How to prove that a/(a+1) ≥ (c+d)/(c+d+1) given that d≥0

Discussion in 'Algebra' started by usernameeee, Mar 31, 2021.

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I am working on a proof for a Metric function, and the last condition leads me to the following inequality:
(a+2b)/(1+a+b) ≥ c/(1+c) where a ≥ c and all a, b, c are positive, real numbers.
I think I can say without much proof, that:
(a+2b)/(1+a+b) ≥ a/(1+a),
because the ratio of numerator and denominator increases.

My problem, is that:
a/(1+a) ≥ c/(1+c)
is not that straight forward.
The only solution I could think of is graphical, which is not ideal.

The alternate form (the one in the title) makes it a more algebra-ish problem.
That form is produced by introducing d, the difference of a and c
a = c + d

and so because a ≥ c it follows that d ≥ 0

So now, the problem is:
a/(1+a) ≥ (c+d)/(c+d+1)

Edit: a and c were mixed up in the last part

Last edited: Mar 31, 2021

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