Question

C/m đẳng thức:

\(\left[\dfrac{2}{3x}-\dfrac{2}{x+1}.\left(\dfrac{x+1}{3x}-x-1\right)\right]:\dfrac{x-1}{x}=\dfrac{2x}{x-1}\)

Answer 1

\(\left(\dfrac{2}{3x}-\dfrac{2}{x+1}\cdot\left(\dfrac{x+1}{3x}-x-1\right)\right):\dfrac{x-1}{x}\)

\(=\left(\dfrac{2}{3x}-\dfrac{2}{x+1}\cdot\dfrac{x+1-3x\left(x+1\right)}{3x}\right)\cdot\dfrac{x}{x-1}\)

\(=\dfrac{2x+2-2\left(x+1\right)\left(1-3x\right)}{3x}\cdot\dfrac{x}{x-1}\)

\(=\dfrac{2\left(x+1\right)-2\left(x+1\right)\left(1-3x\right)}{3}\cdot\dfrac{x}{x-1}\)

\(=\dfrac{2\left(x+1\right)\left(1-1+3x\right)}{3}\cdot\dfrac{x}{x-1}\)

\(=\dfrac{6x\left(x+1\right)}{3}\cdot\dfrac{x}{x-1}=\dfrac{2x^2\left(x+1\right)}{x-1}\)