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Math Help - can't figure this inequality

  1. #1
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    can't figure this inequality

    (M^2+2MN-N^2 )^2<(M^2-2MN-N^2 )^2

    I went about it like this

    M^2+2MN-N^2<M^2-2MN-N^2\ \ \ \ \ \ \ \ (1)
    M^2+2MN-N^2>-M^2+2MN+N^2\ \ \ \ \ \ (2)

    (1)

    M^2+2MN-N^2<M^2-2MN-N^2

    2MN<-2MN

    \implies\ \ \ \ \ M<0<N\ \ \ or\ \ \ M>0>N

    (2)

    M^2+2MN-N^2>-M^2+2MN+N^2

    M^2-N^2>-M^2+N^2

    2M^2-2N^2>0

    (M+N)(M-N)>0

    so (M+N) and (M-N) have the same sign

    (M+N)\ \ and\ \ (M-N)\ \ >0\ \ \ \ (3)

    (M+N)\ \ and\ \ (M-N)\ \ <0\ \ \ \ (4)

    (3)

    (M+N)\ \ and\ \ (M-N)\ \ >0

    \implies\ \ \ \ \ \ M>|N|

    (4)

    (M+N)\ \ and\ \ (M-N)\ \ <0

    \implies\ \ \ \ \ \ M<|N|

    so altogether I have

    M>0>N
    M<0<N
    M>|N|
    M<|N|

    which seems to imply that all M and N fit apart from M=N

    let M=2 N=1

    (M^2+2MN-N^2 )^2=7\ \ \ \ \(M^2-2MN-N^2 )^2=1

    but this has (M^2+2MN-N^2 )^2>(M^2-2MN-N^2 )^2

    What am I doing wrong ?
    Last edited by MathFan; May 25th 2012 at 03:38 PM.
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  2. #2
    MHF Contributor
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    Re: can't figure this inequality

    2mn < -2mn
    1 < -1

    stop!
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  3. #3
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    Re: can't figure this inequality

    2mn<-2mn

    m=2
    n=-1
    2mn=-4
    -2mn=4
    2mn<-2mn
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  4. #4
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    Re: can't figure this inequality

    Multiply out both sides and collect terms and you arrive at

    MN(M^{2}-N^{2}) < 0

    which clearly is not the case for all M and N.

    If M and N are both positive you would need to have M<N.
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  5. #5
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    Re: can't figure this inequality

    Thanks Bob, seems easier than the way I was going about it.

    MN(M^2-N^2)<0

    M\ne0

    M>0\ \ \ \implies \ \ \ 0<M<N\ \ or\ \ -M<N<0

    M<0\ \ \ \implies \ \ \ N<M<0\ \ or\ \ -M>N>0

    Does this cover all of them ?
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