Question

The solution set of the inequation $||x|-2|<|2-x|,x\in R$ is

• A. $\varphi$
• B. $(-\infty ,0)$
• C. $(-\infty ,-2)\cup (2,\infty )$
• D. $(-\infty ,-2)$

Answer 1

The correct option is B $(-\infty ,0)$

$||x|-2|<|2-x|,x\in R$
Let $f\left(x\right)=||x|-2|$ and $g\left(x\right)=|2-x|$
$\Rightarrow f\left(x\right)=|2-|x||$
$\Rightarrow f\left(x\right)=\{\begin{array}{cc}|2-x|,& x\ge 0\\ |2+x|,& x<0\end{array}$


So, $f\left(x\right)=g\left(x\right)\forall x\ge 0$
and $f\left(x\right)<g\left(x\right)\forall x<0$
Hence, required solution is $x\in (-\infty ,0)$