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 August 19th, 2013, 06:17 AM #1 Newbie   Joined: May 2013 Posts: 26 Thanks: 0 Inequality Is this statement true $\frac{1}{2y-x}+\frac{1}{2x-y}\geq \frac{4}{x+y}$ for any x,y side of triangle
August 19th, 2013, 08:17 AM   #2
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Re: Inequality

Quote:
 Originally Posted by masterofmath for any x,y side of triangle
That doesn't make sense. Any x,y are sides of a triangle. But not every x,y,z is that, but that's not the matter.

 August 19th, 2013, 10:55 AM #3 Senior Member   Joined: Jul 2013 From: Croatia Posts: 180 Thanks: 11 Re: Inequality Actually the inequality is true for every positive x and y. Using arithmetic harmonic mean inequality we have $\frac{\frac{1}{2x-y}+\frac{1}{2y-x}}{2} \geq \frac{2}{2x-y+2y-x}$ $\frac{1}{2x-y}+\frac{1}{2y-x} \geq \frac{4}{2x-y+2y-x}$ $\frac{1}{2x-y}+\frac{1}{2y-x} \geq \frac{4}{x+y}$
 August 21st, 2013, 04:17 AM #4 Newbie   Joined: May 2013 Posts: 16 Thanks: 0 Re: Inequality "Solution: 1/(2y+x)+1/(2x+y)=4/(x+y) if x, y side of triangle then this statement is not true."
 August 21st, 2013, 04:49 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,753 Thanks: 2136 It's true if x = y and you have minus (instead of plus) signs in the denominators on the left-hand side. Nobody has suggested it works with the plus signs!
 August 21st, 2013, 11:59 AM #6 Newbie   Joined: May 2013 Posts: 26 Thanks: 0 Re: Inequality So what is the final solution?!
August 21st, 2013, 12:21 PM   #7
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Re: Inequality

Quote:
 Originally Posted by crom the inequality is true for every positive x and y.

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