using variables
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
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
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
Solutions
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
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
1-1
Relate: ten more than twice a number
Write: 2 • y + 10
Define: Let y = the number.
Define a variable and write an algebraic expression for “ten
more than twice a number.”
2y + 10
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
1-1
Write: i = 5 • t
Define: Let t = the number of tickets sold.
Let i = the total income
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.
i = 5t
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
1-1
Define: Let m = the number of miles traveled.Let g = the number of gallons.
Relate: Miles traveled equals 20 times the number of gallons.
Write an equation for the data in the table.
Write: m = 20 • g
m = 20g
Gallons
Miles
4 6 8 10
80 120 160 200
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
1-1
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
pages 6–8 Exercises
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
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
19. = total length in feet,
n = number of tents,
= 60n
20. = number of slices left,
e = number of slices eaten,
= 8 – e
n817 k
1-1
n82
n611n
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
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
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
25. 9 + k – 17
26. 5n + 6.7
27. 37t – 9.85
28.
29. 15 +
30. 7 – 3v
31. 5m –
32. +
33. 8 – 9r
3b 4.5
60w
t 7
p14
q3
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
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. Samples are given.
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
12
14
ALGEBRA 1 LESSON 1-1ALGEBRA 1 LESSON 1-1
Using VariablesUsing Variables
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
54. 0.9
55. 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
ALGEBRA 1 LESSON 1-1ALGEBRA 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
Using VariablesUsing Variables
1-1
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
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
1-2
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
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
Solutions
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.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
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.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
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 taxC = p + 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.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
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.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
1-2
Evaluate each expression for x = 11 and z = 16.
a. (xz)2
= (176)2 Simplify within parentheses. Multiply. = 11 • 256
= 2816= 30,976 Simplify.
(xz)2 = (11 • 16)2 Substitute 11 for x and 16 for z. xz2 = 11 • 162
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
1-2
b. xz2
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.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
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.
= 3(261) Simplify the fraction.
= 783 Multiply.
The total area of all three decks is 783 ft2.
A = 3 ( )pa2
= 3 ( ) 60 • 8.7
2Substitute 60 for p and 8.7 for a.
= 3 ( )5222
Simplify the numerator.
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
1-2
Use the formula A = 3 ( ) to find the total area of all three decks.pa2
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
pages 12–15 Exercises
1. 59
2. 22
3. 7
4. 113
5. 32
6. 29
7. 21
8. 27
9. 124
10. 15
11. 180
12. 169
13. $37.09
14. $347.10
15. 22
16. 3
17. 44
18. 5
19. 16
20. 4
21. 704
22. 7744
23. 185
24. 361
25. 57
26. 9
27. 1936
28. 3872
29. 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. 111
44. 1
45. 51
46. 50.4
47. 242
48. 61
49. 71.6
50. 58
51.
52. 7
53. 1
54. 7
13
56
1-2
7 15
58
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
55. a. left side = 1 right side = 1
b. left side = 4right side = 2
c. Answers may vary. Sample: For a = 2 andb = 3, left side = 25,right side = 13
d. No; as seen in part (b), (a + b)2 = a2 + b2 is not true for all valuesof a and b.
56. 17
57. 9
58. 143
59. 135
60. 143
61. 27
62. 14
63. 308
64. a. 135 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
34
13
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
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
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
1-2
82. a. 38 ft2
b. Yes; 2h = 2 h .
No; h = 2h
since b2 = 0.
Yes; h = h =
2 h .
ALGEBRA 1 LESSON 1-2ALGEBRA 1 LESSON 1-2
Exponents and Order of OperationsExponents and Order of Operations
b1 + b2
2
b1 + b2
22b1 + b2
2/ b1 + b2
2/
2(b1 + b2)2
2(b1 + b2)2
b1 + b2
2
83. D
84. F
85. B
86. H
95. 95%
96. 145%
97. 6%
98. 43
99. 18.9
100. 1.25
101. 60.3
87. B
88. F
89. c + 2
90. 36m
91. t – 21
92.
93. 50%
94. 34%
y5
102–106. Answers may vary. Samples are given.
