Example 2 Given $f(x)=|1-2 x|, x \in[0$, 1]. How many real solutions does the equation $f(f(f(x)))=\frac{x}{2}$ have?
Solution: To find the number of real solutions using the graphical method. First, draw the graph of $f(x) = |2x - 1|$, and then double the y-coordinates of all points while keeping the x-coordinates unchanged. Next, shift the obtained graph down by 1 unit (as shown in Figure 2), and then reflect the part of the graph below the x-axis to above the x-axis, to get the graph of $f(f(x)) = |2f(x) - 1|$ (as shown in Figure 3).

Using the same method, we can draw the graph of the function $f(f(f(x))) = |2f(f(x)) - 1|$ (as shown in Figure 4), which intersects the line $y = \frac{x}{2}$ at 8 points in the interval $[0,1]$. Therefore, the original equation has 8 real solutions.