Properties of Numbers

1
Properties of Numbers
PRE-ALGEBRA LESSON 2-1
At the Red Valley Sports Camp, 15 kids went
horseback riding, 14 played tennis, 23 went
hiking, and the rest of the campers stayed
in their cabins. If 83 kids were in the camp, how
many stayed indoors?
31
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
(For help, go to Lesson 1-5.)
Simplify. 1. 18 (7) 2. 32 (3)
3. (13) 6 4. 2 48
Check Skills Youll Need
2-1
3
Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Solutions 1. 18 (7) 25 2. 32 (3)
32 3 35 32 (3)
35 3. (13) 6 7 4. 2 48 2
(48) 46 2 48 46
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Carlos spent 42 on his golf game. He then
bought a bottle of water for 2 and a chefs
salad for 8. What was the total cost for his
golf game and meal?
You can use the Associative Property of Addition
to find the total cost in two different ways.
42 (2 8) 42 10 52 Add 2 and 8 first.
(42 2) 8 44 8 52 Add 42 and 2 first.
Carloss total cost was 52.
Quick Check
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Name each property shown.
a. 17 x 3 17 3 x
Commutative Property of Addition
b. (36 ? 2)10 36(2 ? 10)
Associative Property of Multiplication
c.  km km 1
Identity Property of Multiplication
d.  (103 26) 4 103 (26 4)
Associative Property of Addition
Quick Check
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Use mental math to simplify (48 7) 2.
(48 7) 2
(7 48) 2 Use the Commutative Property of
Addition.
7 (48 2) Use the Associative Property of
Addition.
7 50 Add within parentheses.
57 Add.
Quick Check
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7
Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Suppose you buy school supplies costing .45,
.65, and 1.55. Use mental math to find the cost
of these supplies.
0.45 0.65 1.55
0.65 0.45 1.55 Use the Commutative
Property of Addition.
0.65 (0.45 1.55) Use the Associative
Property of Addition.
0.65 2.00 Add within parentheses.
2.65 Add.
The cost of the school supplies is 2.65.
Quick Check
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Use mental math to simplify (20 13) 5.
(20 13) 5 (13 20) 5 Use the
Commutative Property of Multiplication.
13 (20 5) Use the Associative Property of
Multiplication.
13 100 Multiply within parentheses.
1,300 Multiply.
Quick Check
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Properties of Numbers
PRE-ALGEBRA LESSON 2-1
Name each property shown. 1. d ? 1 d
2. 3 (2 5) (3 2) 5 3. f ? g
g ? f 4. 32 45 102 45 32 102
Identity Property of Multiplication
Associative Property of Addition
Commutative Property of Multiplication
Commutative Property of Addition
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Evaluate each expression. a. 24 6 2 4 36
b. 6 (3 1) 18 c. 10 6 5 4 2
38
48
30
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
(For help, go to Lesson 1-2.)
Simplify each expression. 1. 3 7 9 2. (9
5)6 3. 8 2 6 4. 2(6 3) 5. 4 5
4 3 6. 3 2 1 2
Check Skills Youll Need
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Use the Distributive Property to find 15(110)
mentally.
15(110) 15(100 10) Write 110 as (100 10).
15 100 15 10 Use the Distributive
Property.
1,500 150 Multiply.
1,650 Add.
Quick Check
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Ms. Thomas gave 5 pencils to each of her 37
students. What is the total number of pencils she
gave to the students?
(37)5 (40 3)5 Write 37 as (40 3).
40 5 3 5 Use the Distributive Property.
200 15 Multiply.
185 Subtract.
Ms. Thomas gave the students 185 pencils.
Quick Check
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Simplify 11(23) 11(7).
11(23) 11(7) 11(23 7) Use the Distributive
Property.
11(30) Add within parentheses.
330 Multiply.
Quick Check
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Use algebra tiles to multiply 4(3x 4).
So 4(3x 4) 12x 16.
Quick Check
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Multiply.
a. 9(2 8y)
18 (72y) Multiply.
18 72y Simplify.
b.  (5m 6)11
55m 66 Multiply.
Quick Check
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The Distributive Property
PRE-ALGEBRA LESSON 2-2
Use the Distributive Property to
simplify. 1. 15(203) 2. 7(180) 3. 6(26)
6(16) 4. (8a 4)9 5. 2(14 y)
3,045
1,260
60
72a 36
28 2y
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Use numbers to write the following word
expression eight and one-third minus five and
two-sevenths.
8 5
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
(For help, go to Lesson 2-2.)
Simplify each expression. 1. 5(b 4) 2. 3(2x
5) 3. 4(8 3q) 4. 6(2b 7)
Check Skills Youll Need
2-3
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Solutions 1. 5(b 4) 5(b) 5(4) 2. 3(2x
5) 3(2x) (3)(5)
5b 20 6x
15 3. 4(8 3q) 4(8) 4(3q) 4. 6(2b
7) 6(2b) (6)(7) 32
12q 12b 42
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Name the coefficients, the like terms, and the
constants in 7x y 2x 7.
