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Inequalities in algebra

  1. May 3, 2010 #1
    I have no idea how to solve this inequality without using AM-GM inequality:


    for any positive integers p,q,x,y show that
    (pq+xy)(px+qy)>=4pqxy
     
  2. jcsd
  3. May 5, 2010 #2
    Multiply out the brackets on the left hand side, and then divide everything by pqxy.

    You are left with: p/y + y/p + q/x + x/q on the left. All there is left to do to prove the result is to show that for any positive integers x and y: x/y + y/x >= 2.
     
  4. May 5, 2010 #3

    mathman

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    Science Advisor

    In general for x > 0, x + 1/x ≥ 2.
    This is trivially proven by x2 -2x + 1 = (x-1)2 ≥ 0.
     
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