～ましょうか　vs　～ませんか


I have been working on these types of sentences, but I am having trouble
figuring out the difference between:

In this context it is an invitation to go eat.

> [食]{た}べましょうか。

and

> [食]{た}べませんか。

What is the difference?




One key difference is the context in which the suggestion is made, and the
assumption(s) the asker has about the listener's response. Take the following
situation where a man is asking a woman on a date _(note that I'll use the
plain forms ～ない and ～よう for such a familiar situation)_ :

> お茶にでも[行かない]{LLLH}？ → "Won't we go get tea or something?" → "Would you like
> to go have tea (or whatever) with me?"

From this, the most likely context is that the man isn't certain of the
woman's response, and thus, he's probably asking her out for the first time.
Now compare that to:

> お茶にでも行こうか？ → "Should/Shall we go get tea or something?"

In this case, we can probably infer either, 1) He has already asked her out at
some time prior, she agreed, and they are just trying to solidify the plan, or
2) They're already a couple, and he is making a suggestion of something to do
together.

* * *

So I think whether the topic of the suggestion has been previously discussed
or not makes a big difference of which one you should use. If it's a "brand
new" topic, you'd use ～ませんか; if the topic has already been approached and
you're suggesting one possible option, then you use ～ましょうか.

So in your case, 食べませんか tells me that the suggestion is "brand new", out-of-
the-blue to the listener, while 食べましょうか tells me that they're already planning
to do _something_ , so "How about let's go eat?"

