Using Variables

1
Using Variables
ALGEBRA 1 LESSON 1-1
Write the operation (, , ?, ) that corresponds
to each phrase. 1. divided by 2. difference 3. mo
re 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
2
Using Variables
ALGEBRA 1 LESSON 1-1
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
Solutions
1-1
3
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
Using Variables
ALGEBRA 1 LESSON 1-1
Define a variable and write an algebraic
expression for ten more than twice a number.
2y 10
1-1
5
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.
i 5t
1-1
6
Using Variables
ALGEBRA 1 LESSON 1-1
Write an equation for the data in the table.
m 20g
1-1
7
Using Variables
ALGEBRA 1 LESSON 1-1
pages 68  Exercises
19. total length in feet, n number of
tents, 60n 20. number of slices left,
e number of slices eaten, 8 e
11. 9 n 12. 13. 5n 14. 13
2n 15. 16. 17. c total cost, n number of
cans, c 0.70n 18. p perimeter, s length
of a side, p 4s
1. p 4 2. y 12 3. 12 m 4. 15c 5. 6.
7. x 23 8. v 3 916. Choice of variable
may vary. 9. 2n 2 10. n 11
n 6
n 8
11 n
17 k
1-1
8
Using Variables
ALGEBRA 1 LESSON 1-1
2124. 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
3438. 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
Using Variables
ALGEBRA 1 LESSON 1-1
1-1
10
Using Variables
ALGEBRA 1 LESSON 1-1
4547. 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 Anabels age in years
and b Barrys 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 54. 0.9 5
5. 1.04 56. 1.35 57. 1.46 58. 0.63 59. 0.09
60. 0.14 61. 0.93 62. 1 63. 0.23 64. 3 65. 0.
11 66. any four of 23, 29, 31, 37, 41, 43,
and 47
1-1
11
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 4 2. 7 7 3. 5
5 4. 9 9 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
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 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
14
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
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 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
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.
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
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
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)2 418 (5)2 First
simplify (2 9) and (15 3).
418 25 Simplify the power.
443 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.
The total area of all three decks is 783 ft2.
1-2
21
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
pages 1215  Exercises 1. 59 2. 22 3. 7 4. 113
5. 32 6. 29 7. 21 8. 27 9. 124 10. 15 11. 180
23. 185 24. 361 25. 57 26. 9 27. 1936 28. 3872 2
9. 18 30. 186 31. 0 32. 7 33. 81 34. 36
35. 8 cm3 36. 91 in.3 37. 21 ft3 38. 28
cm3 39. 15,000 ft3 40. 2596 m3 41. 15 42. 7 43. 1
11 44. 1 45. 51 46. 50.4
47. 242 48. 61 49. 71.6 50. 58 51. 52. 7 53.
1 54. 7
12. 169 13. 37.09 14. 347.10 15. 22 16. 3 17. 4
4 18. 5 19. 16 20. 4 21. 704 22. 7744
1 3
5 6
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 58. 14
3
59. 135 60. 143 61. 27 62. 14 63. 308 64. a. 13
5 in.2 b. The frame is 2 in. wide, so it
adds 2 in. on each side. 65. .16 66. 407.72
cm3
67. a. 523.60 cm3 b. 381.70 cm3 c. about
73 68. 38 69. 127 70. 9 71. 10 72. 10 73. a.
23.89 in.3 b. 2.0 in.3 c. 47.38 in.2
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 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.
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
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
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 96. 145 97. 6 98. 43 99. 18.9 100. 1
.25 101. 60.3
105. 60, 150, 240 106. 26, 39, 52
102106. Answers may vary. Samples are
given. 102. 8, 24, 56 103. 55, 100,
250 104. 44, 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
1-2
25
Exponents and Order of Operations
ALGEBRA 1 LESSON 1-2
Simplify each expression. 1. 50 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. 1. 0.5 2. 0.05 3. 3.25 4.
0.325 5. 6. 7. 8. 3
1-3
27
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
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.
The statement is false.
1-3
31
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
Write , , and , in order from
least to greatest.
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 2023  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 lt 6 24. gt 25. lt 26.
gt
1-3
34
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
1-3
35
Exploring Real Numbers
ALGEBRA 1 LESSON 1-3
51. 52. gt 53. lt 54. gt 55. 56. gt 57. 6 58.
9 59. a 60. 28 61. 48 62. 5
67. Answers may vary. Sample 25, 25
25 50 68. sometimes 69. sometimes 70. sometim
es 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
92. 25 93. 4 94. 195 9598.  Choices of
variables may vary. 95. n number of tickets,
6.25n 96. i cost of item, i 3.98 97. n
number of hours, d distance traveled, d
7n 98. c total cost, n number of
books, c 3.5n
77. a. 2.75 , 19.3 , 10.5
, 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 88. 315 89. 7 90. 0 91. 7
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. 2.7 2. 11 3. 16 Use lt,
, or gt to compare. 4. 5. 6. Find .
rational numbers
irrational numbers
natural numbers, whole numbers integers,
rational numbers
3 4
5 8
gt
lt


7 12
1-3
38
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
(For help, go to skills handbook page 726.)
