2-1 Properties of Numbers PRE-ALGEBRA LESSON 2-1 At the Red Valley Sports Camp, 15 kids went...
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Transcript of 2-1 Properties of Numbers PRE-ALGEBRA LESSON 2-1 At the Red Valley Sports Camp, 15 kids went...
2-1
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
At the Red Valley Sports Camp, 15 kids went horseback riding, 14played tennis, 23 went hiking, and the rest of the campers stayed intheir cabins. If 83 kids were in the camp, how many stayed indoors?
31
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-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 You’ll Need
2-1
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-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
2-1
Carlos spent $42 on his golf game. He then bought
a bottle of water for $2 and a chef’s salad for $8. What was
the total cost for his golf game and meal?
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
42 + (2 + 8) = 42 + 10 = 52 Add 2 and 8 first.
Carlos’s total cost was $52.
You can use the Associative Property of Addition to find the total cost in two different ways.
(42 + 2) + 8 = 44 + 8 = 52 Add 42 and 2 first.
Quick Check
2-1
Name each property shown.
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
a. 17 + x + 3 = 17 + 3 + x
b. (36 2)10 = 36(2 10)
c. km = km • 1
d. (103 + 26) + 4 = 103 + (26 + 4)
Commutative Property of Addition
Associative Property of Multiplication
Identity Property of Multiplication
Associative Property of Addition
2-1
Quick Check
Use mental math to simplify (48 + 7) + 2.
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
(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.
2-1
Quick Check
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.
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
2-1
Quick Check
Use mental math to simplify (20 • 13) • 5.
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-ALGEBRA LESSON 2-1
(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.
2-1
Quick Check
Properties of NumbersProperties of NumbersPRE-ALGEBRA LESSON 2-1PRE-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
Commutative Property of Multiplication
Identity Property of Multiplication
Associative Property of Addition
Commutative Property of Addition
2-1
2-2
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
Evaluate each expression.
a. 24 – 6 ÷ 2 • 4 + 36 b. 6 • (3 – 1) + 18 c. 10 + 6 • 5 – 4 ÷ 2
48 30 38
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-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 You’ll Need
2-2
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
Solutions
1. 3 • 7 – 9 2. (9 – 5) • 6
21 – 9 4 • 6
12 24
3. 8 + 2 • 6 4. 2 • (6 – 3)
8 + 12 2 • 3
20 6
5. 4 • 5 – 4 • 3 6. 3 • 2 – 1 • 2
20 – 12 6 – 2
8 4
2-2
Use the Distributive Property to find 15(110) mentally.
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
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
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?
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
(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.
2-2
Quick Check
Simplify 11(23) + 11(7).
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
11(23) + 11(7) = 11(23 + 7) Use the Distributive Property.
= 11(30) Add within parentheses.
= 330 Multiply.
2-2
Quick Check
Use algebra tiles to multiply 4(3x – 4).
So 4(3x – 4) = 12x – 16.
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
Model four groups of 3x – 4.
Group like tiles.
2-2
Quick Check
Multiply.
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-ALGEBRA LESSON 2-2
a. –9(2 – 8y)
= –18 – (–72y) Multiply.
= –18 + 72y Simplify.
b. (5m + 6)11
= 55m + 66 Multiply.
–9(2 – 8y) = –9(2) – (–9)(8y) Use the Distributive Property.
(5m + 6)11 = (5m)11 + (6)11 Use the Distributive Property.
2-2
Quick Check
The Distributive PropertyThe Distributive PropertyPRE-ALGEBRA LESSON 2-2PRE-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)
60
3,045 1,260
28 + 2y
72a – 36
2-2
2-3
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-ALGEBRA LESSON 2-3
Use numbers to write the following word expression: eight andone-third minus five and two-sevenths.
8 – 513
27
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-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 You’ll Need
2-3
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-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
2-3
Name the coefficients, the like terms, and the
constants in 7x + y – 2x – 7.
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-ALGEBRA LESSON 2-3
Coefficients: 7, 1, –2 Like terms: 7x, –2x Constant: –7
Quick Check
2-3
Simplify 9 + 4f + 3 + 2f.
