Question

|x| < 2/x please answer this

Answer 1

$\text{Case 1:\ }x>0$

$\implies |x|=x$

$\implies x<\dfrac{2}{x}$

Multiplying $x$ on both sides, since $x>0$,

$\implies x^2<2$

$\implies x^2-2<0$

$\implies -\sqrt{2} <x<\sqrt{2}$

$\therefore 0<x<\sqrt{2}$

$\text{Case 2:\ }x=0$

$\implies |x|=0$

$\implies 0<\dfrac{2}{0}$

Clearly, it cannot be $x=0$. So, $x\neq 0$.

$\text{Case 3:\ }x<0$

$\implies |x|=-x$

$\implies -x<\dfrac{2}{x}$

Multiplying $x$ on both sides, since $x<0$,

$\implies -x^2>2$

$\implies x^2+2<0$

Clearly, it is false since the square of real numbers is greater than or 0.

The union of three inequality is $0<x<\sqrt{2}$. This is the required answer.