Transcript Document
Chapter 9
Equations,
Inequalities and
Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
9.7
Linear Inequalities and
Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Linear Inequalities
An inequality is a statement that contains of the
symbols: < , >, ≤ or ≥.
Equations
x=3
Inequalities
x>3
12 = 7 – 3y
12 ≤ 7 – 3y
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Graphing Solutions
Graphing solutions to linear inequalities in one
variable
• Use a number line
• Use a closed circle at the endpoint of a interval
if you want to include the point
• Use an open circle at the endpoint if you DO
NOT want to include the point
Represents the set {xx 7}
7
Represents the set {xx > – 4}
-4
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Example
Graph: 2 x 5
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Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c
are equivalent inequalities.
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Multiplication Property of Inequality
1. If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc
are equivalent inequalities.
2. If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc
are equivalent inequalities.
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Solving Linear Inequalities
To Solve Linear Inequalities in One Variable
Step 1: If an inequality contains fractions, multiply both sides by
the LCD to clear the inequality of fractions.
Step 2: Use distributive property to remove parentheses if they
appear.
Step 3: Simplify each side of inequality by combining like
terms.
Step 4: Get all variable terms on one side and all numbers on the
other side by using the addition property of inequality.
Step 5: Get the variable alone by using the multiplication
property of inequality.
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Example
Solve: 3x + 8 ≥ 5. Graph the solution set.
3x 8 5
3x 8 8 5 8
3 x 3
3 x 3
3
3
x 1
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Example
Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5(x – 1)
3x + 9 ≥ 5x – 5
Apply the distributive property.
3x – 3x + 9 ≥ 5x – 3x – 5
9 ≥ 2x – 5
9 + 5 ≥ 2x – 5 + 5
14 ≥ 2x
7≥x
The graph of solution set
is{x|x ≤ 7}.
Subtract 3x from both sides.
Simplify.
Add 5 to both sides.
Simplify.
Divide both sides by 2.
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Example
Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set.
7(x – 2) + x > –4(5 – x) – 12
7x – 14 + x > –20 + 4x – 12
Apply the distributive property.
8x – 14 > 4x – 32
Combine like terms.
8x – 4x – 14 > 4x – 4x – 32
Subtract 4x from both sides.
4x – 14 > –32
Simplify.
4x – 14 + 14 > –32 + 14
Add 14 to both sides.
4x > –18
Simplify.
9
x
2
Divide both sides by 4.
The graph of solution set
is {x|x > –9/2}.
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Example
You are having a catered event. You can spend at most
$1200. The set up fee is $250 plus $15 per person, find the
greatest number of people that can be invited and still stay
within the budget.
Let x represent the number of people
Set up fee + cost per person × number of people ≤ 1200
250 +
15x
≤ 1200
continued
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continued
You are having a catered event. You can spend at most $1200. The set up fee is
$250 plus $15 per person, find the greatest number of people that can be invited
and still stay within the budget.
250 15x 1200
15 x 950
15 x 950
15
15
x 63.3
The number of people who can be invited must be 63 or
less to stay within the budget.
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