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Math Help - Basic inequality proof

  1. #1
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    Basic inequality proof

    Hi all,

    I'm trying to do some self-studying to refresh and improve my maths skills but I'm struggling with an exercise to provide a proof in a specific way:
    Noting that x=x+y-y, apply the theorem |x+y|<=|x|+|y| together with the fact that |-y|=|y| to prove that |x+y|>=|x|-|y|

    I can see intuitively that it's true, but I haven't quite got the knack of manipulating magnitude inequalities. Any help would be much appreciated.

    Thanks,

    Chris
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  2. #2
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    You have

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

    You can write it like this

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

    Now, you set x=u+v and y=-v

    |u+v-v|-|-v| \leq |u+v|

    using that |-v|=|v|

    |u| - |v| \leq |u+v|

    what is you want to prove.
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  3. #3
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    \left| x \right| = \left| {x - y + y} \right| \leqslant \left| {x - y} \right| + \left| y \right|
    Can you finish?
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  4. #4
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    aha, so I can also just say |x| = |x + y + (-y)| <= |x + y| + |-y| and thus |x| - |y| <= |x + y|
    Thanks a lot to you both!
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