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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?

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    $\begingroup$ Absolute value is not a monotonic function $\endgroup$ – J. W. Tanner Apr 2 at 2:42
  • $\begingroup$ You dont know which number has a greater distance from 0. Talking about x (or-x) and y. $\endgroup$ – randomgirl Apr 2 at 2:47
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    $\begingroup$ As you have demonstrated, this is nonsensical. $x < y$ in no way implies $|x| < |y|$. "Taking the absolute value" of both sides of an inequality is a misleading phrase since you are not justified in doing so without additional context. It may be helpful to mentally replace the absolute value function with an arbitrary function $f$ and consider what it would be to apply $f$ to both sides of an inequality. $\endgroup$ – Brian Apr 2 at 2:47
  • $\begingroup$ So if I assumed that $x>y>0$, for example, then I would know that $x$ is far from $0$ than $y$ and I would be able to do somthing about it? There must be more information some sort of relation between this numbers? $\endgroup$ – Pinteco Apr 2 at 2:52
  • $\begingroup$ I'll give some context and put in the post. $\endgroup$ – Pinteco Apr 2 at 2:55
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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.

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