8.5 $\quad \frac{x^{2}(2 x-9)(x-1)^{3}}{(x+4)^{5}(2 x-6)^{4}} \leq 0$.
8.5 Assuming $x \neq 0$ and $x \neq 3$, divide both sides of the inequality by the positive fraction $\frac{x^{2}}{(2 x-6)^{4}}$ and immediately note that $x=0$ satisfies the given inequality, while $x=3$ does not. Moreover, replace the factors with odd exponents with the corresponding first-degree factors (it is clear that the sign of the expression on the left side of the inequality will not change). As a result, we obtain a simpler inequality, equivalent to the given one for all $x \neq 0$ and $x \neq 3$:

$$
\frac{(2 x-9)(x-1)}{x+4} \leq 0
$$

By drawing the sign curve, we shade the intervals that satisfy this inequality and mark the points $x=0$ and $x=3$ on the same axis (Fig. 8.5). Considering that the value $x=0$ is a solution to the given inequality but does not belong to the shaded interval, we include it in the answer. The value $x=3$ is not a solution to the inequality but belongs to the shaded interval; therefore, it must be excluded.

Answer: $(-\infty,-4) \cup[1,3) \cup(3 ; 4.5] \cup 0$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-270.jpg?height=110&width=586&top_left_y=241&top_left_x=307)

Fig. 8.5