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 Dependent Variable

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# 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.