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## Integrated math 1

### Course: Integrated math 1>Unit 2

Lesson 6: Compound inequalities

# Compound inequalities review

A compound inequality is an inequality that combines two simple inequalities. This article provides a review of how to graph and solve compound inequalities.

## What is a compound inequality?

A compound inequality is an inequality that combines two simple inequalities. Let's take a look at some examples.

### Example with "OR"

So, for example, the numbers $0$ and $6$ are both solutions of the compound inequality, but the number $4$ is not a solution.

### Example with "AND"

This compound inequality is true for values that are both greater than zero and less than four. Graphically, we represent it like this:
So, in this case, $2$ is a solution of the compound inequality, but $5$ is not because it only satisfies one of the inequalities, not both.
Note: If we wanted to, we could write this compound inequality more simply like this:
$0

## Solving compound inequalities

### Example with "OR"

Solve for $x$.
Solving the first inequality for $x$, we get:
$\begin{array}{rl}2x+3& \ge 7\\ \\ 2x& \ge 4\\ \\ x& \ge 2\end{array}$
Solving the second inequality for $x$, we get:
$\begin{array}{rl}2x+9& >11\\ \\ 2x& >2\\ \\ x& >1\end{array}$
Graphically, we get:
So our compound inequality can be expressed as the simple inequality:
$x>1$

### Example with "AND"

Solve for $x$.
Solving the first inequality for $x$, we get:
$\begin{array}{rl}4x-39& >-43\\ \\ 4x& >-4\\ \\ x& >-1\end{array}$
Solving the second inequality for $x$, we get:
$\begin{array}{rl}8x+31& <23\\ \\ 8x& <-8\\ \\ x& <-1\end{array}$
Graphically, we get:
Strangely, this means that there are no solutions to the compound inequality because there's no value of $x$ that's both greater than negative one and less than negative one.
Solve for $x$.