1. Write the operation (+, –, , ÷) that corresponds to each phrase.
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.
more than less than 13
divided by twice 25
1-1

2. 1. divided by: ÷ 2. difference: – 3. more than: + 4. product: 
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: = less than 13: 13 – 8 = 5
divided by 4: 16 ÷ 4 = twice 25: 2(25) = 50
1-1

3. Write an algebraic expression for “the sum of n and 8.”
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

4. Relate: ten more than twice a number
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

5. Relate: The total income is 5 times the number of tickets sold.
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

6. Write an equation for the data in the table.
Using Variables
ALGEBRA 1 LESSON 1-1
Write an equation for the data in the table.
Gallons
Miles
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:        m = • g
m = 20g
1-1

7. e = number of slices eaten, = 8 – e 1. p + 4 2. y – 12 3. 12 – m
Using Variables
ALGEBRA 1 LESSON 1-1
pages 6–8  Exercises
11. 9 – n
12.
13. 5n n
15.
16.
17. c = total cost,
n = number of cans,
c = 0.70n
18. p = perimeter,
s = length of a side,
p = 4s
= total length in feet,
n = number of tents,
= 60n
= number of slices left,
e = number of slices eaten,
= 8 – e
1. p + 4
2. y – 12
3. 12 – m
4. 15c 5. 6.
7. x – 23
8. v + 3
9–16. Choice of variable may vary.
9. 2n + 2
10. n – 11 n 82 n 6 n 8 11 n 17 k
1-1

8. 21–24. Choices of variables may vary. Samples are given.
Using Variables
ALGEBRA 1 LESSON 1-1
21–24. Choices of variables may vary. Samples are given.
21. w = number of workers,
r = number of radios,
r = 13w
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,
p = pay,
p = 8n
k – 17
26. 5n + 6.7
27. 37t – 9.85
28.
30. 7 – 3v
31. 5m –
32. +
33. 8 – 9r 3b
4.5 60 w t 7 p 14 q 3
34–38. Answers may vary.
Samples are given.
34. 5 more than q
35. the difference of 3 and t
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
1-1

9. 39–40. Choices of variables may vary. Samples are given.
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),
c = 0.165n
40. t = time in months,
= length in inches,
= 4.1t
41. a. i. yes; 6 = 3 • 2
ii. yes; 6 = 3 + 3
b. Answers may vary. Sample: i; it makes sense that an equation
relating lawns mowed and hours
worked would be a multiple
of the number of lawns mowed.
42–44.  Choices of variables may vary.
42. a. Let a = the amount in dollars
and n = the number of quarters,
a = 0.25q.
b. $3.25
43. a. d = drop height (ft),
f = height of first bounce (ft), f = d
b. 10 ft
44. s = height of second bounce (ft),
s = d 1 2 4
1-1

10. 45. You jog at a rate of 5 miles per hour. How far do you jog
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
49. H
50. A
51. G
52. D
53. F
62. 1
64. 3
66. any four of 23,
29, 31, 37, 41,
43, and 47
1-1

11. Write an algebraic expression for each phrase. 1. 7 less than 9
Using Variables
ALGEBRA 1 LESSON 1-1
Write an algebraic expression for each phrase.
1. 7 less than 9
2. the product of 8 and p
3. 4 more than twice c
Define variables and write an equation to model each situation.
4. The total cost is the number of sandwiches times $3.50.
5. The perimeter of a regular hexagon is 6 times the length of one side.
9 – 7 8p
2c + 4
Let c = the total cost and s = the number of sandwiches; c = 3.5s
Let p = the perimeter and s = the length of a side; p = 6s
1-1

12. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Find each product.
1. 4 • • • • 9
Perform the indicated operations.
– • 1 ÷ 2
7. 4 – – 5 – 4
9. 5 • ÷ 6 • 2
1-2

13. 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
– 7 = (3 + 12) – 7 = 15 – 7 = 8
6. 6 • 1 ÷ 2 = (6 • 1) ÷ 2 = 6 ÷ 2 = 3
7. 4 – = (4 – 2) + 9 = = 11
– 5 – 4 = (10 – 5) – 4 = 5 – 4 = 1
9. 5 • = (5 • 5) + 7 = = 32
÷ 6 • 2 = (30 ÷ 6) • 2 = 5 • 2 = 10
1-2

14. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify – 14 • 3.
– 14 • 3 = – 14 • 3 Simplify the power: 62 = 6 • 6 = 36.
= – 42 Multiply 14 and 3.
= 68 – 42 Add and subtract in order from left to right.
= 26 Subtract.
1-2

15. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Evaluate 5x = 32 ÷ p for x = 2 and p = 3.
5x + 32 ÷ p = 5 • ÷ 3 Substitute 2 for x and 3 for p.
= 5 • ÷ 3 Simplify the power.
= Multiply and divide from left to right.
= 13 Add.
1-2

16. 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      sales tax
C = p + r • p
sales tax rate
C = p + r • p
= • 32 Substitute 32 for p. Change 8% to 0.08 and substitute 0.08 for r.
= Multiply first.
= Then add.
The total cost of the jeans is $34.56.
1-2

