Question

Solve each inequality. Then graph the solution set.$\mathbf{\left|}\frac{\mathbf{2}\mathbf{t}\mathbf{+}\mathbf{6}}{2}\mathbf{\right|}\mathbf{>}\mathbf{10}$

Short answer

The solution set of inequality is $\left\{t|t<-13ort>7\right\}$.

The solution set on number line is

Step-by-step solution

Step 1. Write two cases of absolute value inequality.

For solving absolute value inequalities, there are two cases

Case 1- The expression inside the absolute value symbols is nonnegative.

Case 2- The expression inside the absolute value symbols is negative.

Step 2. Write given inequality for case 1 and solve.

According to case 1, $\left|\frac{2t+6}{2}\right|>10$is nonnegative.

$\begin{array}{c}\frac{2t+6}{2}>10\\ 2\left(\frac{2t+6}{2}\right)>2\left(10\right)\\ 2t+6-6>20-6\\ \frac{2t}{2}>\frac{14}{2}\\ t>7\end{array}$

Step 3. Write given inequality for case 2 and solve.

According to case 2,$\left|\frac{2t+6}{2}\right|>10$ is negative.

$\begin{array}{c}-\left(\frac{2t+6}{2}\right)>10\\ 2\left(\frac{2t+6}{2}\right)<-2\left(10\right)\\ 2t+6-6<-20-6\\ \frac{2t}{2}<\frac{-26}{2}\\ t<-13\end{array}$

Therefore, the solution is$t<-13$ or $t<-13$.

Step 4. Write solution set and graph the solution.

The solution set of given inequality is $\left\{t|t<-13ort>7\right\}$.

Open circles at 7 and$-13$ shows these points are excluded from the solution set.