Question

Solve each inequality. Then graph the solution set.

$\left|f-9\right|\mathbf{\ge }\mathbf{2}$

Short answer

The solution for the given inequality$\left|f-9\right|\ge 2$ is $\mathit{f}\in \left(-\infty ,\mathbf{7}\right]\cup \left[\mathbf{11},\infty \right)$.

The graph of the solution set which is$f\in \left(-\infty ,7\right]\cup \left[11,\infty \right)$ is:

Step-by-step solution

Step 1. Solve the given inequality .

The solution of the given inequality$\left|f-9\right|\ge 2$ is:

Case 1:$f-9$ is non-negative.

$\begin{array}{c}f-9\ge 2\\ f-9+9\ge 2+9\\ f\ge 11\\ f\in \left[11,\infty \right)\end{array}$

Case 2:$f-9$ is negative.

$\begin{array}{c}-\left(f-9\right)\ge 2\\ \left(-1\right)\left(-\left(f-9\right)\right)\le \left(-1\right)\left(2\right)\\ f-9\le -2\\ f-9+9\le -2+9\\ f\le 7\\ f\in \left(-\infty ,7\right]\end{array}$

The solution of the inequality $\left|x\right|\ge a$is $x\le -a$or $x\ge a$.

That implies the solution of the inequality$\left|x\right|\ge a$ is the union of the solutions of the inequalities$x\le -a$ and $x\ge a$.

Find the union of the solutions of the inequalities$f-9\ge 2$ and$f-9\le -2$ to find the solution of the inequality $\left|f-9\right|\ge 2$.

The union of the solutions of the inequalities$f-9\ge 2$ and$f-9\le -2$ is:

$f\in \left(-\infty ,7\right]\cup \left[11,\infty \right)$

Therefore, the solution of the inequality$\left|f-9\right|\ge 2$ is $f\in \left(-\infty ,7\right]\cup \left[11,\infty \right)$.

Step 2. Draw the graph of the solution set which is f∈(−∞,7]∪[11,∞).

The graph of the solution set which is$f\in \left(-\infty ,7\right]\cup \left[11,\infty \right)$ is: