〜ませんか
won't you…? would you like to…?
Verb ます-stem + ませんか
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A negative question makes a polite invitation: 一緒に行きませんか, "would you like to come along?" Literally "won't you go together?", it leaves room for the other person to say no, which makes it softer than ましょう.
If they agree, the usual reply is ましょう: いいですね、行きましょう. Declining is often done without saying no at all: 明日はちょっと…, "tomorrow's a bit…".
The tone does the work. 行きませんか as an invitation is friendly and rising; a real question about someone's plans ("aren't you going?") is usually worded differently, 行かないんですか.
Examples
週末、映画を見ませんか。
Would you like to see a film this weekend?
一緒にお茶を飲みませんか。
Would you like to have some tea with me?
今度の日曜日、うちに来ませんか。
Would you like to come over this Sunday?
少し休みませんか。
Shall we take a little break?
明日、テニスをしませんか。
Do you want to play tennis tomorrow?