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[Solved] Fractions and inequalities

 
Fractions
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June 14, 2021 11:00 am
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(@armor)
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Given x>0, y>0,    \frac{1}{x}+\frac{9}{y}=1,    show  x+y\geq 12.

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June 14, 2021 11:00 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

Since \small x>0, y>0,  we have \small \frac{1}{x}+\frac{9}{y}\geq 2\sqrt{\frac{1}{x}*\frac{9}{y}}=\frac{6}{\sqrt{xy}}.

Since \small \frac{1}{x}+\frac{9}{y}=1,  we have \small \frac{6}{\sqrt{xy}}\leq 1,   which is equivalent to \small \sqrt{xy}\geq 6.

\small x+y\geq 2\sqrt{xy}\geq 12.

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