Prove $\lvert x\rvert$ = $\lvert-x\rvert$ for all real numbers $x$ Been at this one for a long time. I'm trying to use the fact that $|x|$ = $x$ if $x$ is greater than or equal to 0, and $|x|$ = $-x$ if $x$ is less than 0. Then I want to split the proof into these 2 cases. I'm not sure if I'm supposed to use anything other than that definition for absolute value.

If $x \geq 0$ then $-x \leq 0$, so that $\lvert-x\rvert = -(-x) = x = \lvert x\rvert$.
If $x < 0$ then $-x > 0$, so that $\lvert-x\rvert = -x = \lvert x\rvert$.
