\(\frac{2x+3}{x-1}< x+1\left(x\ne1\right)\)
\(\Leftrightarrow\frac{2x+3}{x-1}-x-1< 0\)
\(\Leftrightarrow\frac{2x+3-x^2+1}{x-1}< 0\)
\(\Leftrightarrow\frac{-x^2+2x+4}{x-1}< 0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}-x^2+2x-4< 0\\x-1>0\end{matrix}\right.\\\left\{{}\begin{matrix}-x^2+2x-4>0\\x-1< 0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}\left[{}\begin{matrix}x< 1-\sqrt{5}\\x>1+\sqrt{5}\end{matrix}\right.\\x>1\end{matrix}\right.\\\left\{{}\begin{matrix}1-\sqrt{5}< x< 1+\sqrt{5}\\x< 1\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x>1+\sqrt{5}\\1-\sqrt{5}< x< 1\end{matrix}\right.\)
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