I'm trying to prove the following inequality
$|\frac{|(x|x|-|x|)|}{|x|}| \leq 1-|x|$
This is the calculation that I'm doing;
$|\frac{|(x|x|-|x|)|}{|x|}|= \frac{|x|^2-|x|}{x} \le \frac{|x|^2+|x|}{x}= |x|+1$
How should I proceed from here?
2026-04-03 12:50:54.1775220654
Modulus Inequalities
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I think you can start from this $$ \left|\frac{(| x|x| - |x||)}{|x|}\right| = \left|\left( \left|\frac{x|x|-|x|}{|x|}\right|\right)\right| = \left| \left|x-1\right|\right| =| 1-x| \le 1-|x|. $$ Last inequality hold because of $x \in (0,1)$.