Answer
$\dfrac{5-2x}{2(x-1)}$
Work Step by Step
Factoring the given expression, $
\dfrac{2}{x-1}-\dfrac{3x}{3x-3}+\dfrac{1}{2x-2}
,$ results to
\begin{array}{l}\require{cancel}
\dfrac{2}{x-1}-\dfrac{3x}{3(x-1)}+\dfrac{1}{2(x-1)}
.\end{array}
Using the $LCD=
6(x-1)
$, the expression above simplifies to
\begin{array}{l}\require{cancel}
\dfrac{6(2)-2(3x)+3(1)}{6(x-1)}
\\\\=
\dfrac{12-6x+3}{6(x-1)}
\\\\=
\dfrac{15-6x}{6(x-1)}
\\\\=
\dfrac{3(5-2x)}{6(x-1)}
\\\\=
\dfrac{\cancel{3}(5-2x)}{\cancel{3}\cdot2(x-1)}
\\\\=
\dfrac{5-2x}{2(x-1)}
.\end{array}
