# Solving Compound Inequalities

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*compound inequality*. This can be done in more than one way, yielding different compound inequalities.

## Compound Inequality

Combining two or more inequalities with the word $``$and" or $``$or" yields what is called a compound inequality.

Compound inequality | Is read as |
---|---|

$x < 5$ or $x > 8$ | $x$ is less than $5$ or greater than $8.$ |

$x > 2$ and $x \leq 4$ | $x$ is greater than $2$ and less than or equal to $4.$ |

Compound inequalities using the word $``$and" are commonly written without the $``$and." For example, $x > 2 \ \text{and} \ x \leq 4,$ can be expressed as a compound inequality by rewriting the first inequality as $2 < x$ — the statement $``x$ is greater than $2"$ is equivalent to $``2$ is less than $x."$ The inequality is then $2 < x \text{ and } x \leq 4,$ which is commonly condensed into

$2 < x \leq 4.$## Solution Set of a Compound Inequality

The solution set of a compound inequality consists of the solution sets of the individual inequalities. For compound inequalities with $``$or," a solution of **either** individual inequality is a solution of the compound inequality. Therefore, the graph of the compound inequality is the union of the graphs of the individual inequalities. These graphs are recognized by the fact that they continue infinitely in either direction.

A solution of a compound inequality with $``$and," however, must be a solution of **both** individual inequalities. Thus, the graph of the compound inequality is the intersection of the graphs of the individual inequalities. These graphs do **not** extend infinitely.

## Solving a Compound Inequality

Solving a compound inequality is done by first separating it into its individual inequalities, which are then solved one at a time as normal. Lastly, their solution sets are combined. As an example, the following inequality can be solved using this method. $\text{-} 3 < 2x - 1 \leq 2$

## Exercises

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