Chọn phương án A.

\(\begin{aligned}
\blacksquare&\,\dfrac{4x+5}{6}< x-3\\
\Leftrightarrow&\,\dfrac{4x+5}{6}-(x-3)<0\\
\Leftrightarrow&\,\dfrac{4x+5-6(x-3)}{6}<0\\
\Leftrightarrow&\,-2x+23<0\\
\Leftrightarrow&\,x>\dfrac{23}{2}\,\,(1)\\
\blacksquare&\,2x+3>\dfrac{7x-4}{3}\\
\Leftrightarrow&\,2x+3-\dfrac{7x-4}{3}>0\\
\Leftrightarrow&\,\dfrac{3(2x+3)-(7x-4)}{3}>0\\
\Leftrightarrow&\,-x+13>0\\
\Leftrightarrow&\,x<13\,\,(2)
\end{aligned}\)

Từ (1) và (2) suy ra \(S=\left(\dfrac{23}{2};13\right)\).