91Ó°ÊÓ

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

Case 1: $x−4$is non-negative.

$\begin{array}{c}x−4<9\\ x−4+4<9+4\\ x<13\\ x∈\left(−∞,13\right)\end{array}$

Case 2: $x−4$is negative.

$\begin{array}{c}−\left(x−4\right)<9\\ \left(−1\right)\left(−\left(x−4\right)\right)>\left(−1\right)\left(9\right)\\ x−4>−9\\ x−4+4>−9+4\\ x>−5\\ x∈\left(−5,∞\right)\end{array}$

The solution of the inequality $\left|x\right|≤a$is $xâ‰\yen −a$and $x≤a$.

That implies the solution of the inequality $\left|x\right|≤a$is the intersection of the solutions of the inequalities $xâ‰\yen −a$and $x≤a$.

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: $x∈\left(−5,13\right)$

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

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