"Lösungen sind möglich."
Maybe I just don't get the joke, but even if "job terminations are possible" may sound as a good solution in the ears of an unsympathetic manager, it does not count as a translation of "Lösungen sind möglich.".
True, you can "einen Arbeitsvertrag auflösen" meaning "to terminate an employment contract", but still that wouldn't be called a "Lösung", except in the sense mentioned above.
As far as I know there are three different meanings of "lösen"
to resolve - "Lösung" would be "answer" or "solution", Auflösung would mean the answer of a riddle.
to free, to release - as in "Bremse lösen", " to release the break", "eine Verlobung (auf)lösen", "to terminate an engagement", "Lösung" would rarely be used in the latter context but rather "Auflösung". A contract could be "gelöst" but "aufgelöst" is less ambiguous, and you would always use "Auflösung" for a noun instead of "Lösung", not to confuse it with the first meaning.
to dissolve - "Lösung" would mean the (chemical) solution, the stuff you could pour from the test tube, "Auflösung" would be the act to dissolve something.
Conclusion: "Lösungen sind möglich." would commonly be understood as the 1st meaning.
If there is a chemical context, I could also imagine the 3rd. If there was a legal context, the 2nd meaning would be possible, but it seems to me plausible only in the singular: "Eine Lösung (des Vertrags) ist möglich". Still you would probably say "Auflösung" then.
I agree! (Though as a native English speaker, I have to defer final judgment.) The broader point is, if a sentence generates confusion/discussion/frustration around "what the heck is this supposed to mean," if it gives learners the impression that secondary meanings are primary (e.g. "termination" vs. "solution" here), if it teaches arbitrary language/meaning/constructions before common/everyday ones, then it's, pedagogically, a poor choice, and should be replaced with something clearer. If the example/learning sentence has a primary meaning that most people can agree is clear, then we can focus on "learning how to say stuff", and not waste energy on "what is it we're saying here, exactly?" This is where, in my opinion, many example sentences in German are "poorly engineered", especially as the level of complexity increases.