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# Solving Compound Inequalities

Objective Learn how to solve compound inequalities.

In this lesson, we are dealing with compound inequalities, expressions in which more than one inequality applies to the same variable. When inequalities are multiplied by a negative number, the inequality symbol must be reversed.

# Compound Inequalities

Let's talk a little about the notation and the terminology.

A compound inequality is an expression like 3 3x + 4 < 9 or 1x 2 7. The two inequality symbols indicate that the variable is to satisfy each part. So, for example, the inequality 2 < 2x - 5 < 4 is read “two is less than 2x - 5 and 2x - 5 is less than four” or “2x - 5 is between 2 and 4”.

## Solving Compound Inequalities

First, remember that solving an inequality (compound or otherwise) means describing its solution set, since there is more than one solution. Compound inequalities can be solved using the same methods that are used to solve single inequalities, the Addition and Multiplication Properties of Inequalities.

Example 1

Solve 3 2 x + 1 8.

Solution

First, subtract 1 from each part so that no constants appear in the expression that contains the variable.

 3 2x + 1 < 8 2 2x < 7 Subtract 1 from each part. 1 x < Divide each part by 2.

The solution set is Why is this considered solving the inequality if, after all, we still have an inequality? Why is this one any better than the original inequality?

The answer is that the inequalities now apply directly to x, not to an algebraic expression involving x. This means that it is much easier to visualize the solution set, as the points to the right of and including 1 and to the left of . Solving compound inequalities means isolating x in the inequalities.

Example 2

Solve -1 3 - x 5.

Solution

First, subtract 3 from each part to remove any constants from the expression that contains x.

 -1 3 - x 5 -4 -x 2 Subtract 3 from each side.

Next, multiply each part of the inequality by -1 in order to isolate x in the middle. Since -1 is negative, reverse the inequality symbols.

4 x -2

The solution set is { x | 4 x -2 }.