- #1

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## Homework Statement

Using the triangle inequality, establish that:

|x| + |y| [itex]\leq[/itex] |x+y| + |x-y|

## Homework Equations

|x + y| [itex]\leq[/itex] |x| + |y|

## The Attempt at a Solution

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

|x + y| [itex]\leq[/itex] |x| + |y|

[itex]\leq[/itex] |x+y-y| + |y-x+x|

[itex]\leq[/itex] |x+y| + |-y| + |y-x| + |x|

[itex]\leq[/itex] |x+y| + |y| + |y-x| + |x| ..........Note that |y-x| = |x-y|

Also,

|x-y| [itex]\leq[/itex] |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)