Can someone explain $|x|=-x$ if $x<0$ . I've proven various theorems in my real analysis text for homework, but I cannot see how $|x|=-x$ if $x<0$ makes sense.
2026-04-05 14:53:09.1775400789
Explanation for |x|=-x if x<0
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This is not a contradiction. Say $x<0$. This means that $x$ is negative, therefore $-x$ is positive. If $x$ is negative, then its absolute value is $-x$.
You can think of $x$ as $x=-r$, where $r>0$. Then $|x| = |-r|=r=-(-r)=-x$.