102. 8, 24, 56
103. 55, 100, 250
104. 44, 66, 121
105. 60, 150, 240
106. 26, 39, 52
1-2
ALGEBRA 1 LESSON 1-2ALGEBRA 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
Exponents and Order of OperationsExponents and Order of Operations
1-2
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
(For help, go to skills handbook page 725.)
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
25
38
23
59
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
1. 0.5 = = =
2. 0.05 = = =
3. 3.25 = 3 = 3 = 3 or
4. 0.325 = = =
5. = 2 ÷ 5 = 0.4
6. = 3 ÷ 8 = 0.375
7. = 2 ÷ 3 = 0.6
8. 3 = 3 + (5 ÷ 9) = 3.5
5 10
12
5 • 15 • 2
5 100
5 • 1 5 • 20
25 100
14
134
25 • 125 • 4
325 1000
25 • 1325 • 40
1340
25
38
23
59
Solutions
1 20
Name the set(s) of numbers to which each number belongs.
a. –13 b. 3.28
integers
rational numbers
rational numbers
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
Which set of numbers is most reasonable for displaying
outdoor temperatures?
integers
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
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.
A negative number can be a fraction, such as – . This is not an integer.23
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
Write – , – , and – , in order from least to greatest.
– = –0.75 Write each fraction as a decimal.
– = –0.583
– = –0.625
34
7 12
58
From least to greatest, the fractions are – , – , and – .34
7 12
58
–0.75 < –0.625 < –0.583 Order the decimals from least to greatest.
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
34
7 12
58
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
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
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. >
1-3
23
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
1-3
27. =
28. 2.001, 2.01, 2.1
29. –9 , –9 , –9
30. – , – ,
31. –1.01, –1.001, –1.0009
32. 0.63, 0.636,
33. , 0.8888,
34. 4
35. 9
36.
37. 0.5
38.
23
34
23
12
56
2225
89
7 11
7 12
39. 0
40. 1295
41.
42. Answers may vary.
Sample:
43. Answers may vary.
Sample:
44. Answers may vary.
Sample:
45. Answers may vary.
Sample:
9 14
35
46. Answers may vary.
Sample:
47. natural numbers, whole numbers, integers, rational numbers
48. natural numbers, whole numbers, integers, rational numbers
49. rational numbers
50. rational numbers
45
15
51
21310
10341000
– 41
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
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
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Exploring Real NumbersExploring Real Numbers
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. –2d. 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
92. 25
93. 4
94. 195
95–98. 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
18
gcm3
gcm3
gcm3
gcm3
12
1-3
ALGEBRA 1 LESSON 1-3ALGEBRA 1 LESSON 1-3
Name the set(s) of numbers to which each given number belongs.
1. –2.7 2. 11 3. 16
Use <, =, or > to compare.
4. 5.
6. Find |– |.
34
58
rational numbers irrational numbers natural numbers, whole numbersintegers, rational numbers
– –
7 12
> <
7 12
Exploring Real NumbersExploring Real Numbers
1-3
34
58
(For help, go to skills handbook page 726.)
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
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. +
10. + 11. + 12. +
15
35
49
79
12
34
38
14
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
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. + = =
10. + = = = 1
11. + = + = = = 1
12. + = + = =
15
35
1 + 35
45
49
79
4 + 79
119
12
34
24
34
2 + 34
29
54
14
38
14
38
28
3 + 28
58
Solutions
Start at –3.
Move right 5 units.
Start at –3.
Move left 5 units.
Start at 3.
Use a number line to simplify each expression.
a. 3 + (–5) b. –3 + 5
c. –3 + (–5)
3 + (–5) = –2 –3 + 5 = 2
–3 + (–5) = –8
Use a number line to simplify each expression.
Move left 5 units.
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
Simplify each expression.
a. 12 + (–23) = –11 b. –6.4 + (–8.6) = –15.0
The difference of the absolute values is 11.
The negative addend has the greater absolute value, so the sum is negative.
Since both addends are negative, add their absolute values.
The sum is negative.
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
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.
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
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.
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
A scuba diver who is 88 ft below sea level begins to ascend
to the surface.
Define: Let r = the number of feet the diver rises.
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
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.
