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Math Help - Proving inequalities using binomial expansion

  1. #1
    Junior Member Kaloda's Avatar
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    Proving inequalities using binomial expansion

    If x and y are real nos. not equal to zero, show that
    \frac{x^2}{y^2}+\frac{16y^2}{x^2}+24\geq\frac{8x}{ y}+\frac{32y}{x}.

    The book hinted that binomial thingy (maybe expansion, I forgot) may be used.

    EDIT:
    BTW, please teach me how to use that \frac, \leq, etc.
    Last edited by Kaloda; June 22nd 2012 at 10:36 AM.
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  2. #2
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    Re: Proving inequalities using binomial expansion

    Quote Originally Posted by Kaloda View Post
    If x and y are real nos. not equal to zero, show that
    \frac{x^2}{y^2}+\frac{16y^2}{x^2}+24\geq\frac{8x}{ y}+\frac{32y}{x}.

    The book hinted that binomial thingy (maybe expansion, I forgot) may be used.

    EDIT:
    BTW, please teach me how to use that \frac, \leq, etc.
    Your code looks alright, but you need to put it inside tex tags.
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  3. #3
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    Re: Proving inequalities using binomial expansion

    Quote Originally Posted by Kaloda View Post
    If x and y are real nos. not equal to zero, show that
     \frac{x^2}{y^2}+\frac{16y^2}{x^2}+24\geq\frac{8x}{  y}+\frac{32y}{x}

    The book hinted that binomial thingy (maybe expansion, I forgot) may be used.

    EDIT:
    BTW, please teach me how to use that \frac, \leq, etc.
    Try multiplying both sides of the inequality by x^2 y^2 (this is positive, so you haven't changed the inequality).
    Then rearrange all the terms on one side of the inequality (so you have \text{bunch of stuff} \geq 0) and see if you can apply the thingy.
    Thanks from Kaloda
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  4. #4
    Junior Member Kaloda's Avatar
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    Re: Proving inequalities using binomial expansion

    Ah. Thank you for the proper syntax.

    So this is the question
    \frac{x^2}{y^2}+\frac{16y^2}{x^2}+24\geq\frac{8x}{ y}+\frac{32y}{x}

    The book farther hinted that you may let \frac{x}{y}=u then use the binomial expansion (a+b)^4
    And BTW, if you know another solution, please post it even though it doesn't follow the book's solution.
    Last edited by Kaloda; June 23rd 2012 at 06:06 AM.
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  5. #5
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    Re: Proving inequalities using binomial expansion

    Well following the book's example we have

    \displaystyle \begin{align*} \frac{x^2}{y^2} + \frac{16y^2}{x^2} + 24 &\geq \frac{8x}{y} + \frac{32y}{x} \\ \left(\frac{x}{y}\right)^2 + \frac{16}{\left(\frac{x}{y}\right)^2} + 24 &\geq 8\left(\frac{x}{y}\right) + \frac{32}{\frac{x}{y}} \\ u^2 + \frac{16}{u^2} + 24 &\geq 8u + \frac{32}{u} \\ u^4 + 16 + 24u^2 &\geq 8u^3 + 32u \\ u^4 - 8u^3 + 24u^2 - 32u + 16 &\geq 0 \\ 1(-2)^0 u^4 + 4(-2)^1 u^3 + 6(-2)^2 u^2 + 4(-2)^3 u^1 + 1(-2)^4 u^0 &\geq 0 \\ (u - 2)^4 &\geq 0  \end{align*}

    which is a true statement and therefore confirms the original statement.
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  6. #6
    Junior Member Kaloda's Avatar
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    Re: Proving inequalities using binomial expansion

    I already proved it by multiplying x^2 y^2 as what awkward said. But thanks Prove it for that brilliant solution.
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