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.
