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Math Help - Prove ||a|-|b||≤ |a-b|

  1. #1
    Junior Member Kaloda's Avatar
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    Question Prove ||a|-|b||≤ |a-b|

    Can someone solve this problem?

    PROVE: For any real numbers a and b ||a|-|b||≤ |a-b|
    Last edited by mr fantastic; September 21st 2010 at 01:43 AM. Reason: Deletd duplicate question, re-titled.
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  2. #2
    MHF Contributor
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    You need to know the triangular inequality.

    |a| = |a - b + b|

     = |(a - b) + b|

     \leq |a - b| + |b|.


    Since |a| \leq |a - b| + |b|

    |a| - |b| \leq |a - b|.


    If we take the modulus of both sides, we have

    \left||a| - |b|\right| \leq \left||a - b|\right|

    \left||a| - |b|\right| \leq |a - b|.
    Last edited by mr fantastic; September 21st 2010 at 01:44 AM.
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