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[Solved] Absolute Value

 
Inequalities
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June 15, 2021 12:26 pm
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Staff 
(@armor)
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If   -1<x<1,     -1<y<1,    show     \left | \frac{x+y}{1+xy} \right |<1. 

 

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June 15, 2021 12:26 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small \left | \frac{x+y}{1+xy} \right |<1\Leftrightarrow (\frac{x+y}{1+xy})^2<1\Leftrightarrow (x+y)^2<(1+xy)^2\Leftrightarrow x^2+y^2<1+x^2y^2\Leftrightarrow (x^2-1)(1-y^2)<0,

which is obviously valid since \small -1<x<1, -1<y<1.

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