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

  • Thread starter Chinnu
  • Start date
  • #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)
 

Answers and Replies

  • #2
I like Serena
Homework Helper
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Hi Chinnu! :smile:

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

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