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.

For an equation to have a solution, you need to be able to substitute all variables in the equation with your solution and obtain a true equality. For example, $x + 3 = 5$ has the one solution of $x = 2$, since $2 + 3 = 5$.
For another example, $x + x = 2x$ has the solutions $x = 5$ and $x = 6$, since $5 + 5 = 10 = 2(5)$ and $6 + 6 = 12 = 2(6)$. In fact, this equation has infinitely many solutions, since this equation is an identity.
In your case, $x = x + 1$ does not have any solutions. For example, $x = 4$ is not a solution, because $4 = 4 + 1$ implies that $4 = 5$, which is not true.
