Example 1. Solve the equation

$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
$$
Solution. Let $\frac{15 x-7}{5}=t$, where $t \in \boldsymbol{Z}$.

From this,

$$
x=\frac{5 t+7}{15}
$$

Then the equation becomes:

$$
\left[\frac{10 t+39}{40}\right]=t
$$

From the definition of "integer part" it follows that:

$$
0 \leqslant \frac{10 t+39}{40}-t<1
$$

Solving this double inequality, we find that

$$
-\frac{1}{30} \leqslant t<\frac{13}{10}
$$

Since $t \in \mathbf{Z}$, the inequality is satisfied by $t_{1}=0$, $t_{2}=1$. Then, respectively, $x_{1}=\frac{7}{15}, x_{2}=\frac{4}{5}$.