Answer
$\dfrac{x-2}{2x-1}$
Work Step by Step
The given expression, $
\dfrac{\dfrac{x+2}{x}-\dfrac{2}{x-1}}{\dfrac{x+1}{x}+\dfrac{x+1}{x-1}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\dfrac{(x-1)(x+2)-x(2)}{x(x-1)}}{\dfrac{(x-1)(x+1)+x(x+1)}{x(x-1)}}
\\\\=
\dfrac{\dfrac{(x-1)(x+2)-x(2)}{\cancel{x(x-1)}}}{\dfrac{(x-1)(x+1)+x(x+1)}{\cancel{x(x-1)}}}
\\\\=
\dfrac{(x-1)(x+2)-x(2)}{(x-1)(x+1)+x(x+1)}
\\\\=
\dfrac{x^2+x-2-2x}{x^2-1+x^2+x}
\\\\=
\dfrac{x^2-x-2}{2x^2+x-1}
\\\\=
\dfrac{(x-2)(x+1)}{(2x-1)(x+1)}
\\\\=
\dfrac{(x-2)(\cancel{x+1})}{(2x-1)(\cancel{x+1})}
\\\\=
\dfrac{x-2}{2x-1}
.\end{array}
