Question

Solve each inequality. Then graph the solution set.

$\left|p+2\right|\mathbf{>}\mathbf{7}$

Short answer

The solution for the given inequality $\left|p+2\right|>7$is $\mathit{p}\in \left(-\infty ,-\mathbf{9}\right)\cup \left(\mathbf{5},\infty \right)$.

The graph of the solution set which is$p\in \left(-\infty ,-9\right)\cup \left(5,\infty \right)$ is:

Step-by-step solution

Step 1. Solve the given inequality |p+2|>7.

The solution of the given inequality$\left|p+2\right|>7$ is:

Case 1:$p+2$ is non-negative.

$\begin{array}{c}p+2>7\\ p+2-2>7-2\\ p>5\\ p\in \left(5,\infty \right)\end{array}$

Case 2:$p+2$ is negative.

$\begin{array}{c}-\left(p+2\right)>7\\ \left(-1\right)\left(-\left(p+2\right)\right)<\left(-1\right)\left(7\right)\\ p+2<-7\\ p+2-2<-7-2\\ p<-9\\ p\in \left(-\infty ,-9\right)\end{array}$

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$p+2>7$ and$p+2<-7$ to find the solution of the inequality $\left|p+2\right|>7$.

The union of the solutions of the inequalities$p+2>7$ and$p+2<-7$ is:

$p\in \left(-\infty ,-9\right)\cup \left(5,\infty \right)$

Therefore, the solution of the inequality$\left|p+2\right|>7$ is $p\in \left(-\infty ,-9\right)\cup \left(5,\infty \right)$.

Step 2. Draw the graph of the solution set which is p∈(−∞,−9)∪(5,∞).

The graph of the solution set which is $p\in \left(-\infty ,-9\right)\cup \left(5,\infty \right)$is: