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.

  1. usernameeee

    usernameeee

    Joined:
    Mar 31, 2021
    Messages:
    2
    Likes Received:
    0
    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
    usernameeee, Mar 31, 2021
    #1
    1. Advertisements

  2. usernameeee

    usernameeee

    Joined:
    Mar 31, 2021
    Messages:
    2
    Likes Received:
    0
    1. Advertisements

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.