\(\dfrac{4}{x-1}-\dfrac{5}{x-2}=-3\left(x\ne1;x\ne2\right)\\ \Leftrightarrow\dfrac{4\left(x-2\right)}{\left(x-1\right)\left(x-2\right)}-\dfrac{5\left(x-1\right)}{\left(x-1\right)\left(x-2\right)}=\dfrac{-3\left(x-1\right)\left(x-2\right)}{\left(x-1\right)\left(x-2\right)}\\ \Leftrightarrow4\left(x-2\right)-5\left(x-1\right)=-3\left(x-1\right)\left(x-2\right)\\ \Leftrightarrow4x-8-5x+5=-3\left(x^2-2x-x+2\right)\\ \Leftrightarrow-x-3=-3\left(x^2-3x+2\right)\\ \Leftrightarrow-x-3=-3x^2+9x-6\\ \Leftrightarrow3x^2-10x+3=0\\ \Leftrightarrow3x^2-9x-x+3=0\\ \Leftrightarrow3x\left(x-3\right)-\left(x-3\right)=0\\ \Leftrightarrow\left(x-3\right)\left(3x-1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=3\\x=\dfrac{1}{3}\end{matrix}\right.\left(tm\right)\)
