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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 3.3
Linear Inequalities
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Basic Concepts
• Symbolic Solutions
• Numerical and Graphical Solutions
• An Application
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Basic Concepts
An inequality results whenever the equals sign in an equation is replaced with any one of the symbols <, ≤, >, or ≥.
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.
Slide 3
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 0 and 1,xf x a a a A linear inequality in one variable is an inequality that can be written in the form
ax + b > 0,
where a ≠ 0. (The symbol > may be replaced with ≥, <, or ≤.)
LINEAR INEQUALITY IN ONE VARIABLE
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 0 and 1,xf x a a a
The solution set for ax + b > 0 with a ≠ 0 is either {x│x < k} or {x│x > k}, where k is the solution to ax + b = 0 and corresponds to the x-intercept for the graph of y = ax + b. Similar statements can be made for the symbols <, ≤, and ≥.
SOLUTION SET FOR A LINEAR INEQUALITY
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Let a, b, and c be real numbers.
1. a < b and a + c < b + c are equivalent. (The same number may be added to or subtracted from each side of an inequality.)
2. If c > 0, then a < b and ac < bc are equivalent. (Each side of an inequality may be multiplied or divided by the same positive number.)
3. If c < 0, then a < b and ac > bc are equivalent. (Each side of an equality may be multiplied or divided by the same negative number provided the inequality symbol is reversed.)
Similar properties exist for ≤ and ≥ symbols.
PROPERTIES OF INEQUALIES
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the inequality. 5x – 2 > 9Solution
5x – 2 > 9
5x – 2 + 2 > 9 + 2
5x > 11 5 11
5 5
x
11
5x
The solution set is . 11
5x x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the inequality. Solution
12 5 1
3n n
12 5 1
3n n
2 3 5 3n n
2 15 3 3n n 4 17 3n
1717 3 74 1n
4 20n
5n
The solution set is {n│n ≤ 5}.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the inequality; 9 – 4x ≤ −x − 6.Solution
Next divide each side by −3. As when we are dividing by a negative number, Property 3 requires reversing the inequality by changing ≤ to ≥.
9 – 4x ≤ −x – 6 9 – 4x – 9 ≤ −x – 6 – 9
−4x ≤ −x – 15 −4x + x ≤ −x – 15 + x
−3x ≤ – 15 3 15
3 3
x
5x
The solution set is {x│x ≥ 5}.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Use the table to find the solution set to each equation or inequality.a. b. c.
x −1 0 1 2 3
3 033
2x
9
23
2
3
2
33 0
2x
33 0
2x
33 0
2x
a. The expression equals 0 when x = 2. Thus the solution set is {x│x = 2}.
33
2x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Use the table to find the solution set to each equation or inequality.a. b. c.
x −1 0 1 2 3
3 033
2x
9
23
2
3
2
33 0
2x
33 0
2x
33 0
2x
b. The expression is positive when x < 2. Thus the solution set is {x│x < 2}.
33
2x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Use the table to find the solution set to each equation or inequality.a. b. c.
x −1 0 1 2 3
3 033
2x
9
23
2
3
2
33 0
2x
33 0
2x
33 0
2x
c. The expression is negative when x > 2. Thus the solution set is {x│x > 2}.
33
2x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Use the graph to find the solution set to each equation or inequality. a.
The graph of crosses the x-axis at x = 2. Thus the solution set is
33 0
2x
33
2y x
2 .x x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Use the graph to find the solution set to each equation or inequality. b.
The graph is above the x-axis when x < 2. Thus the solution set is {x│x < 2}.
33 0
2x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Use the graph to find the solution set to each equation or inequality. c.
The graph is above the x-axis when x > 2. Thus the solution set is {x│x > 2}.
33 0
2x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Use a graph to solve 7 – 4x ≤ x – 8. SolutionThe graph of y1 = 7 – 4x and y2 = x – 8 intersect at the point (3, −5). The graph of y1 is below the graph of y2 when x > 3. Thus 7 – 4x ≤ x – 8 is satisfied when x ≥ 3. Therefore the solution set is {x│x ≥ 3}.
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