How can you tell when an equation in one variable has infinitely many solutions or no solutions?

Answer 1

There is no simple method. The answer depends partly on the variable's domain. For example, 2x = 3 has no solution is x must be an integer, or y^2 = -9 has no solution if y must be a real number but if it can be a complex number, it has 2 solutions.

Answer 2

You can actually solve the equation. With some experience with equations, you will eventually know that equations of a certain form will have exactly one solution (for example, a linear equation); while a polynomial equation of degree 2 typically has 2 solutions, and certain types of trigonometric equations have infinitely many solutions (this is due to the periodicity of the trigonometric functions) - but there is really no simple or universal way to tell in advance how many solutions a given equation has.