One aspect of Korean that seems to cause issues for beginners is how to correctly say they will not/cannot do something. (Personally, I feel I use them correctly, but this is a really common issue I came across when teaching beginners Korean, hence my asking the question for the general public).
What are the proper usages for the endings -지 않다 (지 아니하다), -지 못하다, and -ㄹ/을 수 없다?
For example, if I want to tell a coworker that I have not eaten yet because I was too busy, should I say "너무 바빠서 아직 먹지 못 했어요" or should I say "너무 바빠서 아직 먹지 않았어요"?
(Alternatively in the short form, the sentences above can be "너무 바빠서 아직 못 먹었어요" and "너무 바빠서 아직 안 먹었어요")
How does the -ㄹ/을 수 없다 ending play into this conversation of negating a verb, and when is it appropriate?