Question

Solve the following system of equations in R.
$\frac{|x+2|-x}{x}<0$

Answer 1

We have,

$\frac{|x+2|-x}{x}<0\frac{|x+2|-x}{x}-2<0\frac{|x+2|-x-2x}{x}<0\frac{|x+2|-3}{x}<0...\left(i\right)CaseI:when|x+2|\ge 0i.r.,x\ge -2\Rightarrow \frac{x+2-3x}{x}<0\Rightarrow -2x+2<0\Rightarrow -2x<-2andx>0\Rightarrow x>1...\left(ii\right)CaseII:|x+2|<0ie.,x<-2\Rightarrow -(x+2)-3x<0\Rightarrow -x-2-3x<0\Rightarrow -4x-2<0\Rightarrow -4x<2\Rightarrow x>\frac{-1}{2}...\left(iii\right)andx<0\text{Combining (ii) and (iii) we get}(-\infty ,0)\cup (1,\infty )asthesolutionset.$