How can you determine the number of solutions for an equation?

In general, you cannot: it all depends on the domain.

x + 2 = 0 has no solutions is the set of positive integers but does have one if the domain is the integers.

2x - 3 = 0 has no solutions if the domain is integers, but there is one solution if the domain is the rationals.

x2 - 2 = 0 has no solutions if the domain is the rationals but there are two solutions if the domain is the reals.

x2 + 2 = 0 has no solutions if the domain is the reals but there are two solutions if the domain is the complex numbers.

Cos(x) = 1 has no solutions if the domain is (0, 360) but two solutions for the domain [0, 360].