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I'm inquiring about someone's arm and neck so technically plural and I'm siding with are as in, "Are your arm and neck feeling better?" versus is (singular) however would there be a more formal way of inquiring without the question becoming too ambiguous?

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    Either could be used. "Is" implies that "arm and neck" are being treated as a single concern, which might be the case if, eg, the problem involves a tendon connecting the two. "Are" would refer to the more general case were both the arm and neck were of concern. – Hot Licks Jun 27 '18 at 2:21
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From The Chicago Manual of Style

5.131Agreement in person and number A finite verb agrees with its subject in person and number. When a verb has two or more subjects connected by and, it agrees with them jointly and is plural {Socrates and Plato were wise}. When a verb has two or more singular subjects connected by or or nor, it is singular {Jill or Jan is prepared to speak} {neither Bob nor John has learned his lesson}. When the subject is a collective noun conveying the idea of unity or multitude, the verb is singular {the nation is powerful}. When the subject is a collective noun conveying the idea of plurality, the verb is plural {the faculty were divided in their sentiments}.

Here if you mean them separately, i.e., you want to know about each of the arm and the neck individually, use the plural. If you want to know about a single injury to the arm and neck, use the singular.

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How about "How are your arm and your neck? Are they better?"

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    That doesn’t answer the question - it picks ‘are’ but doesn’t explain why. – Lawrence Jun 27 '18 at 3:41

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