4. Solve the inequality $\left(\log _{x}^{2}(3 x-2)-4\right)(\sin \pi x-1) \leq 0$. (8 points)

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# Solution:

$\left(\log _{x}^{2}(3 x-2)-4\right)(\sin \pi x-1) \leq 0 \Leftrightarrow\left(\log _{x}(3 x-2)-2\right)\left(\log _{x}(3 x-2)+2\right)(\sin \pi x-1) \leq 0$ Domain: $x \in(2 / 3,1) \cup(1,+\infty)$

![](https://cdn.mathpix.com/cropped/2024_05_06_18575ac96ab43cb21cd9g-03.jpg?height=251&width=783&top_left_y=1071&top_left_x=288)

$\left[\begin{array}{l}\left\{\begin{array}{l}x \in \mathbb{R}, \\ \left(\log _{x}(3 x-2)-\log _{x} x^{2}\right)\left(\log _{x} x^{2}(3 x-2)-\log _{x} 1\right) \geq 0 ;\end{array}\right. \\ \left\{\begin{array}{l}x=1 / 2+2 n, n \in \mathbb{Z}, \\ \left(\log _{x}(3 x-2)-\log _{x} x^{2}\right)\left(\log _{x} x^{2}(3 x-2)-\log _{x} 1\right) \leq 0,\end{array}\right.\end{array} \Leftrightarrow\right.$

$\left[\begin{array}{l}\left\{\begin{array}{l}x \in(2 / 3,1) \cup(1,+\infty), \\ \left(x^{2}-3 x+2\right)\left(3 x^{3}-2 x^{2}-1\right) \leq 0 ;\end{array}\right. \\ \left\{\begin{array}{l}x=1 / 2+2 n, n \in \mathbb{Z}, \quad x \in(2 / 3,1) \cup(1,+\infty), \\ \left(x^{2}-3 x+2\right)\left(3 x^{3}-2 x^{2}-1\right) \geq 0,\end{array}\right.\end{array} \Leftrightarrow\right.$

$\left[\begin{array}{l}\left\{\begin{array}{l}x \in(2 / 3,1) \cup(1,+\infty), \\ (x-1)^{2}(x-2)\left(3 x^{2}+x+1\right) \leq 0 ;\end{array}\right. \\ \left\{\begin{array}{l}x=1 / 2+2 n, n \in \mathbb{Z}, \quad x \in(2 / 3,1) \cup(1,+\infty), \\ (x-1)^{2}(x-2)\left(3 x^{2}+x+1\right) \geq 0,\end{array}\right.\end{array} \Leftrightarrow\left[\begin{array}{l}x \in(2 / 3,1) \cup(1,2], \\ x=1 / 2+2 n, n \in \mathbb{N} .\end{array}\right.\right.$

Answer: $x \in(2 / 3,1) \cup(1,2] \cup\{x=1 / 2+2 n, n \in \mathbb{N}\}$.