1-4
39
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
1-4
40
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
3 (5) 2
3 5 2
c. 3 (5)
3 (5) 8
1-4
41
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
42
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
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
44
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 divers
depth below sea level after rising any number of
feet.
88 r
b.  Find the new depth of the scuba diver after
rising 37 ft.
The scuba diver is 51 ft below sea level.
1-4
45
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
1-4
46
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
pages 2730  Exercises 1. 6 (3) 3 2. 1
(2) 3 3. 5 7 2 4. 3 (4)
1 5. 15 6. 11 7. 19 8. 12.14 9. 4 10
. 5 11. 8
1 6
12. 42 13. 2.2 14. 0.65 15. 7.49 16. 1.33 17
. 18. 19. 6 20. 6 21. 0 22.
23. 5 24. 25. 47 12 35, 35 ft below
the surface 26. 8 (5) 3, 3 yd gain 27. 6
13 7, 7F 28. 8.7 29. 1.7 30. 1.7 31. 8.7 3
2. 12.6
13 14
14 15
8 9
3 16
1 8
13 18
1-4
47
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
38. 39. 40. 41. 42. 2.7 43. 13 44.
6.6
33. 5.6 34. 5.6 35. 12.6 3637. Choices of
variable may vary. 36. c change in temp., 8
c a. 1F b. 11F c. 11F 37. c change
in amount of money, 74 c a.
92 b. 45 c. 27
1 1.4 21 23.2
19 24
45. 11 46. 4 47. 3 48. 18.53 49. 20.83 50.
1.72 51. 52. 5 53. 0.8 54. 4 55. 8.8
22 35
18.2 11.6 19.1
0 25 12
1.8 22 7
11 120
1 2
1 3
1-4
48
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
1-4
49
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
77. Answers may vary. Sample 78. The matrices
are not the same size, so they cant be
added. 79. No time and temperature are
different quantities and cant 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 3 0.5
13 5 6 2 18 4 2 2 6 2 0 2
8 3 5 1 10 2 2 1 4 1 0 1
5 2 1 1 8 2 0 1 2 1 0 1
1-4
50
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
1-4
51
Adding Real Numbers
ALGEBRA 1 LESSON 1-4
Simplify. 1. 15 ( 10) 2. 57 ( 11) 3.
42 19 4. 1 ( 3
) 5. 6. Evaluate 25 4x for x 5.
5
68
23
5
1-4
52
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
(For help, go to Lessons 1-3 and 1-4)
1-5
53
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
1-5
54
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Subtract 3 2 using a number line.
3 2 5
1-5
55
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
Subtract 4 (2) using 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) 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) 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 600
P.M. was 8C. Find the temperature at 1000 P.M.
if it fell 7C.
Find the temperature at 1000 P.M. by subtracting
7C from the temperature at 600 P.M.
8 7 8 (7) Add the opposite.
15 Simplify.
The temperature at 1000 P.M. was 15C.
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
15. 13.7 16. 3.5 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
31. 1 32. 1 33. 3 34. 0 35. 7 36. 13 37. 50
.64 38. 9.5 39. 1.5 40. 2 41. 5.5
pages 3436  Exercises
1. 1   2. 2   3. 6   4. 5   5. 1   6. 11   7.
14   8. 3  
1-5
61
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
52. false 1 (7) 6, 1 (7)
8 53. false 2 (1) 3, 3 lt 2 or
1 54. true 55. a. 1992 1997 b. c. Answe
rs may vary. Sample Invest in soccer it is
the only sport that has not lost any
participants.
42. 1.5 43. 1 44. 16 45. 16 46. 11 47. 1 48.
2 49. 12.5 50. 650 ft 51. Answers may vary.
Sample
5.5 8.2 4.9 1.4 3.2 3.9 4.2 3.8 1.3 1.6 5.2 5.
1
6.8 7.9 4.9 1.0 1.8 1.7 5.6 4.1 1.3 1.8 4.9 2.
9
4 6 5 10
5 12 3 7
1 6 8 3


1-5
62
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
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
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), 2 0 60. 61.
73. 9 74. 0.7 75. 4.1 76. 77. 78.
2 2 9 3
1 4
4 0 0 16
1 1
1.2 2.5 5.2
1-5
63
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
7981. Choices of variables may vary. 79. t
total cost, p pounds of pears, t
1.19p 80. p bouquets price, m money
left, m 20 p 81. c check (), s
your share, s
1-5
64
Subtracting Real Numbers
ALGEBRA 1 LESSON 1-5
5
11
12
1
7
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
1-6
67
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
a. 3(11)
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(7.2) Divide within parentheses.
