Question

The inequality $\frac{|x{|}^2-|x|-2}{2|x|-|x{|}^2-2}>0$ holds, if and only if

• A. $-1<x<-\frac{2}{3}$ or $\frac{2}{3}<x<1$
• B. $-2<x<2$
• C. $\frac{2}{3}<x<1$
• D. $x>1$ or $x<-1$ or $-\frac{2}{3}<x<\frac{2}{3}$

Answer 1

The correct option is B $-2<x<2$

$\frac{|x{|}^2-|x|-2}{2|x|-|x{|}^2-2}>0$

$\Rightarrow -\frac{(|x|+1)(|x|-2)}{|x{|}^2-2|x|+2}>0$

$\Rightarrow \frac{(|x|+1)(|x|-2)}{|x{|}^2-2|x|+1+1}<0$

$\Rightarrow \frac{(|x|+1)(|x|-2)}{(|x|-1{)}^2+1}<0$

Now $(|x|+1)>0$ and $(|x|-1{)}^2+1>0$ $\forall x\in R$

$\Rightarrow (|x|-2)<0$

$\Rightarrow x\in (-2,2)$