Motivation: From what I've heard, if you have a system of $n$ equations, featuring $n$ variables, then it will typically be possible to derive a single value for each of the variables (whatever typically would mean in this context). Now, a question about how to prove whether or not this is the case for a system of equations would be interesting, but perhaps too broad/involved for a Q&A here. I have a feeling that for univariable equations, this question is a bit easier, and perhaps even more interesting. Thus, this is my question:
Certain univariable equations do not give a single value for $x$. Take the following two examples:
$$(x^2 = x^2) \implies (x \in \mathcal U) \tag1$$
$$ (x^2 = 5|x| + 3|x|) \implies (x \in \{-8,8\}) \tag2$$
$$(x\times 0 = 5) \implies (x \notin \mathcal U) \tag 3$$
The first equation is a tautology; it offers no information (within any logic that assumes/derives the law of identity), and thus no information about $x$ with which one can pin-point its value. This type of univariable equations is trivially insufficient for giving a single value of $x$. The example I gave is a pure example of it, but any equation in which the variable is subjected to equivalent functions on the LHS and RHS, without the involvement of elucidating non-variables, would constitute an equation of this type, e.g.: $x^2 = (x/2)x + (x/2)x$.
The second equation utilizes functions on both sides of the equation sign that lack the same kind of information; thus, they are not able to make up for each other's deficiency in information.
The last equation contains a contradiction, which makes it undefined, and thus, in a completely different way, deprives $x$ of a single value, although giving it a single "status".
So, my question is this: how do I determine whether a univariable equation (does not) give(s) a single value for $x$ without actually solving for $x$?