8.45 For what values of $m$ are the roots of the equation

$$
4 x^{2}-(3 m+1) x-m-2=0
$$

confined to the interval between -1 and 2?

Solve the inequalities (8.46 - 8.66):

$8.46\left|\frac{3 x+1}{x-3}\right|<3$.
8.45 Let \(x_{1}\) and \(x_{2}\) be the roots of the given equation. According to the condition, \(-10\) and \(f(2)>0\). Thus, we obtain the system of inequalities

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-285.jpg?height=461&width=588&top_left_y=954&top_left_x=306)

Fig. 8.15

\[
\left\{\begin{array}{l}
4 + 3m + 1 - m - 2 > 0, \\
16 - 6m - 2 - m - 2 > 0,
\end{array} \text{ or } \left\{\begin{array}{l}
2m + 3 > 0, \\
-7m + 12 > 0,
\end{array} \text{ or } \left\{\begin{array}{l}
m > -\frac{3}{2}, \\
m < \frac{12}{7}
\end{array}\right.\right.\right.
\]

\[
\text{from which } -\frac{3}{2} < m < \frac{12}{7} \text{.}
\]

Answer: for \(-\frac{3}{2} < m < \frac{12}{7}\).