Quick absolute value inequality explanation, please

Hey everyone, I'm struggling with some of the concepts behind absolute value inequalities and found a certain problem I need some assistance with. I'll explain what I tried, please point out my mistake and explain why it is a mistake. Much appreciation.

Basically $\displaystyle (|x|+1)/(|x|-1) < 1$

if x >= 0

$\displaystyle (x+1-x+1) / x-1 < 0$

which gives x < 1

and if x < 0

$\displaystyle (-x+1)/(-x-1) < 1$

which gives x >= 0

$\displaystyle [0;1)U(1;infinity)$