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Math Help - Solving inequation with 3 absolute values

  1. #1
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    Solving inequation with 3 absolute values

    Hello,
    I have read the sticky thread about solving inequalities, but still I cannot solve this one:
    |x-1|+|x-2|-|x+3|>=x

    Thanks in advance!
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  2. #2
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    You might have to use the triangle inequality here...

    \displaystyle x \leq |x - 1| + |x - 2| - |x + 3|

    \displaystyle x \leq |x| + |-1| + |x| + |-2| - (|x| + |3|), since \displaystyle |a + b| \leq |a| + |b| by the Triangle Inequality

    \displaystyle x \leq |x| + 1 + |x| + 2 - |x| - 3

    \displaystyle x \leq |x|.


    A number is only ever less than its absolute value if it's negative, so

    \displaystyle x \leq 0.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    You might have to use the triangle inequality here...

    \displaystyle x \leq |x - 1| + |x - 2| - |x + 3|

    \displaystyle x \leq |x| + |-1| + |x| + |-2| - (|x| + |3|), since \displaystyle |a + b| \leq |a| + |b| by the Triangle Inequality

    \displaystyle x \leq |x| + 1 + |x| + 2 - |x| - 3

    \displaystyle x \leq |x|.


    A number is only ever less than its absolute value if it's negative, so

    \displaystyle x \leq 0.
    I understand, thanks a lot

    Edit:Unfortunately, I think there is a problem with solving this question with triangle inequation.
    When you write -(|x|+|3|) instead of -|x+3| it ruins the inequation because you subtract bigger number.
    Last edited by msokol89; February 26th 2011 at 11:20 AM. Reason: unresolved question
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