Question

Solve the following inequality:
$\left|\frac{x^2-3x-1}{x^2+x+1}\right|<3$

Answer 1

$\left|\frac{x^2-3x-1}{x^2+x+1}\right|<3$

$-3<\frac{x^2-3x-1}{x^2+x+1}<3$

$-3x^2-3x-3<x^2-3x-1$

$4x^2+2>0$

$2x^2+1>0$

$x\in R$

And

$x^2-3x-1<3x^2+3x+3$

$2x^2+6x+4>0$

$x^2+3x+2>0$

$(x+2)(x+1)>0$

$x\in (-\infty ,-2)\cup (-1,\infty )$

$\therefore x\in (-\infty ,-2)\cup (-1,\infty )$