Question

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

Answer 1

We have,

$\frac{1}{|x|-3}-\frac{1}{2}<0...\left(i\right)CaseI:when|x|\ge 0\Rightarrow x\ge 0\Rightarrow \frac{1}{x-3}-\frac{1}{2}<0\Rightarrow \frac{2-(x-3)}{2(x-3)}<0\Rightarrow \frac{2-x+3}{2x-6}<0\Rightarrow \frac{-x+5}{2x-6}<0\Rightarrow -x+5<0\Rightarrow -x<-5\Rightarrow x>5...\left(ii\right)CaseII:when|x|<0,x<0\Rightarrow \frac{1}{-x-3}-\frac{1}{2}<0\Rightarrow \frac{2-(-x-3)}{2(-x-3)}<0\Rightarrow 2+x+3<0\Rightarrow x+5<0x<-5...\left(iii\right)\text{combining (ii) and (iii) we get}(-\infty ,-5)\cup (-3,3)\cup (5,\infty )\text{as the solution set.}$