Question

Solve the following inequality:
$\frac{x^2-|x|-12}{x-3}⩾2x$

Answer 1

$x\ge 0$

$\frac{x^2-x-12}{x-3}-2x\ge 0$

$\frac{-x^2+5x-12}{x-3}\ge 0$

$\frac{x^2-5x+12}{x-3}\le 0$

$\frac{1}{x-3}\le 0$

$x\in (0,3)$

And

$x<0$

$\frac{x^2+x-12}{x-3}-2x\ge 0$

$\frac{-x^2+7x-12}{x-3}\ge 0$

$\frac{x^2-7x+12}{x-3}\le 0$

$\frac{(x-3)(x+4)}{x-3}\le 0$

$(x+4)\le 0$

$x\in (-\infty ,-4)$

$\therefore x\in (0,3)\cup (-\infty ,-4)$