I would look at 4 different regions
-7 < x <= 3
3< x <= 10
I would find f(x) for each of these regions. You can get rid of the absolute values.
each time f(x) will be a line.
If the gradient is negative then f(the biggest x) will be the minimum sum for that region.
If the gradient is positive then f(the smallest x) will be the minimum sum for that region.
[Graph them and you will see what I mean]
Determine the 4 minimums and then take the lowest minimum value
If you need more help you can ask (be specific about your problem)
but please no one else jump in before hand