Polynomial equations like x^2-3x = 2 and x^2-3x > 4

It sort of depends on the specifics of what you want.

Julia is not itself a CAS and so you’d need to use a package like Symbolics.jl to write your equations. However, this doesn’t have a general purpose equation solver so you’d have to write your own, which would range from fairly straightforward if you are only dealing with quadratics to theoretically impossible if you want to handle arbitrary polynomials.

An alternative would be to use a specific polynomial package like Polynomials.jl and find roots numerically. This section of the docs lists several alternative packages for root-finding with different advantages depending on what you need.

HTH