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Math Help - Inequality in R

  1. #1
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    Inequality in R

    In trying to solve the following inequality in R:

    |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.

    So suppose the above inequality is not true ,then we have:


    |x+y| |y+z|+|z+x| > |x+y+z| |x| +|y| + |z|

    ....................or 2|x| +2|y| +2|z| > |x+y+z| +|x| +|y| +|z|

    .....................or |x| +|y| + |z|> |x+y+z| and here is where i get stuck because i cannot derive a contradiction
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  2. #2
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    Re: Inequality in R

    Quote Originally Posted by psolaki View Post
    In trying to solve the following inequality in R:

    |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|
    [snip]
    Are you trying to prove this?
    If so, it isn't true. For example, try x = y = z = 1.
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    Re: Inequality in R

    Quote Originally Posted by awkward View Post
    Are you trying to prove this?
    If so, it isn't true. For example, try x = y = z = 1.

    |x+y| +|y+z|+|z+x|=|1+1|+|1+1|+|1+1|= 2+2+2 =6

    |x|+|y|+|z| +|x+y+z| = |1|+|1|+|1|+|1+1+1| = 3+3=6

    Hence |x+y|+|y+z|+|z+x| \leq |x|+|y|+|z|+|x+y+z| is true
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  4. #4
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    Re: Inequality in R

    Quote Originally Posted by psolaki View Post
    |x+y| +|y+z|+|z+x|=|1+1|+|1+1|+|1+1|= 2+2+2 =6
    |x|+|y|+|z| +|x+y+z| = |1|+|1|+|1|+|1+1+1| = 3+3=6
    Hence |x+y|+|y+z|+|z+x| \leq |x|+|y|+|z|+|x+y+z| is true
    But that is not the problem you posted!
    What is the correct problem?
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  5. #5
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    Re: Inequality in R

    Classic case of bait & switch ! Ha! Ha!
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    Re: Inequality in R

    Quote Originally Posted by Plato View Post
    But that is not the problem you posted!
    What is the correct problem?
    Quote Originally Posted by psolaki View Post
    In trying to solve the following inequality in R:



    |x+y| +|y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.



    So suppose the above inequality is not true ,then we have:





    |x+y| |y+z|+|z+x| > |x+y+z| |x| +|y| + |z|



    ....................or 2|x| +2|y| +2|z| > |x+y+z| +|x| +|y| +|z|



    .....................or |x| +|y| + |z|> |x+y+z| and here is where i get stuck because i cannot derive a contradiction
    Here is my opening post
    Last edited by psolaki; October 15th 2011 at 03:24 PM.
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    Re: Inequality in R

    Quote Originally Posted by psolaki View Post
    Here is my opening post
    We all can read your OP.
    Quote Originally Posted by psolaki View Post
    In trying to solve the following inequality in R:
    |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.
    Clearly it is |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|
    Let x=y=z=1 then you have
    6=|1+1| |1+1|+|1+1| \leq |1+1+1| |1| +|1| + |1|=5
    So it is false.

    Why can't you see that?
    Or if what you posted is not correct, then why not simply correct it?
    Last edited by Plato; October 15th 2011 at 04:05 AM.
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    Re: Inequality in R

    It is one of those things that no body can explain. I must have gone at least 10 times over my opening post ,yet a terrible mistake has escape my attention .

    I am terribly sorry the correct inequality is:

    |x+y|+|y+z|+|z+x| \leq |x|+|y|+|z|+|x+y+z|
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