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Assuming positive real values of $x,y,z$, and that $ xyz(x+y+z)=1$, how can we prove that $(x+y)(y+z)\ge 2$

I tried using the AM-GM inequality but as if I were looping. I'm not sure if C-S or other inequality methods could help. Thanks for helping me.

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1 Answer 1

up vote 4 down vote accepted

$$(x+y)(y+z)=xy+y^2+zx+yz=y(x+y+z)+zx=\frac y{xyz}+zx=zx+\frac1{zx}\ge 2\sqrt{\frac1{zx}zx}=2$$ using A.M.$\ge$ G.M. for $z,x>0$

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I reached until $zx +\frac{1}{zx}$ but never thought of completing your answer. Thanks! –  Tariq Jan 9 '13 at 11:25
    
@Tariq, welcome, hope I could make things lucid. –  lab bhattacharjee Jan 9 '13 at 11:31

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