6.00
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
17. a. def. of subtr.
b. Dist. Prop.
c. addition
20. –5(7y)
= [–5(7)]y
Assoc. Prop. of Mult.
= –35y mult.
23. 29c + (–29c)
= [29 + (–29)]c Dist. Prop.
= 0 • c Inv. Prop. of Add.
= 0 Mult. Prop. of Zero
18. a. Comm. Prop. of Mult. 21. 8 + 9m + 7
24. 43 1 + 1 = 1 + 1
43
=
9m
+
8
+
7
b. Assoc. Prop. of Mult.
Inv. Prop. of Mult.
Comm. Prop. of Add.
= 2 add.
c. mult.
= 9m + (8 + 7)
25. 2 + g 1 = 2 + 1
Assoc.
Prop.
of
Add.
d. mult.
g
= 9m + 15 add.
Inv. Prop. of Mult.
19. 25 • 1.7 • 4
22. 12x – 3 + 6x
= 3 add.
= 25 • 4 • 1.7
= 12x + (–3) + 6x def. of subtr.
Comm. Prop. of Mult.
= 12x + 6x + (–3)
= (25 • 4) • 1.7
Comm. Prop. of Add.
Assoc. Prop. of Mult.
= (12 + 6)x + (–3) Dist. Prop.
= 100 • 1.7 mult.
= 18x + (–3) add.
= 170 mult.
= 18x –3 def. of subtr.
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
26. 36jkm – 36mjk
27. (32 – 23)(8759)
29. 4 + 6(8 – 3m)
= 36jkm + (–36)mjk
= (9 – 8)(8759) mult.
= 4 + 48 – 18m
def. of subtr.
= [9 + (–8)](8759)
Dist. Prop.
= 36jkm + (–36)jmk
def. of subtr.
= 4 + 48 + (–18m)
Comm. Prop. of Mult.
= 1(8759) add.
def. of subtr.
= 36jkm + (–36)jkm
= 8759
= (4 + 48) + (–18m)
Comm. Prop. of Mult.
Ident. Prop. of Mult.
Assoc. Prop. of Add.
= [36 + (–36)]jkm
= 52 + (–18m) add.
28. (76 – 65)(8 – 8)
Dist. Prop.
= 52 – 18m
= (76 – 65)[8 + (–8)]
= (0)jkm Inv.
def. of subtr.
def. of subtr.
Prop. of Add.
= (76 – 65) • 0
=0
Inv. Prop. of Add.
Mult. Prop. of Zero
=0
Mult. Prop. of Zero
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
30. 5 w – 1 – w(9)
5
= 5 w – 1 – 9w
5
Comm. Prop. of Mult.
= 5(w) – 5 1 – 9w
5
Dist. Prop.
= 5w – 1 – 9w
Inv. Prop. of Mult.
= 5w + (–1) + (–9w)
def. of subtr.
= 5w + (–9w) + (–1)
Comm. Prop. of Add.
= [5w + (–9w)] + (–1)
Assoc. Prop. of Add.
= [5 + (–9)]w + (–1)
Dist. Prop.
= –4w + (–1) add. 40. No; 3 – 5 = –2, while 5 – 3 = 2.
= –4w – 1
41. No; (5 – 3) – 1 = 2 – 1 = 1, while
def. of subtr.
5 – (3 – 1) = 5 – 2 = 3.
31. $52.97
42. No; 1 ÷ 2 = 1 , while 2 ÷ 1 = 2.
2
32. no
33. no
43. No; 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8,
while (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2.
34. no
44. a. Dist. Prop.
35. yes
b. Comm. Prop. of Add.
36. no
c. Assoc. Prop. of Add.
37. no
d. add.
38. yes
e. Dist. Prop.
39. yes
f. add.
1-8
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
45. By the Comm. Prop. of Mult.,
(b + c)a = a(b + c).
By the Dist. Prop.,
a(b + c) = ab + ac.
By the Comm. Prop. of Mult.,
ab + ac = ba + ca,
so (b + c)a = ba + ca.
49.
50.
51.
52.
53.
54.
55.
46. Answers may vary. Sample:
The sandwich tastes the same 56.
whether the peanut butter or
57.
the jelly is on top. This is like
58.
the Comm. Prop. of Add.,
59.
because you can add the
60.
peanut butter and jelly in
either order.
61.
47. both
62.
63.
48. both
both
not mult.; (–2)(–3) = 6
not add.; 1 + 3 = 4
both
A
I
D
G
B
F
6 + 5k
11 – 1 b
3
–10p – 35
28 – 4n
7.4m – 0.05
1-8
64. –18v + 31.2
65. 7 + [m + (–17)]
66. 8 – (9 – t)
67. 1 ( b )
2 4
68. 1 (x + 5.1)
3
69. 7
70. –18.3
71. – 1
4
72. –1 1
5
73. –135
74. 9.2
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Name the property that each equation illustrates.