17. 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

18. 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.
xz2 = 11 • 162
= (176)2
Simplify within parentheses. Multiply.
= 11 • 256
= 30,976
Simplify.
= 2816
1-2

19. 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[ ] Simplify the power.
= 4[43] Add within brackets.
= 172 Multiply.
1-2

20. 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.
Use the formula A = 3 ( ) to find the total area of all three decks. pa 2
A = 3 ( ) pa 2
= 3 ( )
60 • 8.7
Substitute 60 for p and 8.7 for a.
= 3 ( )
522 2
Simplify the numerator.
= 3(261) Simplify the fraction.
= 783 Multiply.
The total area of all three decks is 783 ft2.
1-2

21. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
pages 12–15  Exercises
1. 59
2. 22
3. 7
4. 113
5. 32
6. 29
7. 21
8. 27
9. 124
10. 15
25. 57
26. 9
29. 18
31. 0
32. 7
33. 81
34. 36
35. 8 cm3
in.3
ft3
cm3
39. 15,000 ft3 m3
41. 15
42. 7
44. 1
45. 51
48. 61
50. 58
51.
52. 7
53. 1
54. 7
13. $37.09
14. $347.10
15. 22
16. 3
17. 44
18. 5
19. 16
20. 4 1 3 5 6 7 15 5 8
1-2

22. 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
61. 27
62. 14
64. a. 135 in.2
b. The frame is
2 in. wide, so
it adds 2 in.
on each side.
65. $.16
cm3
67. a cm3
b cm3
c. about 73%
68. 38
70. 9
71. 10
72. 10
73. a in.3
b. 2.0 in.3
c in.2 3 4 1 3
1-2

23. 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
– (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.
5 + 4 – 1 • 2 = 7
25 ÷ (4 • 1) = 8
25 ÷ = 9
42 – (1 + 5) = 10
(42 – 5) ÷ 1 = 11
= 12
2(4 – 1) + 5 = 13
2 • • 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
– 1 = 20
81. Answers may vary. Samples:
2(4 – 1) – 5 = 1
5 + 2 – (1 + 4) = 2
(24 – 1) ÷ 5 = 3
– 4 = 4
2 • 5 – (1 + 4) = 5
(52 – 1) ÷ 4 = 6
1-2

24. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
82. a. 38 ft2
b. Yes; 2h = 2 h .
No; h = 2h
since b2 = 0.
Yes; h = h =
2 h .
95. 95% %
97. 6%
98. 43
, 150, 240
, 39, 52
b1 + b2 2
2b1 + b2 /
2(b1 + b2)
102–106. Answers may vary.
Samples are given.
102. 8, 24, 56
, 100, 250
, 66, 121
83. D
84. F
85. B
86. H
87. B
88. F
89. c + 2
90. 36m
91. t – 21
92.
93. 50%
94. 34% y 5
1-2

25. Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify each expression.
– 4 • 3 + 6
2. 3(6 + 22) – 5
3. 2[(1 + 5)2 – (18 ÷ 3)]
Evaluate each expression.
4. 4x + 3y for x = 2 and y = 4
5. 2 • p2 + 3s for p = 3 and s = 11
6. xy2 + z for x = 3, y = 6 and z = 4 44 25 60 20 51
112
1-2

26. 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. 2 5 3 8 2 3 5 9
1-3

27. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
= = =
= = =
= = = 3 or
= = =
= 2 ÷ 5 = 0.4
= 3 ÷ 8 = 0.375
= 2 ÷ 3 = 0.6
= 3 + (5 ÷ 9) = 3.5 5 10 1 2
5 • 1
5 • 2
100
5 • 20 25 4 13
25 • 1
25 • 4
325
1000
25 • 13
25 • 40 40 3 8 9
Solutions 20
1-3

28. 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

29. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Which set of numbers is most reasonable for displaying outdoor temperatures?
integers
1-3

30. 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 – . This is not an integer. 2 3
The statement is false.
1-3

31. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Write – , – , and – , in order from least to greatest. 3 4 7 12 5 8
– = –0.75 Write each fraction as a decimal.
– = –0.583
– = –0.625 3 4 7 12 5 8
–0.75 < –0.625 < –0.583 Order the decimals from least to greatest.
From least to greatest, the fractions are – , – , and – . 3 4 7 12 5 8
1-3

32. 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

33. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
pages 20–23  Exercises
1. integers, rational numbers
2. rational numbers
3. rational numbers
4. natural numbers,
whole numbers, integers,
rational numbers
5. rational numbers
6. integers, rational numbers
7. whole numbers, integers, rational numbers
8. rational numbers
9. rational numbers
10. irrational numbers
11. Answers may vary.
Sample: –17
12. Answers may vary.
Sample: 53
13. Answers may vary.
Sample: 0.3
14. rational numbers
15. whole numbers
16. integers
17. whole numbers
18. rational numbers
19. true
20. False; answers may vary.
Sample: –
21. False; answers may vary.
Sample: 6
22. true
23. False; answers may vary.
Sample: –6 < | –6 |
24. >
25. <
26. > 2 3
1-3

34. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
27. =
, 2.01, 2.1
29. –9 , –9 , –9
30. – , – ,
31. –1.01, –1.001, –1.0009
, 0.636,
, ,
34. 4
35. 9
36.
38. 2 3 4 1 5 6 22 25 8 9 7 11 12
39. 0
41.
42. Answers may vary.
Sample:
43. Answers may vary.
44. Answers may vary.
45. Answers may vary. 14
46. Answers may vary.
47. natural numbers,
whole numbers,
integers,
rational numbers
48. natural numbers,
49. rational numbers
50. rational numbers
213 10
1034
1000
– 4
1-3

35. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
51. =
52. >
53. <
54. >
55. =
56. >
57. 6
58. 9
59. a
60. 28
61. 48
62. 5
63. a. irrational
b. No; has no final digit.
64. 0
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.
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.
67. Answers may vary. Sample:
25, |25| + | –25| = 50
68. sometimes
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
1-3

36. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3 g
cm3 g
cm3
92. 25
93. 4
95–98.  Choices of variables may vary.
95. n = number of tickets, 6.25n
96. i = cost of item, i
97. n = number of hours,
d = distance traveled,
d = 7n
98. c = total cost,
n = number of books,
c = 3.5n
77. a , ,
, 3.5
b. aluminum, diamond,
silver, gold
78. Answers may vary.
Samples are given.
a. –2.2
b. –2.81
c. –2
d. Yes; find the average
of the two given numbers.
79. A
80. G
81. D
82. G
83. C
84. I
85. C
86. 33
87. 48
89. 7
90. 0
91. 7 g
cm3 g
cm3 1 8 1 2
1-3

37. Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each given number belongs.
1. –
Use <, =, or > to compare.
4. 5.
6. Find |– |.
rational numbers
irrational numbers
natural numbers, whole numbers
integers, rational numbers 3 4 5 8
> 3 4 5 8 –
< – 7 12 7 12
1-3

38. Adding Real Numbers Find each sum. 1. 4 + 2 2. 10 + 7 3. 9 + 5
ALGEBRA 1 LESSON 1-4
(For help, go to skills handbook page 726.)
Find each sum. 1 5 3 4 9 7 2 8
1-4

39. Adding Real Numbers Solutions 1. 4 + 2 = 6 2. 10 + 7 = 17
ALGEBRA 1 LESSON 1-4
= = 17
= = 59
= = 5.2
= 10.9
= 17.1
= =
= = = 1
= = = = 1
= = = 1 5 3
1 + 3 4 9 7
4 + 7 11 2
2 + 3 8
3 + 2
Solutions
1-4

40. Use a number line to simplify each expression.
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Use a number line to simplify each expression.
Use a number line to simplify each expression.
a. 3 + (–5)
b. –3 + 5
Start at 3.
Move left 5 units.
Start at –3.
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

41. Simplify each expression.
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Simplify each expression.
a (–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

42. 6 + (–11) = –5 The water level fell 5 inches.
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

43. 3.6 + (–t) = 3.6 + [–(–1.7)] Substitute –1.7 for t.
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Evaluate (–t) for t = –1.7.
3.6 + (–t) = [–(–1.7)] Substitute –1.7 for t.
= [1.7] –(–1.7) means the opposite of –1.7, which is 1.7.
= 5.3 Simplify.
1-4

44. Relate: 88 ft below sea level plus feet diver rises
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 = – Substitute 37 for r.
= – 51 Simplify.
The scuba diver is 51 ft below sea level.
1-4

45. Add corresponding elements. =
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
7 –5.4 –2 –1 –
2.3 5 –3
Add +
7 –5.4 –2 –1 – –3 +
– (–5.4) 11 + (–2)
–3 + (–1)
Add corresponding elements. = –4
Simplify. =
1-4

46. Adding Real Numbers pages 27–30 Exercises 1. 6 + (–3); 3 12. –42
ALGEBRA 1 LESSON 1-4
pages 27–30  Exercises
(–3); 3
2. –1 + (–2); –3
3. –5 + 7; 2
(–4); –1
5. 15
6. –11
7. –19
9. –4
10. 5
11. –8 1 6
12. –42
14. –0.65
15. –7.49
17.
18. –
19. 6
20. –6
21. 0
22. –
23. 5
24. –
25. – = –35, 35 ft
below the surface
(–5) = 3, 3 yd gain
27. – = 7, 7°F
29. –1.7
31. –8.7 13 14 14 15 8 9 3 16 1 8 13 18
1-4

47. 36–37. Choices of variable may vary. 36. c = change in temp., –8 + c
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
38.
39.
40.
41.
42. –2.7
43. –13
33. –5.6
35. –12.6
36–37. Choices of variable may vary.
36. c = change in temp., –8 + c
a. –1°F
b. –11°F
c. 11°F
37. c = change in amount
of money, 74 + c
a. $92
b. $45
c. $27
–1 1.4 – 19 24
45. 11
46. 4
47. –3
48. –18.53
49. –20.83
50. –1.72
51. –
52. –5
54. 4
55. –8.8
–18.2
11.6
19.1 22 35 25
–12 17 60
1.8 22
– 7 11
120 1 2 1 3
1-4

