text: # 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
explanation: The document contains a math problem and a detailed solution to the problem. The problem asks to prove that \((x+y)(y+z) \ge 2\) given the conditions \(x, y, z > 0\) and \(xyz(x+y+z) = 1\). The solution uses the AM-GM inequality to complete the proof.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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 $z,x>0$
  
  -
  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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem and a detailed solution to the problem. The problem asks to prove that \((x+y)(y+z) \ge 2\) given the conditions \(x, y, z > 0\) and \(xyz(x+y+z) = 1\). The solution uses the AM-GM inequality to complete the proof.
  
  Label: question-and-answer