Answer
$x=\dfrac{5}{3}$
Work Step by Step
The factored form of the given equation, $ \dfrac{x-2}{x^2-7x+10}=\dfrac{1}{5x-10}-\dfrac{1}{x-5} ,$ is \begin{array}{l}\require{cancel} \dfrac{x-2}{(x-5)(x-2)}=\dfrac{1}{5(x-2)}-\dfrac{1}{x-5} .\end{array} Multiplying both sides by the $LCD= 5(x-5)(x-2) ,$ then the solution to the given equation is \begin{array}{l}\require{cancel}
5(x-2)=(x-5)(1)-5(x-2)(1)
\\\\
5x-10=x-5-5x+10
\\\\
5x-x+5x=-5+10+10
\\\\
9x=15
\\\\
x=\dfrac{15}{9}
\\\\
x=\dfrac{\cancel{3}\cdot5}{\cancel{3}\cdot3}
\\\\
x=\dfrac{5}{3}
.\end{array}
