TITLE: If $|x|+|y|>|x+z|$, then is $y > z, y<z$, or $y=z$, or not determinable?
QUESTION [0 upvotes]: If $|x|+|y|>|x+z|$, then is $y > z, y<z$, or $y=z$, or not determinable?
I thought about cases, $x>0,x<0,y>0,y<0,z>0,z<0.$ Plugging numbers is unconvincing to me.

REPLY [0 votes]: Here are simple approaches that don't use numbers:



Putting $x=-z$ , we get



$$|y|+|z|>0$$
Then, you can choose infinitely many pair $y,z\neq 0$, such that all possible cases: $y>z$ or $y<z$ or $y=z$ hold.



Fix $x=0$, you get $|y|>|z|$. This implies, you can also use the same argument. But, note that in this case $y=z$ doesn't hold.
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