Question

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

Answer 1

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

$\left(i\right)whenx>0$

$\frac{1}{x-3}-\frac{1}{2}<0$

$\frac{2-x+3}{2(x-3)}<0$

$\frac{(x-5)}{(x-3)}>0$

$x\epsilon (-\infty ,3)\bigcup (5,\infty )$

$x>0$

$x\epsilon (0,3)\bigcup (5,\infty )$

$\left(ii\right)whenx<0$

$\frac{1}{-x-3}-\frac{1}{2}<0$

$\frac{2-(-x-3)}{-2(x+3)}<0$

$\frac{2+x+3}{x+3}>0$

$\frac{x+5}{x+3}>0$

$x\epsilon (-\infty ,-5)\bigcup (-3,\infty )$

$x<0$

$x\epsilon (-\infty ,-5)\bigcup (-3,0)$

$x\epsilon (-\infty ,-5)\bigcup (-3,0)\bigcup (0,3)\bigcup (5,\infty )$