Coefficients 7, 1, 2
Like terms 7x, 2x
Constant 7
Quick Check
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Simplify 9 4f 3 2f.
Quick Check
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Simplify 2b b 4.
2b b 4 2b 1b 4 Use the Identity
Property of Multiplication.
(2 1)b 4 Use the Distributive Property.
3b 4 Simplify.
Quick Check
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Simplify (7 3x)5 20x.
(7 3x)5 20x 35 15x 20x Use the
Distributive Property.
35 (15x 20x) Use the Associative
Property of Addition.
35 (15 20)x Use the Distributive
Property to combine like terms.
35 5x Simplify.
Quick Check
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Simplifying Variable Expressions
PRE-ALGEBRA LESSON 2-3
Name coefficients, like terms, and
constants. 1. 4f 2f 3 2. z 2y 14
Simplify each expression. 3. 3(a c 1)
2c 4. 4(4v) 4(v 9)
4, 2 4f, 2f 3
1, 2 none 14
12v 36
3a c 3
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
Evaluate 10 for c 10.
8
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
(For help, go to Lesson 1-1.)
Write a variable expression for each
phrase. 1. the sum of x and 46 2. four less
than g 3. t decreased by five 4. the
quotient of z and 26
Check Skills Youll Need
2-4
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
Solutions 1. the sum of x and 46 2. four less
than g x plus 46 g minus 4 x 46 g
4 3. t decreased by five 4. the quotient of
z and 26 t minus 5 z divided by 26 t
5
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
State whether each equation is true, false, or
an open sentence. Explain.
a. 3(b 8) 12
open sentence, because there is a variable
b. 7 (6) 1
c. 9 5 4
true, because 4 4
Quick Check
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
Write an equation for Six times a number added
to the number is the opposite of forty-two. State
whether the equation is true, false, or an open
sentence. Explain.
The equation is an open sentence, because there
is a variable.
Quick Check
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
Is 45 a solution of the equation 120 x 75?
120 x 75
No, 45 is not a solution of the equation.
Quick Check
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33
Variables and Equations
PRE-ALGEBRA LESSON 2-4
A gift pack must hold 20 lb of food. Apples
weigh 9 lb and cheese weighs 5 lb. Can the jar of
jam that completes the package weigh 7 lb?
9 5 j 20
14 j 20 Add.
Quick Check
No, the jar of jam cannot weigh 7 lb.
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Variables and Equations
PRE-ALGEBRA LESSON 2-4
1. Is b 3b 15 true, false, or an open
sentence? 2. Write an equation for A number
added to seven times the number is the opposite
of thirty-two. Is the equation true, false, or an
open sentence? 3. Is 25 a solution of the
equation 175 x 150? 4. Can you stack
eight books that are each 3 in. thick inside a
box that is 18 in. tall?
open sentence
7x x 32 open sentence
yes
no
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Lauralee pitched 12 innings. Maureen pitched 3
fewer innings. How many innings did Maureen
pitch?
8
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
(For help, go to Lesson 2-1.)
Simplify each expression. 1. 3 4 4 2. 7
9 7 3. 8 2 2 4. 6 2 2
Check Skills Youll Need
2-5
37
Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Solutions 1. 3 4 4 2. 7 9 7 3
(4 4) 9 (7 7) 3 0 9 0
3 9 3. 8 2 2 4. 6 2 2 8 (2
2) 6 (2 2) 8 0 6 0 8 6
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Solve y 5 13.
Method 1
Method 2
y 5 13
y 8 Simplify.
y 8
Quick Check
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Larissa wants to increase the number of books in
her collection to 327 books. She has 250 books
now. Find the number of books she needs to buy.
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
(continued)
327 250 x
327 x 250 Use the Commutative Property of
Addition.
327 250 x 250 250 Subtract 250 from each
side.
77 x Simplify.
Larissa needs to buy 77 more books.
Quick Check
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Solve c 23 40.
c 23 40
c 23 23 40 23 Add 23 to each side.
c 17 Simplify.
Quick Check
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Marcys CD player cost 115 less than her DVD
player. Her CD player cost 78. How much did her
DVD player cost?
78 t 115
78 115 t 115 115   Add 115 to each side.
193 t Simplify.
Marcys DVD player cost 193.
Quick Check
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Solving Equations by Adding or Subtracting
PRE-ALGEBRA LESSON 2-5
Solve each equation. 1. y 8 12 2. 7 f
21 20 3. 67 g (36) 4. Ricky
rides his bike 12 miles every day. He stops after
7 miles to rest. How much farther does he have
to ride?
4
6
31
5 mi
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
Divide. Round each quotient to the nearest
hundredth. a.
b.
0.372, which rounds to 0.37
5.006, which rounds to 5.01
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
(For help, go to Lesson 1-9.)
Check Skills Youll Need
2-6
46
Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
288 pens are boxed by the dozen. How many boxes
are needed?
