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Section 2.5. Linear Inequalities. Page 136. Solutions and Number Line Graphs. A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols , or ≥. x > 5, 3 x + 4 < 0, 1 – y ≥ 9 - PowerPoint PPT Presentation
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Copyright © 2013 Pearson Education, Inc.
Section 2.5
Linear Inequalities
Solutions and Number Line Graphs
A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols <, ≤, >, or ≥.
x > 5, 3x + 4 < 0, 1 – y ≥ 9
A solution to an inequality is a value of the variable that makes the statement true. The set of all solutions is called the solution set.
Page 136
Example
Use a number line to graph the solution set to each inequality.
a.
b.
c.
d.
1x
1x
3 x
2x
Page 137
31 .
1 b.
3 .
xc
x
xa
Linear Inequalities in One Variable
)
[
](
Page 137
Interval Notation
Each number line graphed on the previous slide represents an interval of real numbers that corresponds to the solution set to an inequality.Brackets and parentheses can be used to represent the interval. For example:
1x (1, )
1x [1, )
Page 137
Example Interval Notation
Write the solution set to each inequality in interval notation.
a. b.
Solution
a. b.
More examples
6x 2y
(6, ) ( , 2]
5
0.
x
x
d.
c
)5,(
),0[
Page 137
Example Checking a Solution
Determine whether the given value of x is a solution to the inequality.Solution
4 2 8, 7x x
?
?
4 2 <8
4( 87) 2
x
?
28 2 8 ?
26 8 e Fals
Page 138
The Addition Property of InequalitiesPage 139
Example
Solve each inequality. Then graph the solution set.a. x – 2 > 3 b. 4 + 2x ≤ 6 + xSolution
a. x – 2 > 3
x – 2 + 2 > 3 + 2
x > 5
b. 4 + 2x ≤ 6 + x
4 + 2x – x ≤ 6 + x – x
4 + x ≤ 6
4 – 4 + x ≤ 6 – 4
x ≤ 2
Page 139
96 x
Properties of Inequalities
)
[
)3,(
),2[
6 6 3x }3|{ xx
4728 xx2 2
278 xx7x-7x - 2x
}2|{ xx
Page 140
The Multiplication Property of InequalitiesPage 141
Example
Solve each inequality. Then graph the solution set.a. 4x > 12 b.Solutiona. 4x > 12
12
4x
4 2
4 4
1x
3x
b. 1
24x
4 (1
( 2)4
4) x
8 x
Page 141
Example
Solve each inequality. Write the solution set in set-builder notation.a. 4x – 8 > 12 b. Solutiona. 4x – 8 > 12
4 3 4 5x x
4 8 8 12 8x 4 20x
5x
{ | 5}x x
b. 4 3 4 5x x
4 3 3 4 5 3x x x x
4 5x
4 5 5 5x
9 x
{ | 9}x x
Page 142
24
1x
Properties of Inequalities
)8,(
8x }8|{ xx
4 4)
Page 142
186 x(
),3( 6 6
3x
Sign changes
2536 xx
Linear Inequalities
[
),1[
8 8
1
1
x
x
Add 2
xx 538 Add 3x
x88
same
Page 142
2536 xx853 xx
Subtract 6
Subtract 5x
88 x8- 8-
1xSign changes
14)2(31)3(2 xx
Linear Inequalities
1463162 xx
8372 xx Add 8
xx 312 Sub 2x
1
1
x
x
[
),1[
Page 142
Applications
To solve applications involving inequalities, we often have to translate words to mathematical statements.
Page 143
Example
Translate each phrase to an inequality. Let the variable be x.a. A number that is more than 25.
b. A height that is at least 42 inches.
x > 25
x ≥ 42
Page 143
Example
For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.
a. Write a formula that gives the cost C of producing x cases of snacks.
b. Write a formula that gives the revenue R from selling x cases of snacks.
C = 135x + 175,000
R = 250x
Page 144
Example (cont)
For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.
c. Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x cases of snacks.
P = R – C
= 250x – (135x + 175,000)
= 115x – 175,000
Page 144
Example (cont)
For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.
d. How many cases need to be sold to yield a positive profit?
115x – 175,000 > 0
115x > 175,000
x > 1521.74
Must sell at least 1522 cases.
Page 144
DONE
Objectives
• Solutions and Number Line Graphs
• The Addition Property of Inequalities
• The Multiplication Property of Inequalities
• Applications
EXAMPLEGraphing inequalities on a number line
Use a number line to graph the solution set to each inequality.
a.
b.
c.
d.
1x
1x
5x
2x
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