1. ## 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:

$|x|+|x+3|+|x+4|+|x+7|+|x+8|=|x+1| + |x+2|+|x+5|+|x+6|+|x+9|$

2. Originally Posted by DivideBy0
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 ....

$x=-1,-3,-5,-7,-9$