# Let $x,y,z>0; ~ xyz(x+y+z)=1$. Show that $(x+y)(y+z)\ge 2$

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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$$(x+y)(y+z)=xy+y^2+zx+yz=y(x+y+z)+zx=\frac y{xyz}+zx=zx+\frac1{zx}\ge 2\sqrt{\frac1{zx}zx}=2$$ 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
The text contains a mathematical question and a detailed solution to that question, followed by a user's comment on the solution.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Let $x,y,z>0; ~ xyz(x+y+z)=1$. Show that $(x+y)(y+z)\ge 2$

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.

-

$$(x+y)(y+z)=xy+y^2+zx+yz=y(x+y+z)+zx=\frac y{xyz}+zx=zx+\frac1{zx}\ge 2\sqrt{\frac1{zx}zx}=2$$ 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
user
The text contains a mathematical question and a detailed solution to that question, followed by a user's comment on the solution.

TYPE: SINGLE_QUESTION_WITH_ANSWER