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November 12th, 2011, 03:43 AM   #1
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to prove inequality

Suppose x > -1, then x/(x+1) < (x+1)/(x+2)

this needs to be proven ^

I tried to go direct and I ended up with this

x(x+1)(x+2)^2 < (x+1)^3(x+2)

I don't know where to go to from there o.O I tried using induction to continue but it doesn't work...

Thoughts?
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November 12th, 2011, 06:52 AM   #2
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Re: to prove inequality

x/(x + 1) < (x + 1)/(x + 2)

x/(x + 1) - (x + 1)/(x + 2) < 0

(x + 2x - x - 2x - 1)/((x + 1)(x + 2)) < 0

-1/((x + 1)(x + 2)) < 0. The denominator is a parabola that opens upward and has zeros -1 and -2 so the denominator is negative for -2 < x < -1,

so the inequality x/(x + 1) < (x + 1)/(x + 2) is true for x > -1, x < -2.
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November 12th, 2011, 07:17 AM   #3
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Re: to prove inequality

Or


For , true for and



true for all x.

Hence
and
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November 12th, 2011, 09:07 AM   #4
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Re: to prove inequality

Hello, jamesb1!

Quote:








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