The above is my 'proof' of two statements involving $|x|.$ Since they clearly contradict each other, where I have gone wrong in the proof?
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Statement 2 is wrong: $-\vert x \vert = -(x^2)^{1/2} \ne (x^2)^{1/2}$. – Therkel Apr 24 '23 at 07:24
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See 1, 2, 3. – dxiv Apr 24 '23 at 07:26
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Statement 2 is false: the 9th and 13th equalities in your work are false. Those two steps collectively falsely assume that $$y=(y^2)^{\frac12}\iff y\in\mathbb R.$$ In actuality, $$y=(y^2)^{\frac12}\iff y\ge0\\ -y=(y^2)^{\frac12}\iff y\le0.$$
ryang
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