Question

Solve for $x$: $\frac{x^2-|x|-12}{x-3}\ge 2x$

Answer 1

$\frac{x^2-\left|x\right|-12}{x-3}\ge 2x$

$\left(i\right)when(x>0)$

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

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

$\frac{x^2-x-12-2x^2+6x}{x-3}\ge 0$

$\frac{x^2-x-12-2x^2+6x}{x-3}\ge 0$

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

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

$discriminant(b^2-4ac)of(x^2-5x+12)is<0hence(x^2-5x+12)>0$

$\frac{1}{x-3}\le 0,x\epsilon (-\infty ,3)$

$butx>0,x\epsilon (0,3)$

$\left(ii\right)when(x<0)$

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

$\frac{x^2+4x-3x-12}{x-3}\ge 2x$

$\frac{(x-3)(x+4)}{(x-3)}\ge 2x$

$x+4\ge 2x$

$x\le 4$

$butx<0,x\epsilon (-\infty ,0]from\left(i\right)\&\left(ii\right)$

$x\epsilon (-\infty ,0]\bigcup [0,3)$