# Prove Absolute Value Property

1. Feb 26, 2014

### knowLittle

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.

2. Feb 26, 2014

### pasmith

If $-y > y$, so that $0 > 2y$, ie. $y < 0$.

3. Feb 26, 2014

### knowLittle

$-y > y$ would be false even if $y<0$. They would be equal not greater than each other.

I don't see the connection to the proof. Thanks for trying.

4. Feb 26, 2014

### pasmith

Let $y = -1$. Do you agree that $-y = -(-1) = 1 > -1 = y$?

5. Feb 26, 2014

### Ray Vickson

"How is it possible that x-y be ever greater than x+y?" Try x = 1 and y = -1. What is x-y? What is x+y?

6. Feb 27, 2014

### knowLittle

Thank you, I was confused.