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-ALGEBRA LESSON 2-3
9 + 4f + 3 + 2f 6f + 12
2-3
Quick Check
Simplify 2b + b – 4.
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-ALGEBRA LESSON 2-3
2b + b – 4 = 2b + 1b – 4 Use the Identity Property of Multiplication.
= (2 + 1)b – 4 Use the Distributive Property.
= 3b – 4 Simplify.
2-3
Quick Check
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 Propertyto combine like terms.
= 35 + 5x Simplify.
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-ALGEBRA LESSON 2-3
2-3
Quick Check
Simplifying Variable ExpressionsSimplifying Variable ExpressionsPRE-ALGEBRA LESSON 2-3PRE-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)
3a + c – 3
4, –2; 4f, –2f; 3 1, 2; none; –14
12v + 36
2-3
2-4
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
Evaluate 10 – for c = 10.c5
8
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-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 You’ll Need
2-4
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
Solutions
1. the sum of x and 46 2. four less than g x plus 46 g minus 4x + 46 g – 4
3. t decreased by five 4. the quotient of z and 26t minus 5 z divided by 26t – 5 z
26
2-4
State whether each equation is true, false, or an
open sentence. Explain.
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
a. 3(b – 8) = 12
c. –9 + 5 = – 4
b. 7 – (–6) = 1
open sentence, because there is a variable
=/ false, because 13 1
true, because – 4 = – 4
Quick Check
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.
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
The equation is an open sentence, because there is a variable.
Equation 6x + =x –42
six times the numberWords
6x
added to
added to x is
isthe oppositeof forty-two
–42
thenumber
2-4
Quick Check
Is 45 a solution of the equation 120 + x = 75?
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
120 + x = 75
No, 45 is not a solution of the equation.
120 + 450 75 Substitute 45 for x.
165 75=/
2-4
Quick Check
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?
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-ALGEBRA LESSON 2-4
9 + 5 + j = 20
Equation 5 + =j 20
weight of cheese
Words plus
= weight of jam.
is 20 lbweight of
jam
9
j
weight of apples plus
Let
+
14 + j = 20 Add.
14 + 7 20 Substitute 7 for the variable.
No, the jar of jam cannot weigh 7 lb.
21 20 =/
2-4
Quick Check
Variables and EquationsVariables and EquationsPRE-ALGEBRA LESSON 2-4PRE-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?
yes
open sentence
7x + x = –32; open sentence
no
2-4
2-5
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
Lauralee pitched 12 innings. Maureen pitched 3 fewer innings.How many innings did Maureen pitch?
23
8 13
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-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 You’ll Need
2-5
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-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
2-5
Solve y + 5 = 13.
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
Method 1 Method 2
y + 5 = 13 Subtract 5
y + 5 – 5 = 13 – 5 from each side.
y = 8 Simplify.
y + 5 = 13
y = 8
Check y + 5 = 13
8 + 5 13 Replace y with 8.
13 = 13
– 5 = – 5
Quick Check
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.
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
target numberWords plusis 250 number to buy
= number to buy.xLet
Equation 327 + x250=
2-5
(continued)
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
327 = 250 + x
Larissa needs to buy 77 more books.
327 = x + 250 Use the Commutative Property of Addition.
327 – 250 = x + 250 – 250 Subtract 250 from each side.
77 = x Simplify.
Check Is the answer reasonable?250 plus the number of books bought should be a total collection of 327.
250 + 77 = 327
2-5
Quick Check
Solve c – 23 = – 40.
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
c – 23 = – 40
c – 23 + 23 = – 40 + 23 Add 23 to each side.
c = –17 Simplify.
2-5
Quick Check
Marcy’s CD player cost $115 less than her DVD player. Her
CD player cost $78. How much did her DVD player cost?
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-ALGEBRA LESSON 2-5
78 = t – 115
78 + 115 = t – 115 + 115 Add 115 to each side.
193 = t Simplify.
