Solution:

We want to know the value of $c$ so that the graph $|x-20| + |x+18| - x = c$ has one solution. The graph of the function $|x-20| + |x+18| - x$ consists of an infinite section of slope $-3$ for $x \in (-\infty, -18]$, then a finite section of slope $-1$ for $x \in [-18, 20]$, then an infinite section of slope $1$ for $x \in [20, \infty)$. Notice that this graph is strictly decreasing on $(-\infty, 20]$ and strictly increasing on $[20, \infty)$. Therefore any horizontal line will intersect this graph $0$ or $2$ times, except the one that passes through the "vertex" $(20, |20-20| + |20+18| - 20) = (20, 18)$, giving a value of $c = 18$.