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1. Frank doesn't care about his future. And Lilly doesn't care about her future.

How to join up these two by using neither ... nor...?

2. My brothers are going to visit next weekend. Or Jane is going to visit next weekend.

How to join up these sentences by using either... or...?
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In both cases, all you have to do is combine two subjects with the necessary conjunctions. The combined subjects then share the rest of the sentence. Try it and we'll check your work.

Emotion: smile

CJ
1. Neither Frank nor Lilly cares about her / his future.

2. Either my brothers or Jane is going to visit next weekend.

Right? Or maybe:

1. Neither Frank nor Lilly care about their future.

2. Either my brothers or Jane are going to visit next weekend?

The rule says: the verb conjugates in number with the nearest subject. But what about real usage? As far as I can see from literature (publicistic style) the verb mainly takes the plural form in these constructions. Why? What do you think?

Thanks,[:^)]
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1. Neither Frank nor Lilly cares about their future. (the verb is sigular if both subjects are singular)

From that sentence alone there's no way of telling whether they care about their future together or Frank cares about his future and Lily cares about her future. The context in which that sentence is spoken will usually clarify that.

2. Either my brothers or Jane is/are going to visit me/us next weekend.

You will find both 'is and are' in sentences like the above. Whichever verb you choose they are still fundamentally inconsistent. It's best to rewrite sentences like that as

My brothers or Jane are going to visit me/us next weekend.
RusalkaThe rule says: the verb conjugates in number with the nearest subject. But what about real usage? As far as I can see ... the verb mainly takes the plural form ... What do you think?
You are absolutely correct. The rule should certainly be used in any formal situation (including class work), (and in fact it can be used in any situation), but in less formal situations like everyday conversation, people tend to use only the plural, especially with neither ... nor.

CJ
"...but in less formal situations like everyday conversation, people tend to use only the plural, especially with neither ... nor" . I just try to andestand why. What this tendency is determined by? Do people think it's a notional agreement, and if there are two subjects so the verb should take a plural form? Or anything else? On analogy with not only ... but also conjuction? Can this syntactic variation be determined by ethnic and cultural peculiarities of English language usage? By anithing that I as a Russian language speaker do not "feel"?
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RusalkaWhat this tendency is determined by?
This is in the realm of speculation, but I suspect that your idea about two subjects seeming to need (psychologically) a plural verb is probably the best explanation for this tendency.
RusalkaOn analogy with not only ... but also conjuction? Can this syntactic variation be determined by ethnic and cultural peculiarities of English language usage? By anithing that I as a Russian language speaker do not "feel"?
Here I would answer your three questions "no", "no", and "no". Emotion: smile

CJ