Marcy’s DVD player cost $193.
cost of CD player $115 cost of DVD playerWords less thanwas
t = cost of the DVD player.Let
t 115Equation 78 = –
2-5
Quick Check
Solving Equations by Adding or SubtractingSolving Equations by Adding or SubtractingPRE-ALGEBRA LESSON 2-5PRE-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?
5 mi
4 –6
31
2-5
2-6
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
0.59521.6
Divide. Round each quotient to the nearest hundredth.
a. b. 10.51262.1
0.372, which rounds to 0.37 5.006, which rounds to 5.01
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
(For help, go to Lesson 1-9.)
Simplify each quotient.
1. 2.
3. 4.
1818
21–21
–77
–13–13
Check Skills You’ll Need
2-6
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
Solutions
1. = 1 2. = –1
3. = –1 4. = 1
1818
21–21
–77
–13–13
2-6
288 pens are boxed by the dozen. How many boxes
are needed?
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
Let
Equation 288
=
b
number of pensWords times
number of boxes.b
is 12
12
number of boxes
= •
2-6
(continued)
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
288 = 12b
24 = b Simplify.
24 boxes are needed.
Divide each side by 12.28812
12b12
=
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
2-6
Solve –2v = –24.
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
–2v = –24
v = 12 Simplify.
Divide each side by –2.–2v–2
–24–2
=
Check –2v = –24–2 • (12) –24 Replace v with 12.
–24 = –24
2-6
Quick Check
Solve = – 5.
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
x = – 40 Simplify.
x8
= – 5x8x8
8 = 8(–5) Multiply each side by 8.
2-6
Quick Check
Solving Equations by Multiplying or DividingSolving Equations by Multiplying or DividingPRE-ALGEBRA LESSON 2-6PRE-ALGEBRA LESSON 2-6
–6
Solve each equation.
1. 8x = –48 2. –2x = 18
3. 108 = 9x 4. = 14
5. –6 =
v–3
n4
–9
12 –42
–24
2-6
2-7
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-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
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-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 1–4, which result came closest to 181?
Check Skills You’ll Need
2-7
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-ALGEBRA LESSON 2-7
Solutions
1. 158 + 20 = 178 2. 158 + 30 = 188
3. 158 + 25 = 183 4. 158 + 22 = 180
5. |181 – 178| = |3| = 3 |181 – 188| = |–7| = 7
|181 – 183| = |–2| = 2 |181 – 180| = |1| = 1
The result of Exercise 4, 180, is the closest to 181.
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?
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-ALGEBRA LESSON 2-7
You can organize conjectures in a table. As a first conjecture, try both with a price of $1.
2-7
(continued)
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-ALGEBRA LESSON 2-7
Popcorn SodaPrice Price Total Price
$1 $1 79(1) + 96(1) = 79 + 96 The total is too low. Increase = 175 the price of the popcorn only.
$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
2-7
Problem Solving Strategy: Guess, Check, ReviseProblem Solving Strategy: Guess, Check, RevisePRE-ALGEBRA LESSON 2-7PRE-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?166
32 pairs of tennis shoes; 64 pairs of basketball shoes
2-7
2-8
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
Order from least to greatest: , 0.25, , – 0.5, 0.3,– 212
16
–2, – 0.5, , 0.25, 0.3, 12
16
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-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 You’ll Need
2-8
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
Solutions
1. –9, –3, 7
2. –10, –8, –2
3. –5, 0, 3
4. –6, 3, 10
2-8
An open dot shows that –2 is not a solution.
A closed dot shows that –5 is a solution.
Shade all the points to the right of –5.
Shade all the points to the right of –2.
Graph the solutions of each inequality on a number
line.
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
a. x > –2
b. w –5 >–
2-8
(continued)
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
c. k 4
d. y < 6
<– A closed dot shows that 4 is a solution.
Shade all the points to the left of 4.
An open dot shows that 6 is not a solution.
Shade all the points to the left of 6. Quick Check
2-8
Write the inequality shown in each graph.
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
a.
b.
x –3 >–
x < 3
2-8
Quick Check
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.
