1. I'll probably go and check out that apple, actually.

2. I'll probably go and check that apple, actually.

3. I'll probably go and check on that apple, actually.
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Which one is correct? Do we have to use 'out' after 'check' here?

4. The waitress who served in/at the restaurant was very unkind.
(Do we have to use 'in' or 'at'? Are both OK? Then, which one is commonly used?)

1, 2, and 3 -- I'd omit "and" and "actually."

Otherwise, all are fine.

4 -- I'd omit "who served in/at" because the clause is redundant -- if she a waitress in that restaurant, then serving is what she does. You might want to write, "The tall waitress in [or "at"] Mario's was very unkind." The preposition used could be either one.

1. The sentence "I'll probably go and check out that apple, actually" is correct. In this context, "check out" means to examine or inspect something closely. It implies that the speaker wants to take a closer look at the apple.

2. The sentence "I'll probably go and check that apple, actually" is also grammatically correct. However, it has a slightly different meaning. Here, "check" alone means to verify or confirm something. In this case, the speaker intends to verify or confirm something about the apple without necessarily inspecting it closely.

3. The sentence "I'll probably go and check on that apple, actually" is also correct. "Check on" implies that the speaker wants to ensure the well-being or status of the apple and make sure it is okay.

Regarding the use of "out" after "check," it is not mandatory, but it depends on the specific context and intended meaning of the sentence. Using "out" adds emphasis to the action of examining or inspecting something closely.

4. The sentence "The waitress who served in/at the restaurant was very unkind" is grammatically correct. Both "in" and "at" can be used, but they slightly change the meaning.

Using "in" implies that the waitress worked within the physical space of the restaurant, serving customers as part of her job. It suggests that the unkind behavior occurred while she was actually inside the restaurant.

Using "at" implies a broader sense of location and does not necessarily mean she was physically inside the restaurant while serving. It could mean she worked for or was associated with the restaurant but might not have been in the physical space during the unkind behavior.

Both "in" and "at" are commonly used, but "in" is more commonly used when referring to someone working as a waitress within the restaurant.