Fractional equation with a rejected solution

Text

Solve the following equation: $\frac{3}{x-1} -\frac{2}{x^2-1} = \frac{1}{x+1}.$

Solution

Factor $x^2-1=(x-1)(x+1)$; existence conditions $x\neq \pm 1$. Multiplying by $(x-1)(x+1)$: $3(x+1) - 2 = (x-1) \;\Rightarrow\; 3x + 1 = x - 1 \;\Rightarrow\; 2x = -2 \;\Rightarrow\; x = -1.$ But $x=-1$ is excluded from the domain (it makes the denominators vanish): the solution found is not acceptable. $\boxed{\text{No acceptable solution}\ (x=-1 \text{ excluded})}$