Example 7 Solve the equation $\left[\frac{x+1}{4}\right]=\left[\frac{x-1}{2}\right]$.
Solution: By the properties of the Gaussian function, we get
$-1<\frac{x+1}{4}-\frac{x-1}{2}<1$, i.e., $-1<x<7$. Let $y_{1}=\frac{x+1}{4}, y_{2}=$
$\frac{x-1}{2}$. In the same coordinate system, draw the graphs of the two (as shown in Figure 25-1).
In region I, when $-1<x<1$, it does not satisfy the original equation. Let $y_{2}=0$, we get $x=$ 1, so $1 \leqslant x<3$ is the solution.
In region I, let $y_{2}=2$, we get $x=5$, so $3 \leqslant x<5$ is the solution.
In summary, the solution to the original equation is $1 \leqslant x<5$.