Would $x = x+1$ have infinite solutions? If $x = x+1$ and we have the equation $x = x+1$ but since $x = x+1$ would it equal $x+1 = x+1$?
It is a dumb question but someone told me that and I want to make sure it's not false.

You are starting from an inconsistent (or false) proposition, sating that:

if $x=x+1$

As a result, any following steps could lead to false result(s) or inconsistent result(s).
To answer your question, $x=x+1$ has no solution, that is no such $x$ may exist.
Start with a consistent assumption, proceed according to the rules and you will get a consistent result. This is the promise of Mathematics (so, I say :))
