4. (4 points) Solve the equation:

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

where the symbol $[a]$ denotes the integer part of the number $a$.

#
# 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 $\mathrm{x}$ 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

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

or

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

which is satisfied only by two integers: $m=0, m=1$.

Answer: $x=\frac{7}{15} ; x=\frac{4}{5}$.