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
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
2-8
(continued)
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
=
35 mg sodiumWords
number of milligrams of sodium in a serving ofvery low sodium food.
v
has at mosta serving of very low sodium food
Let
Equation 35v <–
2-8
Quick Check
Inequalities and Their GraphsInequalities and Their GraphsPRE-ALGEBRA LESSON 2-8PRE-ALGEBRA LESSON 2-8
x > –30
1. Graph –7 w.
2. Write an inequality for the graph.
3. A child must be at least 50 in. tall to ride on the roller coaster. Write an inequality for this situation.
>–
h 50>–
2-8
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or SubtractingPRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
2-9
Solve each equation.
a. n + 4 = 3 b. 4 – p = 8
n = –1 p = – 4
PRE-ALGEBRA LESSON 2-9PRE-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 You’ll Need
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
PRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
Solutions
1. 2. m + 7 – 7 = 5 – 7 k – 8 + 8 = 11 + 8
3. 4.12 – 12 + h = 21 – 12
m + 7 = 5
m = – 2
12 + h = 21
h = 9
6 = n – 23
29 = n6 + 23 = n – 23 + 23
k – 8 = 11
k = 19
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
Solve each inequality. Graph the solutions.
PRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
a.
b.
4 + s – 4 < 12 – 4 Subtract 4 from each side.
4 + s < 12
s < 8 Simplify.
4 + s < 12
–16 y – 14 >––16 y – 14 > –
–16 + 14 y – 14 + 14 Add 14 to each side.>– –2 y or y –2 Simplify.>– <–
Quick Check
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
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?
PRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
= storage space available.sLet
storage space for our filesWords
is less than or equal toplus
totalspace
storage space left
Inequality 2 s 6+ <–
2 + s 6<–
2 – 2 + s 6 – 2 Subtract 2 from each side.<–
s 4 Simplify.<–
No more than 4 GB are left.
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
Quick Check
Solve –10 < –13 + q.
PRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
–10 < –13 + q
–10 + 13 < –13 + 13 + q Add 13 to each side.
3 < q Simplify.
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
Quick Check
PRE-ALGEBRA LESSON 2-9PRE-ALGEBRA LESSON 2-9
Solve each inequality.
1. e + 4 14 2. –22 g – 6
3. A number q plus the opposite of 5 is less than or equal to 0.
>–<––16 g>–e 10<–
q 5<–
2-9
Solving One-Step Inequalities by Adding or SubtractingSolving One-Step Inequalities by Adding or Subtracting
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or DividingPRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
Solve each equation.
a. q ÷ 3 = 6 b. 7r = 21
q = 18 r = 3
PRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
(For help, go to Lesson 2-6.)
Solve each equation.
1. 6x = 24 2. 63 = –7v
3. = 10 4. = 48 t6
x–2
Check Skills You’ll Need
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
Solutions
1. 2.
3. 4.
–2 = –2(10)
6x = 24
x = 4
= 10
x = –20
= 48
t = 288
6 = 48(6)
63 = –7v
–9 = v
x–2
6x6
246
63–7
–7v–7
x–2
==
t6
t6
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-10PRE-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?
225x 2,000<–
Let x = number of televisions.
Inequality
Wordsnumber oftelevisions
225 lb 2,000 lbtimesis less thanor equal to
x • 225 2,000<–
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
(continued)
Divide each side by 225.<–2,000225
225x225
x 8.8 Simplify. Round the answer down to find a wholenumber of television sets.
<–
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
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing
PRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
Solve –2.z–8
<–
–2z
–8<–
Multiply each side by –8 and reverse the inequality symbol.
–8(–2)z
–8–8 >–
Simplify.z 16>–
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing
Quick Check
>–
Solve each inequality.
1. 3x –27 2. –5w > 15
3. y 4. 0
5. 4f > –12
–x5
<–14
12
<–
PRE-ALGEBRA LESSON 2-10PRE-ALGEBRA LESSON 2-10
w < –3
x 0>–
x –9>–
f > –3
y 2<–
2-10
Solving One-Step Inequalities by Multiplying or DividingSolving One-Step Inequalities by Multiplying or Dividing