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1 3.2 913.4 8 –4
Simplify.=
7 –5.4 –2
11.1 3 –1
–6 8.6 11
2.3 5 –3Add +
7 –5.4 –2
11.1 3 –1
–6 8.6 11
2.3 5 –3+
–6 + 7 8.6 + (–5.4) 11 + (–2)2.3 + 11.1 5 + 3 –3 + (–1)
Add corresponding elements.=
1-4
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
pages 27–30 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
12. –42
13. 2.2
14. –0.65
15. –7.49
16. 1.33
17.
18. –
19. 6
20. –6
21. 0
22. –
1415
89 3 16
18
1318
23. 5
24. –
25. –47 + 12 = –35, 35 ft below the surface
26. 8 + (–5) = 3, 3 yd gain
27. –6 + 13 = 7, 7°F
28. 8.7
29. –1.7
30. 1.7
31. –8.7
32. 12.6
161314
1-4
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
33. –5.6
34. 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
38.
39.
40.
41.
42. –2.7
43. –13
44. 6.6
–1 1.4–21 23.2
–18.211.619.1
025
–12
1.8 22
– 712
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
1924
2235
11 120
13
1760
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
56. 13.8 million people
57. 6.3 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
73. 8.7
74. 0.1
75. –1.9
76. +2
100442
50 221
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
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.
2 0 1–1 3 0.5
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
8 3 5 110 2 2 1
4 1 0 1
5 2 1 18 2 0 12 1 0 1
13 5 6 218 4 2 2
6 2 0 2
82. a. 4
b. –4
1-4
ALGEBRA 1 LESSON 1-4ALGEBRA 1 LESSON 1-4
Adding Real NumbersAdding Real Numbers
1-4
83.
84.
85.
86. –
87.
88. –
89.
8 32 –27
20 6 0
1 –1
2 1
1412
12
12
12
w 10
c2
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., –m is pos. and the sum of two pos. is pos.
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
112. 2.2
113. 18
114. 21
107. <
108. >
109. >
110. =
x 12
t 6 –3m + 1
4m9
58a212b9
x 12
ALGEBRA 1 LESSON 1-4ALGEBRA 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
–4 715
5
Adding Real NumbersAdding Real Numbers
3 –5–2 0
–2 –4 8 –6+
1 – 96 – 6
1-4
15
4 15
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
(For help, go to Lessons 1-3 and 1-4)
Subtracting Real NumbersSubtracting Real Numbers
1-5
Find the opposite of each number.
1. 6 2. –7 3. 3.79 4. –
Simplify.
5. 3 + (–2) 6. 9.5 + (–3.5) 7. 13 + (–8) 8. + – ( )
7 19
23
16
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
1-5
1. opposite of 6: –6 2. opposite of –7: 7
3. opposite of 3.79: –3.79 4. opposite of – :
5. 3 + (–2) = 1 6. 9.5 + (–3.5) = 6
7. 13 + (–8) = 5
8. + – = + – = = =
7 19
23 ( ) ( )1
646
16
4 + (–1)6
36
12
Solutions
7 19
Subtract –3 – 2 using a number line.
–3 – 2 = –5
Start at –3.
Move left 2 units
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
1-5
Subtract 4 – (–2) using tiles.
There are 6 positive tiles left.
4 – (–2) = 6
Start with 4 positive tiles.
Add zero pairs until there are 2 negative tiles.
Remove 2 negative tiles.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
1-5
Simplify –11.6 – (–14).
–11.6 – (–14) = –11.6 + 14 The opposite of –14 is 14.
= 2.4 Add.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
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.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
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.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
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.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
1-5
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
1. –1
2. –2
3. –6
4. –5
5. 1
6. –11
7. 14
8. 3
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. $50.64
38. –9.5
39. –1.5
40. –2
41. 5.5
pages 34–36 Exercises
1-5
1 10
16
1112
1960
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
42. 1.5
43. –1
44. 16
45. –16
46. 11
47. 1
48. –2
49. 12.5
50. 650 ft
51. Answers may vary. Sample:
52. false; –1 – (–7) = 6, –1 + (–7) = –8
53. false; 2 – (–1) = 3, 3 < 2 or –1
54. true
55. a. 1992: 1997: b.
c. Answers may vary. Sample: Invest in soccer; it is the only sport that has not lost any participants.
