2. (3 points) Solve the equation:

$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
$$
# Solution:

By the definition of the integer part of a number, we have the system of inequalities

$$
\left\{\begin{array}{c}
\frac{5+6 x}{8} \geq \frac{15 x-7}{5} \\
\frac{5+6 x}{8} > \frac{41}{90}
\end{array}\right. \\
& x \in\left(\frac{41}{90} ; \frac{9}{10}\right]
$$

Now we need to find those values of \( x \) in this interval for which the number \(\frac{15 x-7}{5}\) is an integer (this restriction follows from the definition of the integer part).

Let \(\frac{15 x-7}{5}=m \in \mathbb{Z}\). Then \(x=\frac{5 m+7}{15}\), and we get the system of inequalities

$$
\frac{41}{90}<\frac{5 m+7}{15} \leq \frac{9}{10}
$$

or

$$
-\frac{1}{42}<5 m \leq \frac{13}{2}
$$

which is satisfied only by two integers: \(m=0, m=1\).

Answer: \(x=\frac{7}{15} ; x=\frac{4}{5}\)