19. For what values of $k$ does the equation
$$
|x-2007|+|x+2007|=k
$$
have $(-\infty,-2007) \cup(2007,+\infty)$ as its solution set?
19. $k>4014$
If $x \in(-\infty,-2007)$, then
$$
-(x-2007)-(x+2007)=k \quad \text { or } \quad x=-\frac{k}{2} .
$$

If $x \in[-2007,2007]$, then
$$
-(x-2007)+(x+2007)=k \quad \text { or } \quad k=4014 .
$$

If $x \in(2007,+\infty)$, then
$$
(x-2007)+(x+2007)=k \quad \text { or } \quad x=\frac{k}{2} .
$$

Thus, the given equation has $(-\infty,-2007) \cup(2007,+\infty)$ as its solution set if and only if
$$
\frac{k}{2}>2007 \quad \text { or } \quad k>4014
$$