5 12–3 7
4 6 5 10–
1 6–8 –3=
5.5 8.2 4.91.4 3.2 3.94.2 3.8 1.31.6 5.2 5.1
6.8 7.9 4.91.0 1.8 1.75.6 4.1 1.31.8 4.9 2.9
1.3 –0.3 0–0.4 –1.4 –2.21.4 0.3 00.2 –0.3 –2.2
1-5
56.
57. [– 0 –3]
58.
59. a. Yes, answers may vary. Sample: n and –n always have the same absolute value.
b. No, answers may vary. Sample:|1| + | –1| | 1 + (–1)|, 2 0
60. –
61.
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
2 –2–9 3
1 –
– 1
62. –4x + 9
63. –12t – m
64.
65.
66. |x| – |y|, |x – y|
and x – y, |x + y|
67. C
68. F
69. C
70. H
71. D
72. 4
73. –9
74. –0.7
75. –4.1
76.
77.
78.
4 0 0 16
1.2–2.5 5.2
1-5
14
9 20
5 12
=/ =/
3760
29r – 3516
–15w – 89
–3
6 –4
12
2–3
13
13
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
Subtracting Real NumbersSubtracting Real Numbers
79–81. Choices of variables may vary.
79. t = total cost, p = pounds of pears, t = 1.19p
80. p = bouquet’s price, m = money left, m = 20 – p
81. c = check ($), s = your share, s =
1-5
c6
ALGEBRA 1 LESSON 1-5ALGEBRA 1 LESSON 1-5
–5 11 –12
–1 7
Subtracting Real NumbersSubtracting Real Numbers
Simplify each expression.
1. 7 – 12 2. 3 – (–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 –5 9 0
1-5
ALGEBRA 1 LESSON 1-6ALGEBRA 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, , ,
� � � � � �� � � � � �
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
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: 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
Solutions
Simplify each expression.
a. –3(–11)
–3(–11) = 33 The product of two negative numbers is positive.
The product of a positive number and a negative number is negative.
–6( ) = –34
184
= –412
Write – as a
mixed number.
184
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
b. –6( )34
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.
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
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 ( )72001000
a 1000
Substitute 7200 for a.
= –5.5(7.2) Divide within parentheses.
= –39.6°F Multiply.
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
The change in temperature is –39.6°F.
Use the order of operations to simplify each expression.
a. –0.24
= –0.0016 Simplify.
= 0.0016 Simplify.
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
b. (–0.2)4
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
Write as repeated multiplication.–(0.2 • 0.2 • 0.2 • 0.2)=
Write as repeated multiplication.(–0.2)(–0.2)(–0.2)(–0.2)=
Simplify each expression.
a. 70 ÷ (–5) b. –54 ÷ (–9)
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
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
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
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.xy
–4–2
= – 4(16) Simplify the power.–4–2
= 2 – 64 Divide and multiply.
= –62 Subtract.
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
xy
Evaluate for p = and r = – .
= –2 Simplify.
= p ÷ r Rewrite the equation.pr
= ÷ Substitute for p and – for r.32
34
(– ) 32
34
= Multiply by – , the reciprocal of – .32
43(– ) 4
334
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
pr
32
34
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
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°Fb. –75°Fc. –51°Fd. –31.5°F
32. –1
33. 8
34. –25
35. 81
36. –81
37. –192
38. –5
39. 675
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
12
45
12
231213
12
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
56.
57. –
58. –2
59. 18
60. –
61. –
62. –125
63. 0.75
64. –27
65. 22.32
66. 44
67. –27
1 18
6 25
12 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
15
12
15
5678
1 10
ALGEBRA 1 LESSON 1-6ALGEBRA 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
85. 31¢
86. Yes; whatever the signs of a and b, | ab |,| a |, and | b | are pos., and | ab | = | a | • | b |.
87. a. 4, –8, 16, –329, –27, 81, –243
b. Pos.; the expression will involve an even number of neg. factors.
c. Neg.; the expression will involve an odd number of neg. factors.
88. 0.1 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
1-6
90.
91.
92.
93. [–12 –2 ]
94.
95.