48. is positive, while the sum of 227 and –319 is negative.
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
million people
million people
58. Weaving; add the
numbers in each column.
59. a. =
b. 0.23
c. about 23%
60. 0
61. –2
62. 1
63. –5
64. 7
65. 5
66. –1
67. 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.
69. Answers may vary. Sample:
Although 20 and –20 are
opposite numbers, there is
no such thing as opposite
temperatures.
70. –0.3
71. –13.7
72. –0.6
75. –1.9
76. +2
100
442 50
221
1-4

49. 77. Answers may vary. Sample:
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
77. Answers may vary. Sample:
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.
82. a. 4
b. –4
80. (continued) b.
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
2 0 1 –
1-4

50. Adding Real Numbers 83. 84. 85. 86. – 87. 88. – 89. 8 32 –27 20 6 0
ALGEBRA 1 LESSON 1-4
83.
84.
85.
86. –
87.
88. –
89.
8 32 –27
1 –1
2 1 1 4 2 w 10 c
90. –
91.
92.
93.
94. Pos.; if m is neg.,
–m is pos. and the
sum of two pos.
is pos.
95. Neg.; if n is pos.,
–n is neg. and the
sum of two neg.
is neg.
96. Pos.; if m is neg.,
97. Zero; sum of neg.
and pos. is the
difference of the
abs. values.
|n| = |m| so |n| – |m| = 0.
98. zero; n + (–m) = n + (–n) = 0
99. B
100. F
101. D
102. F
103. C
104. H
105. <
106. =
111. 9
107. <
108. >
109. >
110. = x 12 t 6
–3m + 1 m 9
58a 21 2b
1-4

51. Adding Real Numbers Simplify.
ALGEBRA 1 LESSON 1-4
Simplify.
(– 10) 2. – 57 + (– 11) 3. –
4. – (– ) 5.
6. Evaluate x for x = – 5. 5
–68
–23 1 5 4 15
3 –5
–2 0
–2 –4
8 –6 +
1 – 9
6 – 6 –4 7 15 5
1-4

52. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
(For help, go to Lessons 1-3 and 1-4)
Find the opposite of each number.
– –
Simplify.
(–2) (–3.5) (–8) –
( ) 7 19 2 3 1 6
1-5

53. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
1. opposite of 6: –6 2. opposite of –7: 7
3. opposite of 3.79: – opposite of – :
(–2) = (–3.5) = 6
(–8) = 5
– = – = = = 7 19 2 3
( ) 1 6 4
4 + (–1)
Solutions
1-5

54. 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

55. 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

56. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify –11.6 – (–14).
–11.6 – (–14) = – The opposite of –14 is 14.
= 2.4 Add.
1-5

57. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify |–13 – (–21)|.
|–13 – (–21)| = | – |   The opposite of –21 is 21.
= | 8 | Add within absolute value symbols.
= 8 Find the absolute value.
1-5

58. 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

59. 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) Add the opposite.
= –15 Simplify.
The temperature at 10:00 P.M. was –15°C.
1-5

60. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
9. –4
10. 11
11. –10
12. –4
13. –2.1
14. –9.2
17. –
18. –1
19.
20. –
21. 3
22. 8
23. 6
24. 2
25. 13
26. 2
27. 6
28. 3
29. –10
30. 7 19 60
31. 1
32. 1
33. 3
34. 0
35. –7
36. –13
37. $50.64
38. –9.5
39. –1.5
40. –2
pages 34–36  Exercises
1. –1  
2. –2  
3. –6  
4. –5  
5. 1  
6. –11  
7. 14  
8. 3   1 6 1 10 11 12
1-5

61. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
43. –1
44. 16
45. –16
46. 11
47. 1
48. –2 ft
51. Answers may vary. Sample:
52. false; –1 – (–7) = 6, –1 + (–7) = –8
53. false; 2 – (–1) = 3, 3 < 2 or –1
54. true
55. a : 1997: b.
c. Answers may vary. Sample: Invest in soccer;
it is the only sport that has not lost any participants.
1.3 –0.3 0
–0.4 –1.4 –2.2
0.2 –0.3 –2.2
5 12
–3 7
4 6
5 10
1 6
–8 –3 – =
1-5

62. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
56.
57. [–   0   –3]
58.
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)|,
60. –
61.
62. –4x + 9
63. –12t – m
64.
65.
66. |x| – |y|, |x – y|
and x – y, |x + y|
67. C
68. F
69. C
70. H
71. D
72. 4
73. –9
74. –0.7
75. –4.1
76.
77.
78.
2 –2
–9 3 1 4
29r – 35 16
–15w – 8 9 5 12
1 –
– 1
4 0
0 16
1.2
–2.5
5.2 –3
6 –4 1 2 3 = / = / 9 20 37 60
1-5

