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Triangle Inequalities Relationship

  1. Jun 6, 2012 #1
    I know the following
    [tex]|x|-|y| \leq |x+y| \leq |x| + |y|[/tex]
    where does [tex] |x-y| [/tex] fit in the above equation?
  2. jcsd
  3. Jun 6, 2012 #2
    How about [itex]x+(-y)[/itex]?

    also, notice that there is a sort of better result, but I like the way you wrote it, makes it easier to remember and figure out what might be needed in a problem. But sometimes the left inequality is written:


    Just so that you understand when many other people write this.
  4. Jun 6, 2012 #3

    [itex]|x|-|y| \leq |x+y| \leq |x| + |y|[/itex]


    [itex]|x|-|y| \leq |x-y| \leq |x| + |y|.[/itex]

    I think we can't say anything about the relationship between[itex]|x+y|[/itex] and [itex]|x-y|,[/itex]
    and in between [itex]||x|-|y|| [/itex]and [itex]|x|-|y|.[/itex]
  5. Jun 6, 2012 #4
  6. Jun 6, 2012 #5
    You can prove this pretty quickly by plugging numbers in, or just notice that the replacement ##y \mapsto -y## yields the other, hence the only possible relation is equality, which is clearly false.
  7. Jun 16, 2012 #6
    if x and y have like signs then lx+yl ≥ lx-yl if unlike signs then lx+yl≤ lx-yl , check it out and always llxl-lyll >= lxl-lyl
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