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:
$\displaystyle |x|+|x+3|+|x+4|+|x+7|+|x+8|=|x+1| + |x+2|+|x+5|+|x+6|+|x+9|$
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:
$\displaystyle |x|+|x+3|+|x+4|+|x+7|+|x+8|=|x+1| + |x+2|+|x+5|+|x+6|+|x+9|$
One way is to consider cases:
$\displaystyle x\geq 0$ then all are $\displaystyle \geq 0$.
Then,
$\displaystyle x<0 \mbox{ and }x+1\geq 0 \implies 0>x\geq -1$
And get another case.
And so on ....
It is not that bad.
$\displaystyle x=-1,-3,-5,-7,-9$
I think, I did not really do this in detail.