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Math Help - Inequality Question

  1. #1
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    Inequality Question

    Hi, I am new to the forum, and I hope I am posting this in the right place:


    Question from our grade 12 exchange student:

    a and b are integers such that

    0 < a < b

    A = sqroot (ab)


    He was able to answer the first question as follows:

    1) show that a < A < b

    a < b

    a*a < b*a

    a^2 < ba

    a < sqroot(ba)

    therefore

    a < A < b


    he was unable to answer the following:

    2) show that

    a < 2ab/(a+b) < A


    Thanks in advance!
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  2. #2
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    Re: Inequality Question

    Quote Originally Posted by chet View Post
    he was unable to answer the following:
    2) show that
    a < 2ab/(a+b) < A
    By looking at (b-a)^2>0 we see that if b>a then b^2+a^2>2ab.
    From that we get a+b>2\sqrt{ab}=\frac{2ab}{\sqrt{ab}}.

    Now from that \frac{2ab}{a+b}<A.

    Can he finish?
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  3. #3
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    Re: Inequality Question

    I will ask him if he can finish. Thanks!!
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