Download Section 2.6 – Linear Inequalities 1 Section 2.6 Linear Inequalities A

Section 2.6
Linear Inequalities
A inequality in the variable x is linear if each term is a constant or a multiple of x.
The inequality will contain an inequality symbol:
<

>

is less than
is less than or equal to
is greater than
is greater than or equal to
Let’s first review inequality notation, their graphs and interval notation.
( the number is not included
Inequality Notation
[ the number is included
Graph
Interval Notation
x5
x5
x5
x5
5  x  5
5  x  5
5  x  5
5  x  5
All real Numbers
To solve an inequality containing a variable, find all values of the variable that make the inequality true. Solve
them like solving a linear equation, but if you multiply or divide both sides of the inequality by a negative
number YOU MUST CHANGE THE DIRECTION OF THE INEQUALITY.
Section 2.6 – Linear Inequalities
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Let’s see why…
Given: 1   x  3
Example 1: Solve each of the following inequalities. Graph the solution set and write the solution set in
interval notation.
a. 2(7 – 4x) > 2 + 8x
b. -3 < -2x + 1 < 7
Example 2: Solve each of the following inequalities. Write the solution set in interval notation.
a
5 1
1
 x  ( x  5)
12 3
6
Section 2.6 – Linear Inequalities
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b
3( x  1) 7
 x
8
2
c. 35  5 x  5( x  7) / 2  70
Section 2.6 – Linear Inequalities
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Sometimes linear inequalities may have no solution or infinitely many solutions. Let’s look at a couple of
problems.
Example 3: Solve each of the following inequalities, if possible.
a. 2( x  3)  5 x  3 x  8
b. x  5  3x  4( x  1)
Section 2.6 – Linear Inequalities
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