(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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Prove this is greater than or equal to 8

**Physics Forums | Science Articles, Homework Help, Discussion**