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Math Help - triangle inequality

  1. #1
    Senior Member sfspitfire23's Avatar
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    triangle inequality

    Could I use thhe triangle inequality to perform the following:

    \frac{|a-b|}{2}\leq \frac{|a|}{2} - \frac{|b|}{2}?

    I cant seem to find a proof of why you can do this is if you can...
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  2. #2
    Junior Member nimon's Avatar
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    I don't think so. The triangle inequality implies that

     |a| = |(a-b)+b| \leq |a-b|+|b|

    so that

     <br />
|a|-|b| \leq |a-b| \Rightarrow \frac{|a|}{2}-\frac{|b|}{2} \leq \frac{|a-b|}{2},<br />

    which is the opposite of what you wanted!

    In fact you can show that

    |a|-|b| \leq |a-b|,|a+b| \leq |a|+|b|,

    which is quite a nice way to remember them all.
    Last edited by nimon; April 24th 2010 at 09:18 AM. Reason: answering the question
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Ah, I must be able to do this then by the triangle inequality:

    \frac{|a-b|}{2}\leq \frac{|a|}{2} + \frac{|b|}{2}

    yes?
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  4. #4
    Junior Member nimon's Avatar
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    Yep! To see why:

     |a-b| = |a+ (-b)| \leq |a|+|-b| = |a|+|b|.

    For practice, try also to show that

    |a|-|b| \leq |a+b|,

    the four inequalities will come in handy for the rest of your life!
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