# Reorganization of an inequality

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• Apr 5th 2012, 01:20 PM
pul
Reorganization of an inequality
Hi everyone, the origin of the problem is not really "pre-university" but the math behind it should be trivial. Here we go:

$\frac{(n-1)x}{n}+y>\frac{(x+y)x}{2x+y}+\left(\frac{n-1}{n}\right)\frac{x^{3}}{(2x+y)^{2}}+\left(\frac{x }{2x+y}\right)y$

I simply need to rearrange this for illustrative purposes (I'd like to show quickly for which n the above condition is met; it's from a paper on auction theory).

Now Wolfram Alfra returns me a rather striking alternate form ("Alternate form assuming n, x and y are positive"):

$(n-3)x^2+(2n-1)y+ny^2>0$

(link: http://www.wolframalpha.com/input/?i=\frac{(n-1)x}{n}%2By>\frac{(x%2By)x}{2x%2By}%2B\left(\frac{ n-1}{n}\right)\frac{x^{3}}{(2x%2By)^{2}}%2B\left(\fr ac{x}{2x%2By}\right)\cdot+y)

Does anyone have a clue how to get to that?

Any help is appreciated.

Cheers,
Patrick
• Apr 5th 2012, 01:57 PM
a tutor
Re: Reorganization of an inequality
Take all the terms to the left and get a common denominator?