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Homework Help: Triangle inequality

  1. Sep 6, 2007 #1
    1. The problem statement, all variables and given/known data

    Show that if |a-5| < 1/2 and |b-8| < 1/2 then |(a+b)-13| < 1. Hint: use the triangle inequality.

    2. Relevant equations
    Triangle Inequality:

    |a+b| <= |a|+|b|

    3. The attempt at a solution

    I really don't know how to use the triangle inequality so I was hoping someone could clear up for me exactly how it is used my book doesn't really make it clear it just states what it is which is what I have stated above. I understand why it is true, I just do not understand how you would use it in a problem. I plugged the first parts into it to get |(a-5)+(b-8)| <= |a-5| + |b-8| I'm not really sure how to simplify this though it should simplify to |a+b-13| but I can't get that everything is just canceling out for me.
    Last edited: Sep 6, 2007
  2. jcsd
  3. Sep 6, 2007 #2


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    Staff Emeritus
    Science Advisor

    Apply the triangle inequality to (a-5) and (b-8). Then |(a-5)+(b-8)|=|a+b-13|[itex]\leq[/itex]|a-5|+|b-8|<1/2+1/2=1

    The first inequality is the triangle inequality, and the second is from the original information.
    Last edited: Sep 6, 2007
  4. Sep 6, 2007 #3
    You're almost there: expand out the brackets on the left hand, and add a further inequality to the right, using the information that you've been given, but which you've not yet used.
  5. Sep 6, 2007 #4
    k so I ended up getting |a+b-13|<= |a+5 + |b-8| < 1 thanks a lot guys.
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