# can't figure this inequality

• May 25th 2012, 03:33 PM
MathFan
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 ?
• May 25th 2012, 06:44 PM
Wilmer
Re: can't figure this inequality
2mn < -2mn
1 < -1

stop!
• May 26th 2012, 05:10 AM
MathFan
Re: can't figure this inequality
2mn<-2mn

m=2
n=-1
2mn=-4
-2mn=4
2mn<-2mn
• May 26th 2012, 07:23 AM
BobP
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.\$
• May 26th 2012, 10:08 AM
MathFan
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 ?