Thực hiện phép tính:
a, \(\dfrac{4x}{4x-2}\) + \(\dfrac{1}{2x\left(1-2x\right)}\)
b, \(\dfrac{3x}{5x+5y}\) + \(\dfrac{-x}{10x-10y}\)
c, \(\dfrac{x+3}{x^2-1}\) + \(\dfrac{x+1}{x-x^2}\)
d, \(\dfrac{4+x^2}{x-2}\) - \(\dfrac{4x}{x-2}\)
e, \(\dfrac{1}{y\left(x-y\right)}\) - \(\dfrac{1}{x\left(x-y\right)}\)
g, \(\dfrac{x+9y}{x^2-9y^2}\) - \(\dfrac{3y}{x^2+3xy}\)
a: \(\dfrac{4x}{4x-2}+\dfrac{1}{2x\left(1-2x\right)}\)
\(=\dfrac{4x}{2\left(2x-1\right)}-\dfrac{1}{2x\left(2x-1\right)}\)
\(=\dfrac{4x^2-1}{2x\left(2x-1\right)}=\dfrac{\left(2x-1\right)\left(2x+1\right)}{2x\left(2x-1\right)}=\dfrac{2x+1}{2x}\)
b: \(\dfrac{3x}{5x+5y}+\dfrac{-x}{10x-10y}\)
\(=\dfrac{3x}{5\left(x+y\right)}-\dfrac{x}{10\left(x-y\right)}\)
\(=\dfrac{3x\cdot2\left(x-y\right)-x\cdot\left(x+y\right)}{10\left(x+y\right)\left(x-y\right)}\)
\(=\dfrac{6x^2-6xy-x^2-xy}{10\left(x+y\right)\left(x-y\right)}=\dfrac{5x^2-7xy}{10\left(x+y\right)\left(x-y\right)}\)
c: \(\dfrac{x+3}{x^2-1}+\dfrac{x+1}{x-x^2}\)
\(=\dfrac{x+3}{\left(x-1\right)\left(x+1\right)}-\dfrac{x+1}{x\left(x-1\right)}\)
\(=\dfrac{x\left(x+3\right)-\left(x+1\right)^2}{x\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{x^2+3x-x^2-2x-1}{x\left(x-1\right)\left(x+1\right)}=\dfrac{x-1}{x\left(x-1\right)\left(x+1\right)}=\dfrac{1}{x\left(x+1\right)}\)
d: \(\dfrac{4+x^2}{x-2}-\dfrac{4x}{x-2}\)
\(=\dfrac{x^2-4x+4}{x-2}\)
\(=\dfrac{\left(x-2\right)^2}{x-2}=x-2\)
e: \(\dfrac{1}{y\left(x-y\right)}-\dfrac{1}{x\left(x-y\right)}\)
\(=\dfrac{x}{xy\left(x-y\right)}-\dfrac{y}{xy\left(x-y\right)}\)
\(=\dfrac{x-y}{xy\left(x-y\right)}=\dfrac{1}{xy}\)
g: \(\dfrac{x+9y}{x^2-9y^2}-\dfrac{3y}{x^2+3xy}\)
\(=\dfrac{x+9y}{\left(x-3y\right)\left(x+3y\right)}-\dfrac{3y}{x\left(x+3y\right)}\)
\(=\dfrac{x\left(x+9y\right)-3y\left(x-3y\right)}{x\left(x+3y\right)\left(x-3y\right)}\)
\(=\dfrac{x^2+9xy-3xy+9y^2}{x\left(x+3y\right)\left(x-3y\right)}=\dfrac{\left(x+3y\right)^2}{x\left(x+3y\right)\left(x-3y\right)}=\dfrac{x+3y}{x\left(x-3y\right)}\)
