2. (3 points) Solve the equation:

$$
\left[\frac{9 x-4}{6}\right]=\frac{12 x+7}{4}
$$
# Solution:

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

$$
\left\{\begin{array}{c}
\frac{9 x-4}{6} \geq \frac{12 x+7}{4} \\
\frac{9 x-4}{6}-\frac{41}{18}
\end{array}\right. \\
& x \in\left(-\frac{41}{18} ;-\frac{29}{18}\right]
\end{aligned}
$$

Now we need to find those values of \( x \) in this interval for which the number \(\frac{12 x+7}{4}\) is an integer (this restriction follows from the definition of the integer part).

Let \(\frac{12 x+7}{4}=m \in \mathbb{Z}\). Then \(x=\frac{4 m-7}{12}\), and we obtain the system of inequalities

$$
-\frac{41}{18}<\frac{4 m-7}{12} \leq-\frac{29}{18}
$$

or

$$
-\frac{61}{12}<m \leq-\frac{37}{12}
$$

which is satisfied only by two integers: \(m=-4, m=-5\).

Answer: \(x=-\frac{9}{4} ; x=-\frac{23}{12}\).