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Ta có :$\left\{ \begin{array}{l}
\frac{3}{2} + \sin x \ne 1\\
\frac{1}{2} + \sin \,x \ge 0
\end{array} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \sin \,x > - \frac{1}{2}$
Suy ra : $\frac{3}{2} + \sin \,x > 1$, từ đó $\sqrt{\frac{1}{2}+\sin x}>0$
. $\begin{array}{l}
(1)\Leftrightarrow {\log _{\frac{3}{2} + \sin x}}\left( {{{\sin }^2}x + \frac{{6\sqrt 3 + 9}}{4}} \right) - 2 \ge 0\\
\Leftrightarrow {\sin ^2}x + \frac{{6\sqrt 3 + 9}}{4} \ge {\left( {\frac{3}{2} + \sin x} \right)^2}\\
\Leftrightarrow 3\sin x-\frac{3\sqrt3}{2}\leq 0\\
\Leftrightarrow - \frac{1}{2} < \sin \,x \leq \frac{{\sqrt 3 }}{2}\\
\Leftrightarrow \left[ \begin{array}{l}
- \frac{\pi }{6} + 2k\pi < x \le \frac{\pi }{3} + 2k\pi \\
\frac{{2\pi }}{3} + 2k\pi \le x < \frac{{7\pi }}{6} + 2k\pi
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(k \in Z)
\end{array}$
Vậy hệ đã cho có nghiệm $\left[ \begin{array}{l}
- \frac{\pi }{6} + 2k\pi < x \le \frac{\pi }{3} + 2k\pi \\
\frac{{2\pi }}{3} + 2k\pi \le x < \frac{{7\pi }}{6} + 2k\pi
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(k \in Z)$
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