Solve Rational Equations - College Algebra

A student attempts to solve $\frac{x+2}{x-3} = \frac{x-1}{x+1} + \frac{8}{(x-3)(x+1)}$ and finds that $x = 3$ and $x = -1$ . What should the student conclude?
Answer: Both solutions are extraneous due to domain restrictions, so there is no solution The original equation has denominators $x-3$ , $x+1$ , and $(x-3)(x+1)$ . This means the domain excludes $x = 3$ and $x = -1$ . Since both algebraic solutions $x = 3$ and $x = -1$ make denominators zero in the original equation, both are extraneous solutions that must be rejected. Therefore, the equation has no solution. This illustrates the critical importance of checking solutions against domain restrictions when solving rational equations.