Question

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

• A. $-1<x<-\frac{-2}{3}\text{or}\frac{2}{3}<x<1$
• B. $-2<x<2$
• C. $\frac{2}{3}<x<1$
• D. none of these

Answer 1

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

Let $|x|=t$
$\Rightarrow \frac{t^2-t-2}{2t-t^2-2}>0$
$\Rightarrow \frac{(t-2)(t+1)}{t^2-2t+2}<0$
$\Rightarrow \frac{(t-2)(t+1)}{\left(\right(t-1{)}^2+1)}<0$

Here $(t-1{)}^2+1>0\forall t\in R$

$\Rightarrow (t-2)(t+1)<0$
$\Rightarrow -1<t<2$
$\Rightarrow -1<|x|<2$
$\Rightarrow -2<x<2$