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If a and b have opposite signs then |a + b| < |a| + |b|

  1. Mar 4, 2016 #1
    1. The problem statement, all variables and given/known data
    If a and b have opposite signs then |a + b| < |a| + |b|

    2. Relevant equations
    No equations.

    3. The attempt at a solution

    Well first start with "a" positive and "-b" negative.

    We have :

    |a|=a

    |-b|=b

    |a-b|=a-b

    We begin with 0 < |a| + |-b|

    Then : 0 < |a| + |-b| -a

    a < |a| + |-b|

    a + (-b) < |a| + |-b|

    which gives us : |a-b| < |a| + |-b|

    We see that the result stays the same when we have -a and b.

    |-a|=a

    |b|=b

    |b-a|=b-a

    We begin with 0 < |b| + |-a|

    Then : 0 < |b| + |-a| -b

    b < |b| + |-a|

    b + (-a) < |b| + |-a|

    which gives us : |b-a| < |b| + |-a|

    Thus wee see that when the signs are different the following inequality holds :

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

    Is it any good ?
     
    Last edited by a moderator: Mar 4, 2016
  2. jcsd
  3. Mar 5, 2016 #2

    Mark44

    Staff: Mentor

    No, this is wrong. In the problem statement, you're given that a and b have opposite signs.
    For your first case, a > 0, and b < 0.

    To say, as you did, ' "-b" negative ', means that b > 0, if -b < 0.
    No.
    The problem asks you to show that |a + b| < |a| + |b|. It has nothing to do with |a - b|.
    Again, the problem has nothing to do with |a - b|.
     
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