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Math Help - triangle inequality

  1. #1
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    triangle inequality

    Hello,
    I had a question regarding triangle inequalities.
    I proved the inequality abs(x+y) <= abs(x) + abs(y).

    Second part of question is asking to use the triangle inequality to prove "shortest path" version:
    for all x,y,z in R, abs(x-y) <= abs(x-z) + abs(z-x).
    I've got no idea on how to start this part.
    Should there be a y in the right half of the inequality? Am I supposed to assume something about y S.T it doesn't need to be
    in the right half?
    Thanks all
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  2. #2
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    Re: triangle inequality

    I assume the inequality should be

    abs(x - y) <= abs(x - z) + abs(z - y).

    Note that the abs arguments in the right-hand side add up to the abs argument in the left-hand side: (x - z) + (z - y) = x - y. This is similar to the original triangle inequality. Does this help?
    Thanks from jys89 and topsquark
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  3. #3
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    Re: triangle inequality

    Quote Originally Posted by jys89 View Post
    I proved the inequality abs(x+y) <= abs(x) + abs(y).

    Second part of question is asking to use the triangle inequality to prove "shortest path" version:
    for all x,y,z in R, abs(x-y) <= abs(x-z) + abs(z-x).
    As written it is false. Counterexample: x=1,~y=5,~\&~z=2.

    You can show that |x-y|\le|y-z|+|z-y| simply by noting that (x-y)=(x-z)+(z-y),
    Thanks from jys89 and topsquark
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  4. #4
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    Re: triangle inequality

    Thanks to both of you guys. I knew something about the problem was wrong.
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