Question

Solve each inequality. Then graph the solution set.

$\mathbf{|}\mathbf{p}\mathbf{-}\mathbf{5}\mathbf{|}\mathbf{<}\mathbf{3}$

Short answer

The solution for the given inequality $\left|p-5\right|<3$is $\mathit{p}\mathbf{\in }\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{8}\mathbf{)}$.

The graph of the solution set which is $p\in \left(2,8\right)$is:

Step-by-step solution

Step 1. Solve the given inequality |p−5|<3.

The solution of the given inequality $\left|p-5\right|<3$is:

Case 1: $p-5$is non-negative.

$\begin{array}{c}p-5<3\\ p-5+5<3+5\\ p<8\\ p\in \left(-\infty ,8\right)\end{array}$

Case 2:$p-5$ is negative.

$\begin{array}{c}-\left(p-5\right)<3\\ \left(-1\right)\left(-\left(p-5\right)\right)>\left(-1\right)\left(3\right)\\ p-5>-3\\ p-5+5>-3+5\\ p>2\\ p\in \left(2,\infty \right)\end{array}$

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

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

Find the intersection of the solutions of the inequalities$p-5<3$ and$p-5>-3$ to find the solution of the inequality $\left|p-5\right|<3$.

The intersection of the solutions of the inequalities$p-5<3$ and$p-5>-3$ is:

$p\in \left(2,8\right)$

Therefore, the solution of the inequality $\left|p-5\right|<3$is $p\in \left(2,8\right)$.

Step 2. Draw the graph of the solution set which is p∈(2,8).

The graph of the solution set which is$p\in \left(2,8\right)$ is: