Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as [latex]4

Example 7: Solving a Compound Inequality

Solve the compound inequality: $3\le 2x+2<6$ .

Solution

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

\begin{array}{lll} 3 \leq 2 x + 2 & and & 2 x + 2 < 6 \\ 1 \leq 2 x & 2 x < 4 \\ \frac{1}{2} \leq x & x < 2 \end{array}

$\begin{array}{lll}3\le 2x+2\hfill & \text{and}\hfill & 2x+2<6\hfill \\ 1\le 2x\hfill & \hfill & 2x<4\hfill \\ \frac{1}{2}\le x\hfill & \hfill & x<2\hfill \end{array}$

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

\frac{1}{2} \leq x < 2

$\frac{1}{2}\le x<2$

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

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

\begin{array}{ll} 3 \leq 2 x + 2 < 6 \\ 1 \leq 2 x < 4 & Isolate the variable term, and subtract 2 from all three parts . \\ \frac{1}{2} \leq x < 2 & Divide through all three parts by 2 . \end{array}

$\begin{array}{ll}3\le 2x+2<6\hfill & \hfill \\ 1\le 2x<4\hfill & \text{Isolate the variable term, and subtract 2 from all three parts}.\hfill \\ \frac{1}{2}\le x<2\hfill & \text{Divide through all three parts by 2}.\hfill \end{array}$

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

Try It 7

Solve the compound inequality $4<2x - 8\le 10$ . Solution

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

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

Solution

Lets try the first method. Write two inequalities:

\begin{array}{lll} 3 + x > 7 x - 2 & and & 7 x - 2 > 5 x - 10 \\ 3 > 6 x - 2 & 2 x - 2 > - 10 \\ 5 > 6 x & 2 x > - 8 \\ \frac{5}{6} > x & x > - 4 \\ x < \frac{5}{6} & - 4 < x \end{array}

$\begin{array}{lll}3+x> 7x - 2\hfill & \text{and}\hfill & 7x - 2> 5x - 10\hfill \\ 3> 6x - 2\hfill & \hfill & 2x - 2> -10\hfill \\ 5> 6x\hfill & \hfill & 2x> -8\hfill \\ \frac{5}{6}> x\hfill & \hfill & x> -4\hfill \\ x< \frac{5}{6}\hfill & \hfill & -4< x\hfill \end{array}$

The solution set is [latex]-4 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: $3y<4 - 5y<5+3y$ . Solution

Linear Inequalities and Absolute Value Inequalities