Q: Taking the absolute value in inequalities. If I've a expression: $-4<3$ and take the absolute value $|-4|<|3|\implies 4<3$ which is false. So I though that maybe the inequality sign would change. But $|-2|<|3| \implies 2>3$ which is also false.
My problem is that I have the inequality $-x<y$ where $x,y>0$. What will happen if I take the absolute value of it: $|-x|<|y|.$ I though it would be $0<|x|<|y|\implies 0<x<y$. 
But from the examples above it seems that this ain't true.
EDIT:
I have a increasing sequence $(x_1>x_2>x_3...$ etc.), $x_1,x_2,...,x_n$ where all $x's$ are positive and a constant $m=1/2$. The inequality: $-x_n<1/2$ is always valid because the $x's$ are positive. But $x_3=1$. So for all $n \geq 3$, $|-x_n|>1/2$ .
Given this context helps in solving my problem in taking the absolute value?
Taking the absolute value on both side we have for all $n>3, |-x_n|>|1/2|>0 \implies x_n>1/2>0$.
Is this correct?

A: No. Apparently, you are trying to remove the absolute value from both sides from the inequality. But that's would mean you apply the inverse function to both sides, but absolute value does not have an inverse.
