1) ĐKXĐ: \(x\notin\left\{-1;2\right\}\)
Ta có: \(\frac{1}{x+1}-\frac{5}{x-2}=\frac{15}{x+2-x^2}\)
\(\Leftrightarrow\frac{x-2}{\left(x+1\right)\left(x-2\right)}-\frac{5\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}=\frac{-15}{\left(x-2\right)\left(x+1\right)}\)
Suy ra: \(x-2-5\left(x+1\right)=-15\)
\(\Leftrightarrow x-2-5x-5+15=0\)
\(\Leftrightarrow-4x+8=0\)
\(\Leftrightarrow-4x=-8\)
hay x=2(loại)
Vậy: \(S=\varnothing\)
2: ĐKXĐ: \(x\notin\left\{2;4\right\}\)
Ta có: \(\frac{x-3}{x-2}+\frac{x-2}{x-4}=3\frac{1}{5}\)
\(\Leftrightarrow\frac{5\left(x-3\right)\left(x-4\right)}{5\left(x-2\right)\left(x-4\right)}+\frac{5\left(x-2\right)^2}{5\left(x-2\right)\left(x-4\right)}=\frac{16\left(x-2\right)\left(x-4\right)}{5\left(x-2\right)\left(x-4\right)}\)
Suy ra: \(5\left(x-3\right)\left(x-4\right)+5\left(x-2\right)^2=16\left(x-2\right)\left(x-4\right)\)
\(\Leftrightarrow5\left(x^2-7x+12\right)+5\left(x^2-4x+4\right)=16\left(x^2-6x+8\right)\)
\(\Leftrightarrow5x^2-35x+60+5x^2-20x+20=16x^2-96x+128\)
\(\Leftrightarrow10x^2-55x+80-16x^2+96x-128=0\)
\(\Leftrightarrow-6x^2+41x-48=0\)
\(\Leftrightarrow-6x^2+32x+9x-48=0\)
\(\Leftrightarrow-2x\left(3x-16\right)+3\left(3x-16\right)=0\)
\(\Leftrightarrow\left(3x-16\right)\left(-2x+3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}3x-16=0\\-2x+3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}3x=16\\-2x=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{16}{3}\left(nhận\right)\\x=\frac{3}{2}\left(nhận\right)\end{matrix}\right.\)
Vậy: \(S=\left\{\frac{16}{3};\frac{3}{2}\right\}\)