(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that, if a, b, c >0 and a+b+c = 1, then,

([itex]\frac{1}{a}[/itex]-1)([itex]\frac{1}{b}[/itex]-1)([itex]\frac{1}{c}[/itex]-1) [itex]\geq[/itex] 8

2. Relevant equations

The usual math is ok

3. The attempt at a solution

I multiplied them out to get:

([itex]\frac{1}{abc}-\frac{1}{ac}-\frac{1}{bc}+\frac{1}{c}-\frac{1}{ab}+\frac{1}{a}+\frac{1}{b}[/itex]-1)

= ([itex]\frac{1}{abc}-\frac{b}{abc}-\frac{a}{abc}+\frac{ab}{abc}-\frac{c}{abc}+\frac{bc}{abc}+\frac{ac}{abc}[/itex]-1)

= ([itex]\frac{1-b-a+ab-c+bc+ac-abc}{abc}[/itex])

= ([itex]\frac{ab+bc+ac}{abc}[/itex]) = ([itex]\frac{1}{a}+\frac{1}{b}+\frac{1}{c}[/itex]), .........as 1-(a+b+c) = 0

abc is a small number as a,b,c are all less than 1, and it is and order smaller than

anything in the numerator. OR, since a, b, c < 1,

([itex]\frac{1}{a}+\frac{1}{b}+\frac{1}{c}[/itex]) > 3

Now I don't know what to do. I'm not even sure I'm on the right track...

There may be some other logic I should be using to make this easier, but I don't see it.

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# Prove this is greater than or equal to 8

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