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Study Guides > College Algebra

Understanding Compound Inequalities

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].


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.
[latex]\begin{array}{lll}3\le 2x+2\hfill & \text{and}\hfill & 2x+2<6\hfill \\ 1\le 2x\hfill & \hfill & 2x<4\hfill \\ \frac{1}{2}\le x\hfill & \hfill & x<2\hfill \end{array}[/latex]
Then, we can rewrite the solution as a compound inequality, the same way the problem began.
[latex]\frac{1}{2}\le x<2[/latex]
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.
[latex]\begin{array}{ll}3\le 2x+2<6\hfill & \hfill \\ 1\le 2x<4\hfill & \text{Isolate the variable term, and subtract 2 from all three parts}.\hfill \\ \frac{1}{2}\le x<2\hfill & \text{Divide through all three parts by 2}.\hfill \end{array}[/latex]
We get the same solution: [latex]\left[\frac{1}{2},2\right)[/latex].

Try It 7

Solve the compound inequality [latex]4<2x - 8\le 10[/latex]. Solution

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].


Lets try the first method. Write two inequalities:
[latex]\begin{array}{lll}3+x> 7x - 2\hfill & \text{and}\hfill & 7x - 2> 5x - 10\hfill \\ 3> 6x - 2\hfill & \hfill & 2x - 2> -10\hfill \\ 5> 6x\hfill & \hfill & 2x> -8\hfill \\ \frac{5}{6}> x\hfill & \hfill & x> -4\hfill \\ x< \frac{5}{6}\hfill & \hfill & -4< x\hfill \end{array}[/latex]
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.
A number line with the points -4 and 5/6 labeled. Dots appear at these points and a line connects these two dots. Figure 3

Try It 8

Solve the compound inequality: [latex]3y<4 - 5y<5+3y[/latex]. Solution

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