problem: Find all values of $x$ such that $\displaystyle\frac{1}{x-1} + \frac{2x}{x - 1} = 5$.
solution: We can combine the two terms on the left side to get $\dfrac{1+2x}{x-1} = 5$.  We then multiply both sides of this equation by $x-1$ to get rid of the fractions.   This gives us $1+2x = 5(x-1)$. Expanding the right side gives $1+2x = 5x -5$.  Subtracting $5x$ from both sides   gives $1-3x = -5$, and subtracting 1 from both sides of this equation yields $-3x = -6$.  Dividing both sides   of this equation by $-3$ gives us our answer, $x = \boxed{2}$.