1. 1m = m
2. (– 3 + 4) + 5 = – 3 + (4 + 5)
Iden. Prop. Of Mult.
3. –14 • 0 = 0
Assoc. Prop. Of Add.
Mult. Prop. Of Zero
4. Give a reason to justify each step.
a. 3x – 2(x + 5) = 3x – 2x – 10
Distributive Property
b.
= 3x + (– 2x) + (– 10)
Definition of Subtraction
c.
= [3 + (– 2)]x + (– 10)
Distributive Property
d.
= 1x + (– 10)
Addition
e.
= 1x – 10
Definition of Subtraction
f.
= x – 10
Identity Property of Multiplication
1-8
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
(For help, go to Lesson 1-3.)
Graph each number on a number line.
1. 6
2. –5
3. 2.7
4. 0
Write the coordinate of each point on the number line below.
5. A
6. B
7. C
1-9
8. D
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Solutions
1.
2.
3.
4.
5. A: 1
6. B: –2.5
7. C: 3.5
1-9
8. D: –4
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Name the coordinates of point A in the graph.
Move 2 units to the left of the origin. Then move
3 units up.
The coordinates of A are (–2, 3).
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Graph the point B(–4, –2) on the coordinate plane.
Move 4 units to the left of the origin.
Then move 2 units down.
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
In which quadrant or on which axis would you find each point?
a. (2, –5)
b. (6, 0)
Since the x-coordinate is
positive and the y-coordinate
is negative, the point is in
Quadrant IV.
Since the y-coordinate is 0, the
point is on the x-axis.
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
The table shows the
number of hours worked and
the amount of money each
person earned. Make a scatter
plot of the data.
Name
Hours
worked
Janel
6
Roscoe
12
Victoria
11
Alex
9
Jordan
15
Jennifer
10
Amount
earned
$25.50
$51.00
$46.75
$38.25
$63.75
$42.50
For 6 hours worked and earnings of $25.50,
plot (6, 25.50).
The highest amount earned is $63.75. So a
reasonable scale on the vertical axis is from
0 to 70 with every 10 points labeled.
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Use the scatter plot in your answer for Additional Example 4
to answer the following question: Is there a positive correlation,
negative correlation, or no correlation between the number of hours
worked and the amount earned? Explain.
As the number of hours worked increases, the
earnings increase. There is a positive correlation
between hours worked and earnings.
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
pages 62–64 Exercises 11. IV
17. neg. correlation
1. (4, 5)
12. y–axis
2. (2, –2)
13. I
3. (–5, 0)
14. II
4. (5, 4)
15. No; the point is on
the y-axis, not in
Quadrant III.
5-8.
18. pos. correlation
19. no correlation
20. (–3, 4)
21. (5, 0)
22. (6, –6)
23. (–3, –2), (1, 3), (2, –4)
16.
24. a. Answers may vary.
Sample: (–1, 1),
(–2, 2), (10, –10)
b. Answers may vary.
Sample: (1, 1),
(–2, 2), (–10, –10)
9. II
10. x–axis
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
27.
25.
isosceles triangle
square
30. Pos. correlation;
the greater the
number of cars,
the higher the
pollution levels
for a city.
28.
26.
29. Neg. correlation;
the more classes
you take, the more
work you have, so
the less free time
you have.
rhombus
31. No correlation;
baby’s length at
birth is not related
to its birthday.
rectangle
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
32. Pos. correlation;
the more you
exercise, the
more calories
you burn.
35. a.
34. a. Neg. correlation;
rain or snow makes
b.
travel more difficult
or inconvenient, so
voters would be less
33. Answers may vary.
likely to go to the polls.
Samples: Pos. correlation; b. Answers may vary.
the number of hours a
Sample: In general,
person earning an hourly
the weather has the
wage works and the size
same effect on both
of the paycheck. Neg.
sides, but candidates
correlation; the number
usually want as many
of people working on a
votes as possible, so
project and the time it
they should be concerned.
takes to complete the
project. No correlation;
person’s height and the
length of his or her hair.
1-9
neg. correlation
Answers may vary.
Sample: No; it is
generally not
reasonable to
conclude that
correlation
between two trends
implies cause.
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
36. a, b.
36. (continued)
d. Answers may vary.
Sample: about 200 cal
37. 20 units; 21 square units
38. 7.5 square units
b. pos. correlation
c. In general, it appears that
the greater the number of
grams of fat in a serving
of food, the greater the
number of calories.
39. Answers may vary. Sample:
There is a circle with radius 5
and center (2, 3) that contains
the points (–1, –1), (–2, 0),
(–2, 6), (–1, 7), (5, 7), (6, 6), (6, 0),
and (5, –1). (It also contains
the points (–3, 3), (2, 8), (7, 3),
and (2, –2).)