–15 21 –9
–22 10 18 –12 8 –6
23
4.7 –1.3 0.79–0.02 6.4 0
–8 –10.6 46.2 0 12
1 –
0
14
3 16
29
23
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
ALGEBRA 1 LESSON 1-6ALGEBRA 1 LESSON 1-6
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
96. a. 139 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. 312.5 ft
c. 4687.5 ft
98. –23°C
99. –
100.
101. –
102.
103. –
104. –8
105. , or 11
106. –
107. –
108.
109. A
110. F
111. C
112. H
113. B
18
1 16
1 32
1 64
1 64
454
9 16 5 48
12
114. –2
115. –20
116. 0.8
117. 9.8
118. 2
119.
120. 4.95
121. 56
122. 4.59
123.
124. 23
125. 121
126. 196
127. $34.93
14
14
172
34
ALGEBRA 1 LESSON 1-6ALGEBRA 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 = – .
56 – 22
–7 27 12
312
Multiplying and Dividing Real NumbersMultiplying and Dividing Real Numbers
1-6
8mn
13
12
ALGEBRA 1 LESSON 1-7ALGEBRA 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)
The Distributive PropertyThe Distributive Property
1-7
12
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
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
5. t(10 – 4) = t(6) = (6)t = • 6 t = 3t
6. m(–3 – 1) = m(–4) = –4m
12
12
12
12
Solutions
Use the Distributive Property to simplify 26(98).
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
26(98) = 26(100 – 2) Rewrite 98 as 100 – 2.
= 26(100) – 26(2) Use the Distributive Property.
= 2600 – 52 Simplify.
= 2548
The Distributive PropertyThe Distributive Property
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.
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
1-7
Simplify 3(4m – 7).
3(4m – 7) = 3(4m) – 3(7) Use the Distributive Property.
= 12m – 21 Simplify.
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
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.
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
1-7
Simplify –2w2 + w2.
–2w2 + w2 = (–2 + 1)w2 Use the Distributive Property.
= –w2 Simplify.
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
1-7
Relate: –6 times the quantity 7 minus m
Write: –6 • (7 – m)
Write an expression for the product of –6 and the quantity 7
minus m.
–6(7 – m)
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
1-7
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
1-7
pages 50–52 Exercises
1. 2412
2. 663
3. 5489
4. 2448
5. 686
6. 2997
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
21. 1.5q + 8
22. 18n – 42
23. – r
24. –4.5b + 13.5
25. 2w + 4
26. –36 + 16n
27. –x – 3
28. –x + 3
29. –3 – x
30. –3 + x
31. –6k – 5
32. –7x + 2
33. –2 + 7x
34. –4 + z
45
52
1516 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)
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
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 – )
407
507
58
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
14
12
43
1120
1330
1112
1-7
17 z – 34
7 100
13
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
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
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.
1-7
7142
3 20
107120
3x – 10 3
ALGEBRA 1 LESSON 1-7ALGEBRA 1 LESSON 1-7
The Distributive PropertyThe Distributive Property
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
115. 12.127
116. 45.3
117. 45.7
118. 2.57
119. 0.92
120. –9.6
121. –7.7
122. 23
123. a. 4 +
b. 7; 5; 8
1-7
16
5415
m3
ALGEBRA 1 LESSON 1-7ALGEBRA 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
47
35b –( )
The Distributive PropertyThe Distributive Property
1-7
47
35
ALGEBRA 1 LESSON 1-8ALGEBRA 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
Properties of Real NumbersProperties of Real Numbers
1-8
ALGEBRA 1 LESSON 1-8ALGEBRA 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
Properties of Real NumbersProperties of Real Numbers
Solutions
1-8
Name the property each equation illustrates.
a. 3 • a = a • 3
b. p • 0 = 0
c. 6 + (–6) = 0
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
1-8
Commutative Property of Multiplication, because the order of the factors changes
Multiplication Property of Zero, because a factor multiplied by zero is zero
Inverse Property of Addition, because the sum of a number and its inverse is zero
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.
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
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
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
1-8
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
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.
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 numberoutside parentheses is distributed to the two terms inside the parentheses.
8. Assoc. Prop. of Mult.; the grouping of the factors changes.
pages 56–57 Exercises 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
1-8
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
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.
