Analysis: Prove |x|+|y| is less than or equal to |x+y|+|x-y|

1. Sep 26, 2011

Chinnu

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

Using the triangle inequality, establish that:

|x| + |y| $\leq$ |x+y| + |x-y|

2. Relevant equations

|x + y| $\leq$ |x| + |y|

3. The attempt at a solution

I have tried a few things, here are those that seem like they would be most useful:

|x + y| $\leq$ |x| + |y|

$\leq$ |x+y-y| + |y-x+x|

$\leq$ |x+y| + |-y| + |y-x| + |x|

$\leq$ |x+y| + |y| + |y-x| + |x| ..........Note that |y-x| = |x-y|

Also,

|x-y| $\leq$ |x| + |-y| = |x| + |y|

which might be able to be used in the middle inequality above.

I'm not sure what to do from here (or if I'm on the right track)

2. Sep 26, 2011

I like Serena

Hi Chinnu!

Try substituting x=u+v and y=u-v.
Note that you can find a u and a v for any x and y.