Question

Solve each inequality. Then graph the solution set.$|\mathbf{2}f+\mathbf{7}|\ge \mathbf{21}$

Short answer

The solution set of inequality is $\mathbf{\left\{}\mathbf{f}\mathbf{|}\mathbf{f}\mathbf{\le }\mathbf{-}\mathbf{14}\mathbf{or}\mathbf{f}\mathbf{\ge }\mathbf{7}\mathbf{\right\}}$.

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, $|2f+7|\ge 21$ is nonnegative.

$\begin{array}{c}2f+7\ge 21\\ 2f+7-7\ge 21-7\\ \frac{2f}{2}\ge \frac{14}{2}\\ f\ge 7\end{array}$

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

According to case 2, $|2f+7|\ge 21$is negative.

$\begin{array}{c}-(2f+7)\ge 21\\ 2f+7\le -21\\ 2f+7-7\le -21-7\\ \frac{2f}{2}\le \frac{-28}{2}\\ f\le -14\end{array}$

Therefore, the solution is $f\ge 7$or $f\le -14$.

Step 4. Write solution set and graph the solution.

The solution set of given inequality is $\left\{t|f\le -14orf\ge 7\right\}$.

Closed circles at 7 and $-14$shows these points are included into the solution set.