Piecewise Linear Functions

Look at $f(x) = |x - 3| + |x - 7| + |x - 5|$. This is continuous and piecewise linear. It has corners at $x = 3, 5, 7$. Between corners, the slope stays constant.

At each corner, the slope increases by $2$. Before $x = 3$, all three terms contribute $-1$ each, giving slope $= -3$. Between $3$ and $5$, the first term flips to $+1$, so slope $= -1$. You don't need $f(x)$ for all $x$. You just need the corners and their slope increments. This insight powers Slope Trick.