Prove Absolute Value Property

1. The problem statement, all variables and given/known data

Prove that for every two real numbers x and y

##|x+y| \leq |x| + |y| ##





2. Relevant equations







3. The attempt at a solution



There are three cases. The easiest ones is when they are both positive and negative.

The third one I have problems with.

The numbers have different sign. Say x>0 and y<0

Divide this into two subcases:

case 3.1

## x+y \geq 0##





## |x| +|y| = x+(-y) = x-y##

Now, so far so good, but my book states the following.

## |x| +|y| = x+(-y) = x-y > x+y = |x+y|##

How is it possible that x-y be ever greater than x+y?



case 3.2

## x+y < 0 ##

This one is easy too.