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Math Help - Tricky absolute equation

  1. #1
    Senior Member DivideBy0's Avatar
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    Tricky absolute equation

    This question came up in a not-so-recent math competition, and, seeing the thread on solving absolute inequalities, I wonder how you would tackle this:

    Solve for all x:

    |x|+|x+3|+|x+4|+|x+7|+|x+8|=|x+1| + |x+2|+|x+5|+|x+6|+|x+9|
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  2. #2
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    Quote Originally Posted by DivideBy0 View Post
    This question came up in a not-so-recent math competition, and, seeing the thread on solving absolute inequalities, I wonder how you would tackle this:

    Solve for all x:

    |x|+|x+3|+|x+4|+|x+7|+|x+8|=|x+1| + |x+2|+|x+5|+|x+6|+|x+9|
    One way is to consider cases:
    x\geq 0 then all are \geq 0.

    Then,
    x<0 \mbox{ and }x+1\geq 0 \implies 0>x\geq -1
    And get another case.

    And so on ....

    It is not that bad.

    x=-1,-3,-5,-7,-9
    I think, I did not really do this in detail.
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