\(\frac{x-4}{\left(2x-1\right)\left(x-3\right)}< 0\)\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-4< 0\\\left(2x-1\right)\left(x-3\right)>0\end{matrix}\right.\\\left\{{}\begin{matrix}x-4>0\\\left(2x-1\right)\left(x-3\right)< 0\end{matrix}\right.\end{matrix}\right.\)
\(TH1:\left\{{}\begin{matrix}x-4< 0\\\left(2x-1\right)\left(x-3\right)>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x< 4\\\left[{}\begin{matrix}\left\{{}\begin{matrix}2x-1>0\\x-3>0\end{matrix}\right.\\\left\{{}\begin{matrix}2x-1< 0\\x-3< 0\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x< 4\\\left[{}\begin{matrix}\left\{{}\begin{matrix}x>\frac{1}{2}\\x>3\end{matrix}\right.\\\left\{{}\begin{matrix}x< \frac{1}{2}\\x< 3\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x< 4\\\left[{}\begin{matrix}x>3\\x< \frac{1}{2}\end{matrix}\right.\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}3< x< 4\\x< \frac{1}{2}\end{matrix}\right.\)
\(TH2:\left\{{}\begin{matrix}x-4>0\\\left(2x-1\right)\left(x-3\right)< 0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x>4\\\left[{}\begin{matrix}\left\{{}\begin{matrix}2x-1>0\\x-3< 0\end{matrix}\right.\\\left\{{}\begin{matrix}2x-1< 0\\x-3>0\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x>4\\\left[{}\begin{matrix}\left\{{}\begin{matrix}x>\frac{1}{2}\\x< 3\end{matrix}\right.\\\left\{{}\begin{matrix}x< \frac{1}{2}\\x>3\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x>4\\\left[{}\begin{matrix}\frac{1}{2}< x< 3\\\left(ktm\right)\end{matrix}\right.\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x>4\\\frac{1}{2}< x< 3\end{matrix}\right.\)\(\left(ktm\right)\)
Vậy phương trình có tập nghiệm là 3 < x < 4
