Answer
$$\frac{{10}}{x} + \frac{1}{{x - 1}}$$
Work Step by Step
$$\eqalign{
& \frac{{11x - 10}}{{{x^2} - x}} \cr
& {\text{factoring}} \cr
& = \frac{{11x - 10}}{{x\left( {x - 1} \right)}} \cr
& {\text{partial fraction decomposition}} \cr
& \frac{{11x - 10}}{{x\left( {x - 1} \right)}} = \frac{A}{x} + \frac{B}{{x - 1}} \cr
& 11x - 10 = A\left( {x - 1} \right) + Bx \cr
& {\text{letting }}x = 0 \cr
& - 10 = A\left( { - 1} \right) \cr
& 10 = A \cr
& {\text{letting }}x = 1 \cr
& 11 - 10 = A\left( {1 - 1} \right) + B\left( 1 \right) \cr
& 1 = B \cr
& {\text{substituting the values}} \cr
& \frac{A}{x} + \frac{B}{{x - 1}} = \frac{{10}}{x} + \frac{1}{{x - 1}} \cr} $$
