Many problems provably don't have a solution. OP didn't specify a problem domain but in both maths and computing the possibility that what you are trying to achieve is mathematically impossible is a real one, and distinguishing the absence of a solution from your own lack of ability to find the solution that does exist is a real challenge with no easy answer.
Real-life problems always have a solution, but sometimes it isn't a technical one. Or a technical solutions exist, but is too expensive. To take your math example, e(x) = 0 have no solution, but in real life, taking x = -100 could be close enough for your purpose.
In real life problems often have multiple solutions and we don't like some of them, possibly the simplest ones. This is much like exceedingly expensive solutions to technical problems. In my experience customers faced with a costly solution reframe their problem and accept to do their business a different way. Their customers won't notice.
