Example 19 Let $f(x)=|1-2 x|, x \in[0,1]$. Then, the number of solutions to the equation $f\{f[f(x)]\}=\frac{1}{2} x$ is $\qquad$ .
Analysis: Let $y=f(x), z=f(y), w=f(z)$, use “ $\rightarrow$ ” to indicate the change, we have
$$
\begin{array}{l}
x: 0 \rightarrow 1, y: 1 \rightarrow 0 \rightarrow 1, \\
z: 1 \rightarrow 0 \rightarrow 1 \rightarrow 0 \rightarrow 1, \\
w: 1 \rightarrow 0 \rightarrow 1 \rightarrow 0 \rightarrow 1 \rightarrow 0 \rightarrow 1 \rightarrow 0 \rightarrow 1 .
\end{array}
$$

Since the “ $\rightarrow$ ” change is linear and occurs within the non-negative real number range, the number of intersections between the graph of $w=f\{f[f(x)]\}$ and $w=\frac{x}{2}$ is 8, which means the number of solutions is 8.