Multiplying boths sides of an equation by $\frac{1}{x}$ I want to know what can happen if we multiply both sides of an equation by  $\frac{1}{x}$, where $x$ is a variable.
I mean, is it possible that we get redundant equations? Or defective equations ?

You might find it useful to argue "by cases". An example would be instructive:

Find all $x$ satisfying $x^2 + x = 0$.

Seeing the common factor of $x$, it is tempting to divide both sides by $x$ (i.e., multiply both sides by $\frac{1}{x}$). Any time you divide by an unknown quantity, however, you should be concerned about the case where that quantity is zero (since division by zero is undefined). You could use cases to organize your thinking: either $x$ is zero or it is not.
Case 1: $x = 0$
Since $x = 0$, we can substitue to find $0^2 + 0$ is indeed $0$, so $x = 0$ is a solution.
Case 2: $x \neq 0$
Since $x \neq 0$, it is perfectly legal to divide both sides of the equation by $x$. We are left with the simpler equation $x + 1 = 0$, which has solution $x = -1$.
Putting the two cases together, we see that both $x = 0$ and $x = -1$ are solutions. Notice that if you had divided both sides by $x$ without thinking in cases, you would have missed the $x = 0$ solution.
