1: \(\dfrac{4x-1}{3x^2y}-\dfrac{7x-1}{3x^2y}\)
\(=\dfrac{4x-1-7x+1}{3x^2y}\)
\(=\dfrac{-3x}{3x^2y}=-\dfrac{1}{xy}\)
2: \(\dfrac{2x-7}{10x-4}-\dfrac{3x+5}{4-10x}\)
\(=\dfrac{2x-7}{10x-4}+\dfrac{3x+5}{10x-4}\)
\(=\dfrac{2x-7+3x+5}{10x-4}=\dfrac{5x-2}{10x-4}=\dfrac{1}{2}\)
3: \(\dfrac{3}{2x+6}-\dfrac{x-6}{2x^2+6x}\)
\(=\dfrac{3}{2\left(x+3\right)}-\dfrac{x-6}{2x\left(x+3\right)}\)
\(=\dfrac{3x-x+6}{2x\left(x+3\right)}\)
\(=\dfrac{2x+6}{2x\left(x+3\right)}=\dfrac{2\left(x+3\right)}{2x\left(x+3\right)}=\dfrac{1}{x}\)
4: \(x^2+1-\dfrac{x^4-3x^2+2}{x^2-1}\)
\(=x^2+1-\dfrac{\left(x^2-2\right)\left(x^2-1\right)}{x^2-1}\)
\(=x^2+1-\left(x^2-2\right)=3\)
5: \(\dfrac{1}{x\left(x+y\right)}+\dfrac{1}{y\left(x+y\right)}+\dfrac{1}{x\left(x-y\right)}+\dfrac{1}{y\left(y-x\right)}\)
\(=\dfrac{x+y}{xy\left(x+y\right)}+\dfrac{1}{x\left(x-y\right)}-\dfrac{1}{y\left(x-y\right)}\)
\(=\dfrac{1}{xy}+\dfrac{y-x}{xy\left(x-y\right)}\)
\(=\dfrac{1}{xy}-\dfrac{1}{xy}=0\)
7: \(\dfrac{x^3}{x-1}-\dfrac{x^2}{x+1}-\dfrac{1}{x-1}+\dfrac{1}{x+1}\)
\(=\dfrac{x^3-1}{x-1}-\dfrac{x^2-1}{x+1}\)
\(=\dfrac{\left(x-1\right)\left(x^2+x+1\right)}{x-1}-\dfrac{\left(x-1\right)\left(x+1\right)}{x+1}\)
\(=x^2+x+1-\left(x-1\right)=x^2+2\)
8: \(\dfrac{x^2+3x+2}{x^2+9x+18}:\dfrac{x^2-1}{x^2+6x+9}\)
\(=\dfrac{\left(x+1\right)\left(x+2\right)}{\left(x+3\right)\left(x+6\right)}\cdot\dfrac{\left(x+3\right)^2}{\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{\left(x+2\right)\left(x+3\right)}{\left(x+6\right)\left(x-1\right)}\)
