Determining whether or not a relation involving absolute value is transitive

Question

I needed help doing a relation problem the specific problem is

$|x+y|$ = $|x|$ + $|y|$

I first tried thinking of some counter-examples but I couldn't think of any so i tried showing it.

So to show that a relation is transitive

you are given that $|x+y| = |x| +|y|$ and $|y+z| = |y| + |z|$ and need to show that $|x+z| = |x| + |z|$.

I tried isolating the second given for $|y|$ so I have $|y| = |y+z| - |z|$ and then using this I plugged it in the first given to get

$|x+y| = |x| + |y+z| - |z|$

Now this is where the problem rises since these have a absolute value around them I cant just cancel out the variables so I get stuck here and I am not sure what to do.

Answer 1

If $|x+y|=|x|+|y|$, by squaring both sides we get $xy=|xy|$.

Thus either one of the numbers is $0$ or they are both nonzero and have the same sign.

What happens with $x=1$, $y=0$ and $z=-1$?