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
40. a. the distance a car traveled
on the Indiana toll road and
the toll charged
b. The points have the same
x-coordinate; that is, they
lie on the same vertical line.
c. The points have the same
y-coordinate; that is, they lie
on the same horizontal line.
d. Pos. correlation; in general,
as distance increases,
the toll increases.
41. C
44. G
45. A
46. H
47. x – 4(2x + 1) – 3 = x – 8x – 4 – 3 Dist. Prop.
= 1x – 8x – 4 – 3 Ident. Prop. of Mult.
= 1x + (–8x) + (–4) + (–3) def. of subtr.
= [1x + (–8x)] + [(–4) + (–3)]
Assoc. Prop. of Add.
= [1 + (–8)]x + (–4) + (–3) Dist. Prop.
= –7x + (–4) + (–3) add.
= –7x + (–7) add.
= –7x –7 def. of subtr.
42. G
43. D
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
48. 5(8t) + 4(9 – t) – 37 = 40t + 36 – 4t – 37 Dist. Prop.
= 40t + 36 + (–4t) + (–37) def. of subtr.
= 40t + (–4t) + 36 + (–37) Comm. Prop. of Add.
= [40t + (–4t)] + [36 + (–37)] Assoc. Prop. of Add.
= [40 + (–4)]t + [36 + (–37)] Dist. Prop.
= 36t + (–1) add.
= 36t – 1 def. of subtr.
49. 8b + 7a – 4b – 9a
= 8b + 7a + (–4b) + (–9a) def. of subtr.
= 8b + (–4b) + 7a + (–9a) Comm. Prop. of Add.
= [8b + (–4)b] + [7a + (–9)a] Assoc. Prop. of Add.
= [8 + (–4)]b + [7 + (–9)]a Dist. Prop.
= 4b + (–2)a add.
= 4b – 2a def. of subtr.
1-9
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
50. 3m2 – (10m + 3m2)
= 3m2 – 1(10m + 3m2) Mult. Prop. of –1
= 3m2 – 10m – 3m2 Dist. Prop.
= 3m2 + (–10m) + (–3m2) def. of subtr.
= 3m2 + (–3m2) + (–10m) Comm. Prop. of Add.
= [3 + (–3)]m2 + (–10m)
Dist. Prop.
= 0m2 + (–10m) Inv. Prop. of Add.
= 0 + (–10m) Mult. Prop. of Zero
= –10m Ident. Prop. of Add.
51.
9
19
3.1
–4.9
–1
1
2
10 1
2
–0.7
–4.7
–7
21
–2.1
8
1-9
1
2
–11
5
–
–17
30
55. true
56. false; 1 = 12
57. true
–13
–1
52. [16.5 –29 1 –9.5]
53.
54.
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Use the graph for 1 – 6.
Name the coordinates of each point.
1. A
2. D
3. F
(–3, –3)
(–2, 1)
(2, 0)
5. In which quadrant is point E?
II
6. Describe the trend.
positive correlation
1-9
4. G
(3, 2)
Tools of Algebra
ALGEBRA 1 CHAPTER 1
1. n = number, c = cost,
c = 2.3n
2. p = payment,
c = change,
c = 10 – p
3. 4
4. 0
5. 5
9
6. 12
7. 5
8. 6
18. 12d – 30
11. 1 4
5
19–20. Answers may vary.
12. 18
13.
14.
11
2
19. 10x + 3( 1 – x)
–3
5
–1
13
–10
3
15
–1
5
–11
2
–10
15. False, since 1 is a
2
rational number
but is not an integer.
9. –20
16. False, since |0| = 0,
which is not pos.
10. –10
17. –3 + 18a
1-A
= 10x + 1 – 3x
Dist. Prop.
= 1 + 10x – 3x
Comm. Prop. of Add.
= 1 + (10 – 3)x
Dist. Prop.
= 1 + 7x subtr.
20. (33 – 33)(1 – 22)
= (33 + (–33))(1 – 22)
def. of subtr.
= 0(1 – 22)
Inv. Prop. of Add.
= 0 Mult. Prop. of Zero
Tools of Algebra
ALGEBRA 1 CHAPTER 1
21. –10(2 – 11)
5
m+6
23. (p – 5 )( 1 + p)
8
4
24. –10 3
8
22.
25. 2.7
26. a. Sometimes; 5 – 3 = 2,
but 3 – 5 = –2.
b. Never; 3 – (–2) = 5
and 8 – (–10) = 18.
c. Sometimes; –8 – (–3) = –5,
but –3 – (–8) = 5.
d. Never; –3 – (8) = –11
and –7 – (10) = –17.
27. Answers may vary.
Sample:
– 2 , 1.5, 1, – 1
6
3
34. neg. correlation
35. about 41°F
36. If a and b have
different signs,
– 2 , – 1 , 1, 1.5
3
6
28. y-axis
29. Quadrant III
30. 28 square units
31. 2 yd lost
32. $177
33. $19.30
1-A
| a | is positive
b
and a is negative.
b