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$1)\,\,\,\,0 < \frac{x}{{x - 1}} \le {2^{ - 1\,}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
$
$\Leftrightarrow \frac{x}{x-1}>0 và \frac{x+1}{2(x-1)}\leq 0\Leftrightarrow \left\{ \begin{array}{l}
x>1 hoặc x<0 \\
x\in (-1;1) \end{array} \right.$
$\Leftrightarrow $$ - 1 < x < 0$
$2)\,\,\,\,0 < \frac{{3x + 1}}{{x + 1}} \le {\left( {\frac{1}{2}} \right)^{ - 1\,}}\,\,\,\,\,\,\,\,\,\,\,\,$
$\Leftrightarrow \left\{ \begin{array}{l} \frac{3x+1}{x+1}>0\\ \frac{x-1}{x+1}\leq 0 \end{array} \right.\Leftrightarrow \left\{ \begin{array}{l} x>-\frac{1}{3} hoặc x<-1\\ x\in (-1;1)\end{array} \right.$
$\Leftrightarrow$ $ - \frac{1}{3} < x \le 1$
$3)\,\,\,\,0 < \frac{{x - 3}}{{x + 3}} \le {\left( {\frac{1}{4}} \right)^{ - \frac{1}{2}\,}}\,\,\,\,\,\,\,\,\,\,\,\,$
$\Leftrightarrow \left\{ \begin{array}{l} \frac{x-3}{x+3}>0\\ \frac{x+9}{x+3}\geq 0 \end{array} \right.\Leftrightarrow \left\{ \begin{array}{l} x>3 hoặc x<-3 \\ x>-3 hoặc x\leqslant -9 \end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l}
x \leq - 9\\
x > 3
\end{array} \right.$
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