19. 25 • 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 Assoc. Prop. of Mult.= –35y mult.
21. 8 + 9m + 7 = 9m + 8 + 7 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) Comm. Prop. of Add.= (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
24. 43 + 1 = 1 + 1
Inv. Prop. of Mult. = 2 add.
25. 2 + g = 2 + 1
Inv. Prop. of Mult. = 3 add.
143
1g
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
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.
1-8
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
1-8
15
30. 5 w – – w(9)
= 5 w – – 9w
Comm. Prop. of Mult.
= 5(w) – 5 – 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)
Dist. Prop.
= –4w + (–1) add.
= –4w – 1
def. of subtr.
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.
12
15
15
ALGEBRA 1 LESSON 1-8ALGEBRA 1 LESSON 1-8
Properties of Real NumbersProperties of Real Numbers
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.; 1 + 3 = 4
52. both
53. A
54. I
55. D
56. G
57. B
58. F
59. 6 + 5k
60. 11 – b
61. –10p – 35
62. 28 – 4n
63. 7.4m – 0.05
64. –18v + 31.2
65. 7 + [m + (–17)]
66. 8 – (9 – t)
67. ( )
68. (x + 5.1)
69. 7
70. –18.3
71. –
72. –1
73. –135
74. 9.2
13
b4
1213
1415
ALGEBRA 1 LESSON 1-8ALGEBRA 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
Properties of Real NumbersProperties of Real Numbers
1-8
ALGEBRA 1 LESSON 1-9ALGEBRA 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
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
1.
2.
3.
4.
5. A: 1 6. B: –2.5 7. C: 3.5 8. D: –4
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
Solutions
1-9
Name the coordinates of point A in the graph.
Move 2 units to the left of the origin. Then move3 units up.
The coordinates of A are (–2, 3).
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
Graph the point B(–4, –2) on the coordinate plane.
Move 4 units to the left of the origin.
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Then move 2 units down.
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
In which quadrant or on which axis would you find each point?
a. (2, –5)
Since the x-coordinate is positive and the y-coordinate is negative, the point is in Quadrant IV.
b. (6, 0)
Since the y-coordinate is 0, the point is on the x-axis.
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
The table shows thenumber of hours worked andthe amount of money eachperson earned. Make a scatterplot of the data.
Name Hours Amountworked earned
Janel 6 $25.50Roscoe 12 $51.00Victoria 11 $46.75Alex 9 $38.25Jordan 15 $63.75Jennifer 10 $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 from0 to 70 with every 10 points labeled.
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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.
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
25.
square
26.
rhombus
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.
27.
isosceles triangle
28.
rectangle
1-9
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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
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).)
1-9
ALGEBRA 1 LESSON 1-9ALGEBRA 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
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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
ALGEBRA 1 LESSON 1-9ALGEBRA 1 LESSON 1-9
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
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 9 –1319 –1
3.1 –0.7 21–4.9 –4.7 –2.1
–1 –7 8
–11 –
10 –17
5 30
12
1-9
1212
ALGEBRA 1 LESSON 1-9ALGEBRA 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
Graphing Data on the Coordinate PlaneGraphing Data on the Coordinate Plane
1-9
ALGEBRA 1 CHAPTER 1ALGEBRA 1 CHAPTER 1
Tools of AlgebraTools of Algebra
1-A
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
59
18. 12d – 30
19–20. Answers may vary.
19. 10x + 3( – x)
= 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
–1 15 –1113 –1 2
–10 5 –10
11 –32 5
12
13
45
Tools of AlgebraTools of AlgebraALGEBRA 1 CHAPTER 1ALGEBRA 1 CHAPTER 1
1-A
21. –10(2 – 11)
22.
23. (p – )( + p)
24. –10
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.
58
14
5 m + 6
38
34. neg. correlation
35. about 41°F
36. If a and b have
different signs,
| | is positive
and is negative.
ab
ab
27. Answers may vary. Sample:– , 1.5, 1, –
– , – , 1, 1.5
28. y-axis
29. Quadrant III
30. 28 square units
31. 2 yd lost
32. $177
33. $19.30
23
16
23
16