91Ó°ÊÓ

$\left|x−4\right|<2$

The solution to the inequality $\left|x\right|<a$is $x<a$$x>−a$and $x<a$

That implies the solution of the inequality $\left|x\right|<a$is the intersection of the solutions of the inequalities $x>−a$ and $x<a$.

The solution to the given inequality $\left|x−4\right|<2$is

Case I: $x−4$is non-negative.

$\begin{array}{c}x−4<2\\ x−4+4<2+4\\ x<6\\ x∈\left(−∞,6\right)\end{array}$

Case 2: $x−4$is negative.

$\begin{array}{c}−\left(x−4\right)<2\\ \left(−1\right)\left(−\left(x−4\right)\right)>\left(−1\right)\left(2\right)\\ x−4>−2\\ x−4+4>−2+4\\ x>2\\ x∈\left(2,∞\right)\end{array}$

The solution to the inequality$\left|x\right|<a$is $x>−a$and $x<a$

That implies the solution of the inequality $\left|x\right|<a$is the intersection of the solutions of the inequalities $x>−a$and $x<a$.

Find the intersection of the solutions of the inequalities $x−4<2$and $x−4>−2$find the solution of the inequality $\left|x−4\right|<2$.

The intersection of the solutions of the inequalities $x−4<2$and $x−4>−2$is : $x∈\left(2,6\right)$

Therefore, the solution to the inequality $\left|x−4\right|<2$is $x∈\left(2,6\right)$.