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Absolute Value Proof (Need Help)

  1. Feb 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that abs(a+b+c) is less than or equal to abs(a)+abs(b)+abs(c)

    2. Relevant equations


    3. The attempt at a solution

    This makes sense to me that this would always be true, but i just can't seem to figure out how to write it out
  2. jcsd
  3. Feb 13, 2008 #2


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    Write it as abs((a+b)+c) then use the triangle inequality a couple times.
  4. Feb 13, 2008 #3


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    Are you asking to prove |a+b+c| < |a| + |b| + |c| given |a+b| < |a| + |b|? Or do you need to prove |a+b| < |a| + |b| as well?
  5. Feb 13, 2008 #4
    Yes, we are given that the triangle inequality is true, (and we also know how to prove the triangle inequality if that helps.)

    But it doesn't seem that I can prove this the same way.

    I know abs(a+b) <= abs(a) + abs(b)

    So I can very easily get that abs(a+b) + abs(c) <= abs(a) + abs(b) + abs(c)

    But how do I a get the left half to what i want?
  6. Feb 13, 2008 #5


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    Let d = a+b. |d+c| < |d|+|c| by tri. ineq.

    So, |a+b+c| < |a+b|+|c|.

    Now use the tri. ineq. on a+b, then add |c| to both sides.
  7. Feb 13, 2008 #6


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    Think of (a+b) as a single number. That is, look at |x+ y| and then set x= a+b, y= c.
  8. Feb 13, 2008 #7
    Wow...it seems so obvious now...

    Sometimes I feel like a genius, others I feel stupid as hell, guess whats the case now?

    Thanks so much.
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