Answer
{$\frac{1}{5},2$}
Work Step by Step
First, we add the fractions on the left hand side by taking their LCM. Upon inspection, the LCM is found to be $x(x-1)$:
$\frac{2}{x-1}+\frac{1}{x}=\frac{5}{2}$
$\frac{2(x)+1(x-1)}{x(x-1)}=\frac{5}{2}$
$\frac{2x+x-1}{x(x-1)}=\frac{5}{2}$
$\frac{3x-1}{(x^{2}-x)}=\frac{5}{2}$
Now, we cross multiply the two fractions in order to create a linear equation:
$\frac{3x-1}{x(x-1)}=\frac{5}{2}$
$2(3x-1)=5(x^{2}-x)$
$6x-2=5x^{2}-5x$
$5x^{2}-5x=6x-2$
$5x^{2}-5x-6x+2=0$
$5x^{2}-11x+2=0$
Now, we use rules of factoring trinomials to solve the equation:
$5x^{2}-11x+2=0$
$5x^{2}-10x-1x+2=0$
$5x(x-2)-1(x-2)=0$
$(x-2)(5x-1)=0$
$(x-2)=0$ or $(5x-1)=0$
$x=2$ or $x=\frac{1}{5}$
Therefore, the solution is {$\frac{1}{5},2$}.
