A **compound inequality** includes two inequalities in one statement. A statement such as [latex]4<x\le 6[/latex] means [latex]4<x[/latex] and [latex]x\le 6[/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

### Example 7: Solving a Compound Inequality

Solve the compound inequality: [latex]3\le 2x+2<6[/latex].

### Solution

The first method is to write two separate inequalities: [latex]3\le 2x+2[/latex] and [latex]2x+2<6[/latex]. We solve them independently.

Then, we can rewrite the solution as a compound inequality, the same way the problem began.

In interval notation, the solution is written as [latex]\left[\frac{1}{2},2\right)[/latex].

The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.

We get the same solution: [latex]\left[\frac{1}{2},2\right)[/latex].

### Example 8: Solving a Compound Inequality with the Variable in All Three Parts

Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[/latex].

### Solution

Lets try the first method. Write two inequalities**:**

The solution set is [latex]-4<x<\frac{5}{6}[/latex] or in interval notation [latex]\left(-4,\frac{5}{6}\right)[/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.