39.6F Multiply.
The change in temperature is 39.6F.
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
b. (0.2)4
1-6
71
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Simplify each expression.
a. 70 (5)
b. 54 (9)
1-6
72
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Evaluate 4z2 for x 4, y 2, and z
4.
1-6
73
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
Evaluate for p and r .
2 Simplify.
1-6
74
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
1-6
75
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
1-6
76
Multiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6
8084. 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
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. 3. p 4. (mp)3 5.
mnp 6. Evaluate 2a 4b c for a 2, b
, and c .
22
56
7
27
12
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) 5. t(10 4) 6. m(3 1)
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The Distributive Property
ALGEBRA 1 LESSON 1-7
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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
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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.
The total cost of 4 CDs is 51.96.
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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.
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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.
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify 2w2 w2.
2w2 w2 (2 1)w2 Use the Distributive
Property.
w2 Simplify.
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86
The Distributive Property
ALGEBRA 1 LESSON 1-7
Write an expression for the product of 6 and
the quantity 7 minus m.
6(7 m)
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87
The Distributive Property
ALGEBRA 1 LESSON 1-7
1-7
88
The Distributive Property
ALGEBRA 1 LESSON 1-7
47. 2(3c 9) 48. (3 r )(r
7) 49. 44,982 50. 2392 51. 14.021 52. 84.012 53
. 11.82 54. 15.992 55. 30.1 7x 56. d
dh 57. 12.8 n 58. 135b 128
59. 18.6 15m 60. d 42 61. 8.4 300g
512h 62. 12k 5m 62 63. 6p 14q
30w 64. 36x 10.8y 72.9 65. 2 (5
k) 66. 6 (8 p) 67. (b
) 68. 69. 4 (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 74. 4.78d 75. 76p2
20p 9 76. 6.1t 2 13.7t 77. 1.5m 12.5v
5 8
1 4
1 2
4 3
11 20
13 30
40 7
50 7
11 12
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The Distributive Property
ALGEBRA 1 LESSON 1-7
78. n n3 79. k h 80. 7m2
mz 4 81. 19 7t 6y 82. 1.4b 5b2
5c 83. 4xyz 6xy 84. terms 7t, 6v, 7, 19y,
coefficients 7, 6, 19, constant 7 85. a.
(84 10)50 ft2 b. 4700 ft2
86. 10 pennies, 7 nickels, 4 quarters, 1
dime 87. 4(1.02) 3(0.99) 3(0.52) t
8.61 88. 27 3t 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.
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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. 2.3
86 115. 12.127 116. 45.3 117. 45.7 118. 2.57 11
9. 0.92 120. 9.6
121. 7.7 122. 23 123. a. 4 b. 7 5 8
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The Distributive Property
ALGEBRA 1 LESSON 1-7
Simplify each expression. 1. 11(299) 2.
4(x 8) 3. 3(2y 7) 4. (6 p)
5. 1.3a 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
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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
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
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
Solutions
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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
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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.
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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
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
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. 10. 100 11. 740
0 12. 14.95 13. 4200 14. 5 15. 13 16. 6.00
pages 5657  Exercises
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.
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Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
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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   Comm. Prop. of
Mult. 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)   def. of subtr. (76 65)
0   Inv. Prop. of Add. 0   Mult. Prop. of
Zero
29. 4 6(8 3m) 4 48 18m   Dist.
Prop. 4 48 (18m)   def. of subtr. (4
48) (18m)   Assoc. Prop. of Add. 52
(18m)  add. 52 18m   def. of subtr.
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100
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
1-8
101
Properties of Real Numbers
ALGEBRA 1 LESSON 1-8
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
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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 8.
D
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Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
1. 2. 3. 4. 5. A 1 6. B 2.5
7. C 3.5 8. D 4
Solutions
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105
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Name the coordinates of point A in the graph.
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106
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
Graph the point B(4, 2) on the coordinate
plane.
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.
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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.
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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.
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110
Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
pages 6264  Exercises 1. (4, 5) 2. (2,
2) 3. (5, 0) 4. (5, 4) 5-8.  9. II 10.
xaxis
11. IV 12. yaxis 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)
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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 babys length at birth is not
related to its birthday.
27. isosceles triangle 28. rectang
le
25. square 26. rhombus
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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 persons height and
the length of his or her hair.
35. a. neg. correlation b. Answers may vary.
Sample No it is generally not
reasonable to conclude that correlation
between two trends implies cause.
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.
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Graphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9
36. (continued) d. Answers may vary. Sample
about 200 cal 37. 20 units 21 square
units 38. 7.5 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).)
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.
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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.
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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.
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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
11 10 17 5 30
9 13 19 1
3.1 0.7 21 4.9 4.7 2.1 1 7 8
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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
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Tools of Algebra
ALGEBRA 1 CHAPTER 1
1-A
119
Tools of Algebra
ALGEBRA 1 CHAPTER 1
1-A