Solve and discuss the following parametric equation:

$$
\frac{a x-3}{b x+1}=2
$$
In solving the task, many only managed to reach the point that the condition $a \neq 2 b$ must be satisfied. Fewer noticed that this condition is not sufficient, since if $a=-3 b$, then after the transformations, the root obtained is false, as substituting this into the denominator on the left side of the fraction results in a value of 0. Therefore, the equation has a unique solution only if the conditions $a \neq 2 b$ and $a \neq-3 b$ are satisfied; otherwise, there is no solution.