I know that whenever $Ax=\lambda x$, then $\lambda$ is an eigenvalue of $A$ and $x$ is the corresponding eigenvector. So if the assumption is $Ax \neq \lambda x$, does it mean that I should avoid the eigenvectors of $A$ corresponding to the eigenvalue $\lambda$? Well, I want to construct a matrix consisting of eigenvectors of $A$.
Fact i know: If $Ax \neq \lambda x$, then $\lambda$ is not an eigenvalue of $A$.
Quantifiers are so important. You only know $\lambda$ is not an eigenvalue if and only if $\forall x\ne0~ Ax\ne\lambda x$. (Depending on context, you may even have to extend scalars to the algabriac closure and take any $x$ from the correponding vector space over the algabraic closure to be in the scope of the unversal quantifier.) On the other hand, just having for some $x$ that $Ax\ne\lambda x$ just shows that $A$ is not the diagonal matrix with only $\lambda$ on the diagonal. When no quantifiers are explicitly mentioned and you introduce a free variable, most people will innocently assume the existensial quantifier was intended to apply.