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
(continued)
288 12b
24 b Simplify.
24 boxes are needed.
Check  Is the answer reasonable? Twelve times
the number of boxes is the number of pens.
Since 12 ? 24 288, the answer is reasonable.
Quick Check
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
Solve 2v 24.
2v 24
v 12 Simplify.
Quick Check
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
Solve 5.
x 40 Simplify.
Quick Check
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Solving Equations by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-6
6
9
12
42
24
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
Find three different odd numbers whose sum is 21.
List all possibilities.
1, 3, 17 1, 5, 15 1, 7, 13 1, 9, 11 3, 5, 13
3, 7, 11 5, 7, 9
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
(For help, go to Lesson 1-5.)
Simplify. 1. 158 20 2. 158 30 3. 158
25 4. 158 22 5. In Exercises 14, which
result came closest to 181?
Check Skills Youll Need
2-7
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
During the intermission of the play, the Theater
Club sold cups of popcorn and soda. The club sold
79 cups of popcorn and 96 sodas for a total of
271. What was the selling price of a cup of
popcorn? Of a soda?
You can organize conjectures in a table. As a
first conjecture, try both with a price of 1.
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
(continued)
2 1 79(2) 96(1) 158 96 The total is too
low.
254 Increase the price of the soda.
2 2 79(2) 96(2) 158 192 The total is too
high. 350 Decrease the
price of the popcorn.
1 2 79(1) 96(2) 79 192 The total is
correct. 271
The popcorn price was 1, and the soda price was
2.
Quick Check
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Problem Solving Strategy Guess, Check, Revise
PRE-ALGEBRA LESSON 2-7
Solve using any strategy. 1. A sporting goods
store manager ordered twice as many pairs of
basketball shoes as tennis shoes. He ordered 96
pairs in all. How many pairs of each did he
order? 2. Manuel wrote a 512-word paper
over the weekend. He wrote 180 more words on
Sunday than he did on Saturday. How many words
did he write on Saturday?
32 pairs of tennis shoes 64 pairs of basketball
shoes
166
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
Order from least to greatest , 0.25, ,
0.5, 0.3, 2
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
(For help, go to Lesson 1-4.)
Graph each set of numbers on a number line. Order
the numbers from least to greatest. 1. 3, 7,
9 2. 2, 10, 8 3. 0, 3, 5 4. 3, 6,
10
Check Skills Youll Need
2-8
60
Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
2-8
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
Graph the solutions of each inequality on a
number line.
a. x gt 2  
b. w 5 
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
(continued)

c. k 4  
d. y lt 6
Quick Check
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
Write the inequality shown in each graph.
a. 
b. 
x lt 3
Quick Check
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
Food can be labeled very low sodium only if it
meets the requirement established by the federal
government. Use the table to write an inequality
for this requirement.
Label Definition
Sodium-free food Less than 5 mg per serving
Very low sodium food At most 35 mg per serving
Low-sodium food At most 140 mg per serving
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
(continued)
Quick Check
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Inequalities and Their Graphs
PRE-ALGEBRA LESSON 2-8
x gt 30
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Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
Solve each equation. a. n 4 3
b. 4 p 8
n 1
p 4
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Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
(For help, go to Lesson 2-5.)
Solve each equation. 1. m 7 5 2. k 8
11 3. 12 h 21 4. 6 n 23
Check Skills Youll Need
2-9
69
Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
2-9
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Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
Solve each inequality. Graph the solutions.
a.
4 s lt 12
4 s lt 12
4 s 4 lt 12 4 Subtract 4 from each
side.
s lt 8 Simplify.
b.
Quick Check
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71
Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
You have 6 GB on your computer. The files you
have stored on the hard drive occupy at least 2
GB. How much storage space is left for other
files?
Quick Check
No more than 4 GB are left.
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Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
Solve 10 lt 13 q.
10 lt 13 q
10 13 lt 13 13 q Add 13 to each side.
3 lt q Simplify.
Quick Check
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Solving One-Step Inequalities by Adding or
Subtracting
PRE-ALGEBRA LESSON 2-9
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Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
Solve each equation. a. q 3 6
b. 7r 21
r 3
q 18
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75
Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
(For help, go to Lesson 2-6.)
Solve each equation. 1. 6x 24 2. 63 7v
3. 10 4. 48
Check Skills Youll Need
2-10
76
Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
2-10
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Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
A 1-ton truck has the ability to haul 1 ton, or
2,000 lb. At most, how many television sets can
the truck carry if each TV set weighs 225 lb?
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Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
(continued)
At most, the truck can carry 8 television sets.
Check Is the answer reasonable? The total weight
of 8 television sets is 8(225) 1,800 lb, which
is less than 2,000 lb but so close that another
television set could not be carried. The answer
is reasonable.
Quick Check
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Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
Solve 2.
Quick Check
2-10
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Solving One-Step Inequalities by Multiplying or
Dividing
PRE-ALGEBRA LESSON 2-10
w lt 3
f gt 3
2-10