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Math Help - Solving algebraic inequalities

  1. #1
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    Solving algebraic inequalities

    Hi, I'm having trouble solving this inequality:

    'Let p and q be positive real numbers'. Prove that:

    (p + 2)(q + 2)(p + 1) \geq 16pq

    Please help explain how to solve it to me.

    Thanks, BL
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  2. #2
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    Quote Originally Posted by BG5965 View Post
    Hi, I'm having trouble solving this inequality:

    'Let p and q be positive real numbers'. Prove that:

    (p + 2)(q + 2)(p + 1) \geq 16pq
    This is not true for example if p = q = 6. Then the left side is 448 and the right side is 576.
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  3. #3
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    Maybe the inequality should be

    (p+2)(q+2)(p+q)\geq 16pq.

    That can be proved as follows.

    (x - y)^2 \geq 0
    x^2 + y^2 - 2xy \geq 0,
    x^2 + y^2 \geq 2xy.

    Now let x = \sqrt{p} and y = \sqrt{q}, then

    p + q \geq 2\sqrt{pq}.

    Similarly q + r \geq 2\sqrt{qr} and r + p \geq 2\sqrt{rp}.

    Multiplying these three inequalities and putting r = 2, leads to the stated result.
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