Is the sentence below natural to native speaker? I am confused because it is somewhat strange as 잘 and 안 are used together in the same sentence.
이 문은 잘 안 닫혀요. Translation:This door does not close well.
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Yes. The expression of this type is commonly used.
나는 물건을 잘 잃어 버린다. I lost my things well(=frequently)
That is, 잘 decorates the verb. Here 잘 gives an extended meaning of the verb or comparative meaning.
1) 이 문은 잘 닫혀요. This door is locked well (Performance of this door is better that that of other kinds of doors)
2) 이 문은 잘 안 닫히네요. This door is not locked well (Even though this door is locked, it has low capacity when we compare it to other standard.)
3) When we check all doors in the building, we can use 이 문은 잘 닫히네요. 저 문은 잘 안 닫혀요.
(1) 저는 공부를 했어요.
저는 공부를 안 했어요.
저는 공부를 빨리 안 했어요 (A). I did not study quickly.
This has the same form by comparing it to the sentence 이 문은 잘 안 닫혀요.
In further, two sentences 저는 빨리 공부를 안 했어요 (B). 이 문은 잘 닫히지 않아요 share same form.
In the link, (A) and (B) are "a little awkward in Korean", because there is a more natural alternative 저는 공부를 천천히 했어요.
(2) As far as I know, sentence containing 잘 may have alternative but we may use the both.
example : 칼이 잘 안들어요 (alternative : 칼이 무뎌요). This knife is not sharp well.
(3) Even though the form 빨리 + 안 is not used with 공부하다, it can be used :
빨리 안 갈래 ? Go fast.
일이 빨리 안 되고 있다. The work is done slowly.