How do you recognize when an equation has no real solution or an infinite number of solutions?

It depends on the equation. Also, the domain must be such that is supports an infinite number of solutions.

A quadratic equation, for example, has no real solution if its discriminant is negative. It cannot have an infinite number of solutions.

Many trigonometric equations are periodic and consequently have an infinite number of solutions - provided the domain is also infinite.

A function defined as follows:

f(x) = 1 if x is real

f(x) = 0 if x is not real

has no real solutions but an infinite number of solutions in complex numbers.