Sum only positive numbers in series

Question

How can i sum only positive numbers in series like below:

For example i have a series like;
(10-x)
(3-x)
(4-x)
(15-x)

when i sum it up my equation is : (32-4x)

For x=6 result is 8

but i only want to sum positive numbers

(10-6)=4
(3-6)=-3
(4-6)=-2
(15-6)=9

when i sum only positive number result is 13. so 32-4x not working for me. what should my equation be for summing only positive numbers?

Thanks a lot.

Answer 1

This can be obtained with the absolute value

$$\sum_k\frac{(c_k-x)+|c_k-x|}2$$ to cancel out the negative terms.

$$\frac{32-4x+|10-x|+|3-x|+|4-x|+|15-x|}2.$$

As the function is piecewise linear and the constants seem irregular, there is no way to simplify it.

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It is also customary to denote the positive part of an expression with a $+$ exponent, giving

$$\sum_k(c_k-x)^+=\sum_k\max(0,c_k-x).$$

Answer 2

If there are only a finite number of terms of the form $a_i - x$ just look at the different $x$ where $a_i > x$ becomes true and segment the real line into the respective intervals (here the term "intervals" is meant to include such with an infinite border).