I was goofing around with programming some math functions the other day, and I came up with an equality that I can't prove/disprove. It involves absolute values and that is where I am getting stumped.
$$ \lvert(x-y)\rvert(x+y) = \lvert x^2 - y^2 \rvert $$
Is it possible to work this out with algebra or otherwise?
We have that $|x-y| \cdot (x+y) = |x^2-y^2|.$ But $$x^2 - y^2 = (x-y) \cdot (x+y),$$ so $$|x-y| \cdot (x+y) = |x-y| \cdot |x+y|.$$
Therefore either $|x-y| = 0,$ or we can divide by it to find that $|x+y| = x+y.$
Hence the two solutions to this inequality are $x=y,$ or $x+y \geq 0.$