Question

The set of values of $x,$ satisfying the inequality $\left|\frac{1}{|x|-3}\right|>\frac{1}{2}$ is

• A. $(-5,-3)$
• B. $(-3,-1)$
• C. $(1,3)$
• D. $(3,5)$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $(-5,-3)$
B $(-3,-1)$
C $(1,3)$
D $(3,5)$

$\left|\frac{1}{|x|-3}\right|>\frac{1}{2}$
Case $1:$
$\frac{1}{|x|-3}>\frac{1}{2}$
$\Rightarrow \frac{1}{|x|-3}-\frac{1}{2}>0$
$\Rightarrow \frac{5-|x|}{|x|-3}>0$
$\Rightarrow \frac{|x|-5}{|x|-3}<0$
$\Rightarrow |x|\in (3,5)$
$\Rightarrow x\in (-5,-3)\cup (3,5)$

Case $2:$
$\frac{1}{|x|-3}<\frac{-1}{2}$
$\Rightarrow \frac{1}{|x|-3}+\frac{1}{2}<0$
$\Rightarrow \frac{|x|-1}{|x|-3}<0$
$\Rightarrow |x|\in (1,3)$
$\Rightarrow x\in (-3,-1)\cup (1,3)$