63. 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 ($),
s = your share, s = c 6
1-5

64. Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Simplify each expression.
1. 7 – – (–8) 3. –7 – 5
Evaluate each expression for x = 3 and y = 4.
4. x – y 5. | – y – x| 6. –
– 2 x
y 4
x 8 – 5 y –5 11
–12 –1 7
–5 –5
9 0
1-5

65. 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

66. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
1. –2 + (–2) + (–2) + (–2) = 4(–2) = –8
2. –5 + (–5) + (–5) + (–5) + (–5) = 5(–5) = –25
3. –6 – 6 – 6 – 6 = 4(–6) = –24
4. –12 – 12 – 12 – 12 – 12 – 12 = 6(–12) = –72
5. 4 – 2 = 2, and 6 – 4 = 2, so the pattern is to add 2: = 8, = 10, and = 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, = 6, and = 12. –18, –12, –6, 0, 6, 12
Solutions
1-6

67. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
b. –6( ) 3 4
a. –3(–11)
The product of a positive
number and a negative
number is negative.
–6( ) = – 3 4 18
–3(–11) = 33
The product of two negative numbers is positive.
= –4 1 2
Write – as a
mixed number. 18 4
1-6

68. 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

69. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Use the expression –5.5( ) to calculate the change in temperature for an increase in altitude a of 7200 ft. a
1000
–5.5( ) = –5.5 ( )
7200
1000 a
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

70. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Use the order of operations to simplify each expression.
a. –0.24
Write as repeated multiplication.
–(0.2 • 0.2 • 0.2 • 0.2) =
= –0.0016
Simplify.
b. (–0.2)4
Write as repeated multiplication.
(–0.2)(–0.2)(–0.2)(–0.2) = =
Simplify.
1-6

71. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
a. 70 ÷ (–5)
The quotient of a positive number and a negative number is negative.
= –14
The quotient of a negative number and a negative number is positive.
= 6
b. –54 ÷ (–9)
1-6

72. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6 x y
Evaluate – – 4z2 for x = 4, y = –2, and z = –4.
– – 4z2 = – 4(–4)2 Substitute 4 for x, –2 for y, and –4 for z. x y –4 –2
= – 4(16) Simplify the power. –4 –2
= 2 – 64 Divide and multiply.
= –62 Subtract.
1-6

73. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6 p r 3 2 3 4
Evaluate for p = and r = – .
= p ÷ r Rewrite the equation. p r
= ÷ Substitute for p and – for r. 3 2 4
(– )
= Multiply by – , the reciprocal of – . 3 2 4
(– )
= –2 Simplify.
1-6

74. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
1. –15
2. –15
3. 15
4. 15
5. –34.4
6. –2
7. –120
8. –105
9. –80
10. 80
11. –78
12. 81
13. –12
14. 12
15. –15
16. 5
17. –8
18. –13
19. 48
20. 1
21. 12
22. –8
23. –432
24. 1
25. –64
26. –87
27. 4
28. 24
29. 36
30. 12
31. a. –24°F
b. –75°F
c. –51°F
d. –31.5°F
32. –1
33. 8
34. –25
35. 81
36. –81
37. –192
38. –5
40. –2
41. –4
42. 5
43. 6
44. –11
pages 41–44  Exercises
45. 12
46. –2
47. –8
48. –1
49. –7
50. –28
51. 0
52. 4
53. 3
54. 1
55. –15 1 2 4 5 3
1-6

75. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
56.
57. –
58. –2
59. 18
60. –
61. –
62. –125
64. –27
66. 44
67. –27 1 18 6 25 2 8 15
68. –60
69. 30
70. a. i. 2
ii. –6
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.
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.
72. –1
73. –
74. 1
75. –
76. –6
77. –4
78. –
79. 5 12 7 10
1-6

76. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
90.
91.
92.
93. [–12  –2 ]
94.
95. – –9 –
18 –12
8 –6 2 3
4.7 – –
–8 –10.6 4
1 – 1 4 16 9
87. a. 4, –8, 16, –32
9, –27, 81, –243
b. Pos.; the expression
will involve an even
number of neg. factors.
c. Neg.; the expression
will involve an odd
is the multiplicative
inverse of 10 because
0.1(10) = 1. (–10 is the
opposite of 10.)
89. The opposite of a nonzero
number n is –n while the
multiplicative inverse is . 1 n
80–84. Answers may vary.
Samples are given.
80. ac + b
81. b – a + c
82. ab – c
83. –ab + c
84. –bc + a
85. 31¢
86. Yes; whatever the
signs of a and b, | ab |, | a |, and | b | are pos., and | ab | = | a | • | b |.
1-6

77. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
96. a ft; 91 ft
b. Less than 4 s;
for t = 4, h = 155 – 16t 2
= –101, so
h = 0 for some value
less than 4 (about 3.1).
97. a. a = 5000 – 25t
b ft
c ft
98. –23°C
99. –
100.
101. –
102.
103. –
104. –8
, or 11
106. –
107. –
108.
109. A
110. F
111. C
112. H
113. B 1 8 16 32 64 45 4 9 5 48 2
114. –2
115. –20
118. 2
119.
123.
127. $34.93 72 3
1-6

78. Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify.
1. –8(–7) 2. –6(–7 + 10) – 4
Evaluate each expression for m = –3, n = 4, and p = –1.
p 4. (mp)3 5. mnp
6. Evaluate 2a ÷ 4b – c for a = –2, b = – , and c = – . 56
– 22 8m n –7 27 12 1 3 1 2 3 1 2
1-6

79. 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) 3. –1(–9 + 8)
4. –0.5(8 – 6) t(10 – 4) 6. m(–3 – 1) 1 2
1-7

80. The Distributive Property
ALGEBRA 1 LESSON 1-7
( )
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
t(10 – 4) = t(6) = (6)t = • 6 t = 3t
6. m(–3 – 1) = m(–4) = –4m 1 2
Solutions
1-7

81. 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

82. 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 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

83. The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify 3(4m – 7).
3(4m – 7) = 3(4m) – 3(7) Use the Distributive Property.
= 12m – 21 Simplify.
1-7

84. 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

85. The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify –2w2 + w2.
–2w2 + w2 = (–2 + 1)w2 Use the Distributive Property.
= –w2 Simplify.
1-7

86. 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

87. The Distributive Property
ALGEBRA 1 LESSON 1-7
pages 50–52  Exercises
2. 663
5. 686
7. 20,582
8. 24,480
9. $3.96
10. $11.82
11. $29.55
12. $209.51
13. $98.97
14. $2.76
15. 7t – 28
16. –2n + 12
17. 3m + 12
18. b –
19. –2x – 6
20. 4y + 6
q + 8
22. 18n – 42
– r
24. –4.5b
25. 2w + 4
26. – n
27. –x – 3
28. –x + 3
29. –3 – x
30. –3 + x
31. –6k – 5
32. –7x + 2
33. –2 + 7x
34. –4 + z 4 5 2 15 16
35. –3t
36. 20k2
37. 7x
38. 24w
39. 5v2
40. 6m
41. –17q
42. –45x
43. 3(m – 7)
44. –4(4 + w)
45. 2(b + 9)
46. –11(n – 8)
1-7

88. The Distributive Property
ALGEBRA 1 LESSON 1-7
47. 2(3c + 9)
48. (3 + r )(r – 7)
49. 44,982 x
d – dh
– n
b + 128 m
d – 42
61. –8.4 – 300g + 512h
62. –12k – 5m + 62
63. 6p – 14q + 30w
64. 36x y – 72.9
(5 – k)
( p)
(b – )
68.
(x – )
70. 8x + 38
71. No; 2a • 2b = 4ab.
72. The student did not mult.
the second number in
parentheses, 10, by 4 to
get the correct answer,
12x + 40.
73. Answers may vary.
Sample: 2(x + 5) = 2x + 10 d
75. –76p2 – 20p – 9
76. –6.1t t
m – 12.5v 5 8 1 4 1 2 7
100 4 3 11 20 13 30 40 7 50 7 17
z – 34 1 3 11 12
1-7

89. The Distributive Property
ALGEBRA 1 LESSON 1-7
n + n3
79. – k h
80. 7m2 – mz + 4
– 7t + 6y
b – 5b2 + 5c
83. –4xyz + 6xy
84. terms: –7t, 6v, 7, –19y,
coefficients: –7, 6, –19,
constant: 7
85. a. ( )50 ft2
b ft2 71 42
pennies, 7 nickels,
4 quarters, 1 dime
87. 4(1.02) + 3(0.99) + 3(0.52)
= t ; $8.61 t
89. –6r + 37
90. –3m – 9
91. 2a + 2ab + abc
92. 14b + 42b2
93. 5y + 13z
94.
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.
107
120 3 20
3x – 10 3
1-7

90. The Distributive Property
ALGEBRA 1 LESSON 1-7
97. D
98. G
99. A
100. F
101. D
102. H
103. C
104. I
105. –68
106. –4
107. 4
108. 7
109.
110. –
111. –
112. –3
113. –17
114. –2.386
120. –9.6 1 6
121. –7.7
123. a. 4 +
b. 7; 5; 8 5 4 m 3 1 5
1-7

91. The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify each expression.
1. 11(299) (x + 8) – 3(2y – 7)
4. –(6 + p) a + 2b – 4c + 3.1b – 4a
6. Write an expression for the product of and the quantity b minus .
3289
4x + 32
– 6y + 21
– 6 – p
–2.7a + 5.1b – 4c 4 7 3 5 4 7 3 5
b –
( )
1-7

92. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
(For help, go to Lessons 1-4 and 1-6.)
Simplify each expression.
(9 + 2) 2. 3 • (–2 • 5)
4. –4(7)(–5) 5. – (–4) • 3 • 4
x – t – 8 + 3t 9. –5m + 2m – 4m
1-8

93. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
Solutions
(9 + 2) = 8 + (2 + 9) = (8 + 2) + 9 = = 19
2. 3 • (–2 • 5) = 3 • (–10) = –30
= = = 26
4. –4(7)(–5) = –4(–5)(7) = 20(7) = 140
5. – (–4) = –6 + (–4) + 9 = – = –1
• 3 • 4 = 0.25 • 4 • 3 = 1 • 3 = 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

94. 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

95. 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 $ Find the total amount you spent.
= Commutative Property of Addition
= ( ) Associative Property of Addition
= Add within parentheses first.
= Simplify.
The total amount spent was $61.95.
1-8

96. 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

97. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
pages 56–57  Exercises
5. Inv. Prop. of Add.;
a number and its
inverse are added.
6. Comm. Prop. of Mult.;
the order of the factors
changes.
7. Dist. Prop.; a number
outside parentheses
is distributed to the
two terms inside
the parentheses.
8. Assoc. Prop. of Mult.;
the grouping of the
factors changes.
9. Inv. Prop. of Mult.;
a number and its
mult. inverse are
multiplied.
14. –5
15. 13
16. $6.00
1. Ident. Prop. of Add.;
0, the identity for
addition, is added.
2. Comm. Prop. of Add.;
the order of the terms
changes.
3. Ident. Prop. of Mult.;
1, the identity for
multiplication,
is multiplied.
4. Assoc. Prop. of Add.;
the grouping of the
terms changes.
1-8

98. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
17. a. def. of subtr.
b. Dist. Prop.
c. addition
18. a. Comm. Prop. of Mult.
b. Assoc. Prop. of Mult.
c. mult.
d. mult.
• 1.7 • 4
= 25 • 4 • 1.7  
Comm. Prop. of Mult.
= (25 • 4) • 1.7  
Assoc. Prop. of Mult.
= 100 • 1.7  mult.
= 170  mult.
20. –5(7y)
= [–5(7)]y  
= –35y  mult.
m + 7
= 9m   
Comm. Prop. of Add.
= 9m + (8 + 7)  
Assoc. Prop. of Add.
= 9m + 15 add.
22. 12x – 3 + 6x
= 12x + (–3) + 6x  def. of subtr.
= 12x + 6x + (–3)  
= (12 + 6)x + (–3)  Dist. Prop.
= 18x + (–3)  add.
= 18x –3  def. of subtr.
23. 29c + (–29c)
= [29 + (–29)]c  Dist. Prop.
= 0 • c  Inv. Prop. of Add.
= 0  Mult. Prop. of Zero
= 1 + 1  
Inv. Prop. of Mult.
= 2  add.
g = 2 + 1 
= 3  add. 1 43 g
1-8

99. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
26. 36jkm – 36mjk
= 36jkm + (–36)mjk  
def. of subtr.
= 36jkm + (–36)jmk  
Comm. Prop. of Mult.
= 36jkm + (–36)jkm  
= [36 + (–36)]jkm  
Dist. Prop.
= (0)jkm  Inv.
Prop. of Add.
= 0  
Mult. Prop. of Zero
27. (32 – 23)(8759)
= (9 – 8)(8759)  mult.
= [9 + (–8)](8759)  
def. of subtr.
= 1(8759)  add.
= 8759  
Ident. Prop. of Mult.
28. (76 – 65)(8 – 8)
= (76 – 65)[8 + (–8)]  
= (76 – 65) • 0  
Inv. Prop. of Add.
= 0  
Mult. Prop. of Zero
(8 – 3m)
= – 18m  
Dist. Prop.
= (–18m)  
def. of subtr.
= (4 + 48) + (–18m)  
Assoc. Prop. of Add.
= 52 + (–18m)  add.
= 52 – 18m  
1-8

100. Properties of Real Numbers
ALGEBRA 1 LESSON 1-8 1 5
30. 5 w – – w(9)
= 5 w – – 9w  
Comm. Prop. of Mult.
= 5(w) – – 9w  
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)  
= –4w + (–1)  add.
= –4w – 1  
31. $52.97
32. no
33. no
34. no
35. yes
36. no
37. no
38. yes
39. yes
40. No; 3 – 5 = –2, while 5 – 3 = 2.
41. No; (5 – 3) – 1 = 2 – 1 = 1, while
5 – (3 – 1) = 5 – 2 = 3.
42. No; 1 ÷ 2 = , while 2 ÷ 1 = 2.
43. No; 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8,
while (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2.
44. a. Dist. Prop.
b. Comm. Prop. of Add.
c. Assoc. Prop. of Add.
d. add.
e. Dist. Prop.
f. add. 2
1-8

101. 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.
46. Answers may vary. Sample:
The sandwich tastes the same
whether the peanut butter or
the jelly is on top. This is like
the Comm. Prop. of Add.,
because you can add the
peanut butter and jelly in
either order.
47. both
48. both
49. both
50. not mult.; (–2)(–3) = 6
51. not add.; = 4
52. both
53. A
54. I
55. D
56. G
57. B
58. F k
– b
61. –10p – 35
– 4n
m – 0.05
64. –18v
[m + (–17)]
66. 8 – (9 – t)
( )
(x + 5.1)
69. 7
70. –18.3
71. –
72. –1
73. –135 1 3 b 4 2 5
1-8

