Results 1 to 5 of 5

Thread: can't figure this inequality

  1. #1
    Newbie
    Joined
    Dec 2011
    Posts
    16

    can't figure this inequality

    $\displaystyle (M^2+2MN-N^2 )^2<(M^2-2MN-N^2 )^2$

    I went about it like this

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

    $\displaystyle (1)$

    $\displaystyle M^2+2MN-N^2<M^2-2MN-N^2$

    $\displaystyle 2MN<-2MN$

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

    $\displaystyle (2)$

    $\displaystyle M^2+2MN-N^2>-M^2+2MN+N^2$

    $\displaystyle M^2-N^2>-M^2+N^2$

    $\displaystyle 2M^2-2N^2>0$

    $\displaystyle (M+N)(M-N)>0$

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

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

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

    $\displaystyle (3)$

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

    $\displaystyle \implies\ \ \ \ \ \ M>|N|$

    $\displaystyle (4)$

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

    $\displaystyle \implies\ \ \ \ \ \ M<|N|$

    so altogether I have

    $\displaystyle M>0>N$
    $\displaystyle M<0<N$
    $\displaystyle M>|N|$
    $\displaystyle M<|N|$

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

    let M=2 N=1

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

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

    What am I doing wrong ?
    Last edited by MathFan; May 25th 2012 at 03:38 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,184
    Thanks
    80

    Re: can't figure this inequality

    2mn < -2mn
    1 < -1

    stop!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2011
    Posts
    16

    Re: can't figure this inequality

    2mn<-2mn

    m=2
    n=-1
    2mn=-4
    -2mn=4
    2mn<-2mn
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2009
    Posts
    671
    Thanks
    136

    Re: can't figure this inequality

    Multiply out both sides and collect terms and you arrive at

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

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

    If $\displaystyle M$ and $\displaystyle N$ are both positive you would need to have $\displaystyle M<N.$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Dec 2011
    Posts
    16

    Re: can't figure this inequality

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

    $\displaystyle MN(M^2-N^2)<0$

    $\displaystyle M\ne0$

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

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

    Does this cover all of them ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Finding the Angle in this Figure (see the figure)
    Posted in the Geometry Forum
    Replies: 1
    Last Post: Dec 19th 2011, 02:44 PM
  2. Replies: 2
    Last Post: Jan 11th 2011, 08:20 PM
  3. Replies: 3
    Last Post: Dec 12th 2010, 01:16 PM
  4. Replies: 3
    Last Post: Oct 21st 2010, 08:46 AM
  5. Replies: 2
    Last Post: Jun 4th 2010, 07:02 AM

Search Tags


/mathhelpforum @mathhelpforum