# All’s Well That Ends Well

In 2003 Carl Libis of Assumption College in Worcester, Mass., received this solution from a student in an algebra course:

\begin{aligned} \frac{1}{x+1} + \frac{1}{x-2} &= \frac{x+3}{x^2-x-2} \\ \frac{x+1}{1} + \frac{x-2}{1} &= \frac{x^2-x-2}{x+3} \\ x+1+x-2 &= \frac{x^2 - \frac{x}{x}-2}{3} \\ 2x-1 &= \frac{x^2 - 1 - 2}{3} \\ 3(2x-1) &= x^2 - 3 \\ 6x &= x^2 - 2 \\ \frac{6x}{2x} &= \frac{x^2 - 2}{2x} \\ 3 &= \frac{x^2}{x} - \frac{2}{2} \\ 3 &= x-1 \\ 4 &= x \end{aligned}

(Via Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 34:1 [January 2003], 50-54.)