Question

Solve the following system of equations in R.
$\frac{1}{|x|-3}\le \frac{1}{2}$

Answer 1

$\frac{1}{|x|-3}\le \frac{1}{2}\Rightarrow \frac{1}{|x|-3}-\frac{1}{2}\le 0\Rightarrow \frac{2-|x|+3}{2(|x|-3)}\le 0\Rightarrow \frac{5-|x|}{2(|x|-3)}\le 0\Rightarrow \frac{|x|-5}{2(|x|-3)}\ge 0\Rightarrow \frac{|x|-5}{|x|-3}\ge 0\Rightarrow |x|\ge 5or|x|<3\Rightarrow xϵ(-\infty ,-5]\cup [5,\infty )orxϵ(-3,-3)\Rightarrow xϵ(-\infty ,-5]\cup [-3,-3)\cup [5,\infty )Thesolutionsetis(-\infty ,-5]\cup [-3,-3)\cup [5,\infty )$