Example 7 The set of all real solutions to the equation $[2 x]+[3 x]=9 x-\frac{7}{4}$ is $\qquad$ .
According to the properties of the Gaussian function, we have
$$
2 x-1<[2 x] \leqslant 2 x, 3 x-1<[3 x] \leqslant 3 x \text {. }
$$

Therefore, $5 x-2<[2 x]+[3 x] \leqslant 5 x$.
Thus, $5 x-2<9 x-\frac{7}{4} \leqslant 5 x$.
Solving this, we get $-\frac{1}{16}<x \leqslant \frac{7}{16}$.
Hence, $-\frac{37}{16}<9 x-\frac{7}{4} \leqslant \frac{35}{16}$.
Since $9 x-\frac{7}{4}$ is an integer, we have
$$
9 x-\frac{7}{4}=-2,-1,0,1,2 \text {. }
$$

Solving for $x$, we get $x=-\frac{1}{36}, \frac{1}{12}, \frac{7}{36}, \frac{11}{36}, \frac{5}{12}$.
Upon verification, $x=-\frac{1}{36}$ and $x=\frac{7}{36}$ are both solutions to the original equation.