Solving Linear Inequalities
An inequality can be formed by linking two expressions with an
inequality symbol <, ≤, >, or
≥.
A linear inequality in one variable is an inequality that can be written in
the form
ax + b > c where a, b, and c are real numbers, a
≠ 0, x is a variable, and > may be
replaced by ≥, <, or
≤.
Note:
Here are the definitions of the inequality
symbols:
< means "less than", as in 4 < 5.
≤ means "less than or equal to",
as in 3 ≤ 8 or 8
≤ 8.
> means "greater than", as in 6 > 1.
≥ means "greater than or equal to", as in
2
≥ 0 or 2
≥ 2.
Here are some examples of linear inequalities:
3x < 18 
8w  7 > 33 
7x  13 ≤ 2(4x + 5) 
A linear inequality can be solved using the same steps as when solving a
linear equation, but with one important difference:
When you multiply or divide an inequality by a negative number,
you must reverse the direction of the inequality symbol.
For example:
Reverse the inequality symbol.
Simplify. 
2x < 6
x > 3 
Note:
When you divide into a positive number,
do not reverse the inequality.
Example 1
Solve: 8w  7 > 33. Then, graph the solution on a number line.
Solution
Add 7 to both sides.
Simplify.
Divide both sides by 8 and reverse the
direction of the inequality symbol.
Simplify. 
8w  7 > 33 8w  7 + 7 > 33 + 7
8w > 40
w < 5 
To graph the solution, plot an open circle on the number line at 5.
Then, shade the number line to the left of 5.
Note:
When graphing the solution of an
inequality on a number line, remember the
following:
â€¢ Use an open circle, Â°, if the inequality
symbol is < or >.
An open circle indicates the point is
NOT part of the solution.
â€¢ Use a closed circle, â€¢, if the inequality
symbol is ≤ or
≥.
A closed circle indicates the point is
part of the solution.
