Download Chapter 2 Section 8

Chapter 2
Equations,
Inequalities and
Problem Solving
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Bellwork:
A
C
1. Solve Ax  By  C for y. y   x 
B
B
9
5
160
2. Solve F  C  32 for C.
C F
5
9
9

A 
I
3. Solve Q 
for I.
I  A  QL
L


4. SolveY  a  m(x  b) for x. Y  a  b  x
m


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2.8
Linear Inequalities and
Problem Solving
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Objectives:
Graph
inequalities on a number line
Use the addition/multiplication
property to solve inequalities
Solve problems modeled by
inequalities
Write solutions in interval notation
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Linear Inequalities
An inequality is a statement that contains
one 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 bracket (closed circle) at the endpoint
of a interval if you want to include the point
• Use a parenthesis (open circle) at the endpoint
if you DO NOT want to include the point
Represents the set {xx  7}
Represents the set {xx > – 4}
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Graphing Solutions
Graphing solutions to linear inequalities in one
variable
• Use a number line
• Use a bracket (closed circle) at the endpoint
of a interval if you want to include the point
• Use a parenthesis (open circle) at the endpoint
if you DO NOT want to include the point
]
(
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Represents the interval (-∞, 7]
Represents the interval (-4, ∞)
7
Interval Notation
Set Notation
Interval Notation
Graph
x<#
(-∞, #)
x≤#
(-∞,#]
x>#
(#, ∞)
x≥#
[#, ∞)
)
]
(
[
smallest to LARGEST
•Parenthesis when the endpoint is not included
•Brackets when the endpoint is included
•Infinity always has a parenthesis
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Example
Which is the graph of 2  x  5 ?
]
)
-2
5
)
]
-2
5
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Addition Property of Inequality
If a, b, and c are real numbers, then
a < band a + c < b + c
are equivalent inequalities.
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10
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.
stays the
same
2. If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc
are equivalent inequalities.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
sign
flips
11
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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12
Example
Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5(x – 1)
3x + 9 ≥ 5x – 5
3x – 3x + 9 ≥ 5x – 3x – 5
9 ≥ 2x – 5
9 + 5 ≥ 2x – 5 + 5
14 ≥ 2x
7≥x
Apply the distributive property.
Subtract 3x from both sides.
Simplify.
Add 5 to both sides.
Simplify.
Divide both sides by 2.
Interval
Notation
(-∞, 7]
[
7
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13
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.
Interval
9
x
Divide both sides by 4.
Notation
2
(-9/2, ∞)
)
-9/2
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Example
Solve: 3x + 8 ≥ 5. Graph the solution set.
3x  8  5
3x  8  8  5  8
3 x  3
3x 3

3
3
x  1
Interval notation
[-1,∞)
]
-1
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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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Closure:
1.
2.
3.
What does interval notation
represent?
What does a parenthesis mean? a
bracket?
Name 3 words/phrases that mean
“less than or equal to”
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