Can summations distribute across absolute values?

Question

Can I distribute a summation as follows?

$$ k\sum_{x \in X} \left| x - b \right| = \left| \left(k\sum_{x \in X}x \right) - \left( k\sum_{x \in X}b \right) \right| $$

Answer 1

Hint:

Let, $X=\{a_1,a_2, \dots a_n\}$

$b > a_i$ , $1 \le i \le (n-k)$ and $b<a_i , (n-k)<i\le n$, for $k>1$.

Use your distribution and check out why it fails.

Answer 2

We have the triangle inequality $|a+b|\le |a|+|b|$. The equality fails, if $a,b$ have different signs. What you have here is a generalization to more than two summands and hence can in general only be written as inequality, not as equality.