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