\(\frac{2}{3}\le\frac{x^2-mx+1}{x^2-x+1}\le\frac{3}{2}\Leftrightarrow\left\{{}\begin{matrix}2x^2-2mx+2\le3x^2-3x+3\\3x^2-3mx+3\ge2x^2-2x+2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+\left(2m-3\right)x+1\ge0\\x^2-\left(3m-2\right)x+1\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\Delta=\left(2m-3\right)^2-4\le0\\\Delta=\left(3m-2\right)^2-4\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{1}{2}\le x\le\frac{5}{2}\\0\le x\le\frac{4}{3}\end{matrix}\right.\Leftrightarrow\frac{1}{2}\le x\le\frac{4}{3}\)
