A unique solution Find the sum of all values of k so that the system
$$y=|x+23|+|x−5|+|x−48|$$$$y=2x+k$$
has exactly one solution in real numbers. If the system has one solution, then one of the three $x's$, should be 0. Then, there are 3 solutions, when each of the modulus becomes 0. Where am I going wrong?

$$y-2x$$ is a piecewise linear function and is convex. The slope is nowhere zero (slope values are $-5,-3,-1,1$) so that it achieves an isolated minimum, which is the requested value of $k$, at the lowest vertex.
$$\begin{align}-23&\to145\\5&\to61\\48&\to\color{green}{18}\end{align}$$
