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Math Help - Inequality

  1. #1
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    Inequality

    (x+y)*(x+z)*(y+z)>=8xyz

    How to show this, I suppose somehow with the AM–GM inequality, but don't know how.

    Thank you!

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  2. #2
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    Re: Inequality

    Quote Originally Posted by gotmejerry View Post
    (x+y)*(x+z)*(y+z)>=8xyz

    How to show this, I suppose somehow with the AM–GM inequality, but don't know how.

    Thank you!

    1) x+y \geq 2\sqrt {xy}

    2) x+z \geq 2\sqrt {xz}

    3) y+z \geq 2\sqrt {yz}

    Now multiply left hand sides and right hand sides of these three equations .
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  3. #3
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    Re: Inequality

    Thank you!
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  4. #4
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    Re: Inequality

    you are welcome
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  5. #5
    Super Member ILikeSerena's Avatar
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    Re: Inequality

    Just nitpicking...

    For x=1, y=-1, z=-1, you get:

    (1-1)(1-1)(-1-1) \ge 8 \cdot 1 \cdot -1 \cdot -1

    \Rightarrow 0 \ge 8

    which is a contradiction.
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  6. #6
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    Re: Inequality

    Quote Originally Posted by gotmejerry View Post
    (x+y)*(x+z)*(y+z)>=8xyz
    How to show this, I suppose somehow with the AM–GM inequality, but don't know how.
    I assume that x,~y,~\&~z are non-negative.
    We have x^2+y^2\ge 2xy. so x^2z+y^2z\ge 2xyz..

    Expand (x+y)(x+z)(y+z).
    Do you see what to do next?
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  7. #7
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    Re: Inequality

    Sorry, I forgot that x,y,z are postives
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