102. 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)
3. –14 • 0 = 0
4. Give a reason to justify each step.
Iden. Prop. Of Mult.
Assoc. Prop. Of Add.
Mult. Prop. Of Zero
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

103. 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. –
Write the coordinate of each point on the number line below.
5. A 6. B 7. C 8. D
1-9

104. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Solutions 1. 2. 3. 4.
5. A: B: – C: D: –4
1-9

105. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Name the coordinates of point A in the graph.
The coordinates of A are (–2, 3).
Move 2 units to the left of the origin. Then move
3 units up.
1-9

106. 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

107. 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

108. 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 Amount
worked earned
Janel 6 $25.50
Roscoe 12 $51.00
Victoria 11 $46.75
Alex 9 $38.25
Jordan 15 $63.75
Jennifer 10 $42.50
For 6 hours worked and earnings of $25.50,
plot (6, 25.50).
The highest amount earned is $ So a reasonable scale on the vertical axis is from
0 to 70 with every 10 points labeled.
1-9

109. 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

110. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
pages 62–64  Exercises
1. (4, 5)
2. (2, –2)
3. (–5, 0)
4. (5, 4)
5-8. 
9. II
10. x–axis
11. IV
12. y–axis
13. I
14. II
15. No; the point is on
the y-axis, not in
Quadrant III.
16.
17. neg. correlation
18. pos. correlation
19. no correlation
20. (–3, 4)
21. (5, 0)
22. (6, –6)
23. (–3, –2), (1, 3), (2, –4)
24. a. Answers may vary.
Sample: (–1, 1),
(–2, 2), (10, –10)
b. Answers may vary.
Sample: (1, 1),
(–2, 2), (–10, –10)
1-9

111. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
29. Neg. correlation;
the more classes
you take, the more
work you have, so
the less free time
you have.
30. Pos. correlation;
the greater the
number of cars,
the higher the
pollution levels
for a city.
31. No correlation;
baby’s length at
birth is not related
to its birthday.
25.
square
26.
rhombus
27.
isosceles triangle
28.
rectangle
1-9

112. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
32. Pos. correlation;
the more you
exercise, the
more calories
you burn.
33. Answers may vary.
Samples: Pos. correlation;
the number of hours a
person earning an hourly
wage works and the size
of the paycheck. Neg.
correlation; the number of people working on a
project and the time it
takes to complete the
project. No correlation;
person’s height and the
length of his or her hair.
34. a. Neg. correlation;
rain or snow makes travel more difficult
or inconvenient, so
voters would be less
likely to go to the polls.
b. Answers may vary.
Sample: In general,
the weather has the
same effect on both
sides, but candidates
usually want as many
votes as possible, so
they should be concerned.
35. a. neg. correlation
b. Answers may vary.
Sample: No; it is
generally not
reasonable to
conclude that
correlation
between two trends
implies cause.
1-9

113. Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
36. a, b.
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.
36. (continued)
d. Answers may vary. Sample: about 200 cal
units; 21 square units
square units
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

114. 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
42. G
43. D
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.
1-9

115. 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 (–4t) + (–37)  def. of subtr.
= 40t + (–4t) (–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

116. 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.
52. [16.5  –29  1  –9.5]
53.
54.
55. true
56. false; 1 = 12
57. true 1 2 1 2
–11 –
10 –17
5 30 1 2
9 –13 –1
3.1 –0.7 21
–4.9 –4.7 –2.1
–1 –7 8
1-9

117. 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 4. G
5. In which quadrant is point E?
6. Describe the trend.
(–3, –3)
(–2, 1)
(2, 0)
(3, 2) II
positive correlation
1-9

118. Tools of Algebra 1. n = number, c = cost, c = 2.3n 2. p = payment,
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.
6. 12
7. 5
8. 6
9. –20
10. –10
11. 1
12. 18
13.
14.
15. False, since is a
rational number
but is not an integer.
16. False, since |0| = 0,
which is not pos.
17. –3 + 18a 5 9
18. 12d – 30
19–20. Answers may vary.
19. 10x + 3( – x)
= 10x + 1 – 3x  
Dist. Prop.
= x – 3x  
Comm. Prop. of Add.
= 1 + (10 – 3)x  
= 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
–1 15 –11
13 –1 2
–10 5 –10
11 –3
2 5 1 2 3 4
1-A

119. Tools of Algebra 21. –10(2 – 11) 22. 23. (p – )( + p) 24. –10 25. 2.7
ALGEBRA 1 CHAPTER 1
21. –10(2 – 11)
22.
23. (p – )( + p)
24. –10
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. 5 8 1 4
m + 6 3
34. neg. correlation
35. about 41°F
36. If a and b have
different signs,
| | is positive
and is negative. a b
27. Answers may vary.
Sample:
– , 1.5, 1, –
– , – , 1, 1.5
28. y-axis
29. Quadrant III
square units
31. 2 yd lost
32. $177
33. $19.30 2 3 1 6
1-A