Question

Solve each inequality. Then graph the solution set.

$\left|x-4\right|\mathbf{<}\mathbf{9}$

Short answer

The solution for the given inequality$\left|x-4\right|<9$is localid="1647431389629" $\mathit{x}\mathbf{\in }\left(-5,13\right)$.

The graph of the solution set which is$x\in \left(-5,13\right)$is:

Step-by-step solution

Step 1. Solve the given inequality |x−4|<9.

The solution of the given inequality $\left|x-4\right|<9$is:

Case 1: $x-4$is non-negative.

$\begin{array}{c}x-4<9\\ x-4+4<9+4\\ x<13\\ x\in \left(-\infty ,13\right)\end{array}$

Case 2: $x-4$is negative.

$\begin{array}{c}-\left(x-4\right)<9\\ \left(-1\right)\left(-\left(x-4\right)\right)>\left(-1\right)\left(9\right)\\ x-4>-9\\ x-4+4>-9+4\\ x>-5\\ x\in \left(-5,\infty \right)\end{array}$

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 $x-4<9$and $x-4>-9$to find the solution of the inequality $\left|x-4\right|<9$.

The intersection of the solutions of the inequalities $x-4<9$and $x-4>-9$is: $x\in \left(-5,13\right)$

Therefore, the solution of the inequality $\left|x-4\right|<9$is $x\in \left(-5,13\right)$.

Step 2. Draw the graph of the solution set which is x∈(−5,13).

The graph of the solution set which is$x\in \left(-5,13\right)$ is: