Attaching ませんか (or the more informal ないか) to a verb creates a friendly invitation. This construction is similar to asking 'Would you like to do (A)?' or 'How about we do (A)?'.

There is a subtle difference in nuance when compared to ましょうか. ませんか functions as a direct invitation to the listener, while ましょうか is closer to a suggestion that an action would be a good idea for the group.

• 今度(こんど)の日曜日(にちようび)にテニスをしませんか。
  Would you like to play tennis next Sunday? (Polite)
• この後(あと)、カラオケに行(い)かないか。
  How about we go to karaoke after this? (Casual)

Using ませんか is generally the safer option. ないか is very informal and should only be used with close friends, as it can sound overly blunt in other situations.

To emphasize that the speaker is inviting someone to do something 'together', the word 一緒(いっしょ)に is often included in the sentence for clarity.

• 一緒(いっしょ)に晩(ばん)ご飯(はん)を作(つく)りませんか。
  How about we make dinner together?

Let's observe the subtle distinction between ましょうか and ませんか by seeing how they function in the same context.

• ちょっと休(やす)みましょうか。
  Shall we take a little break?
• ちょっと休(やす)みませんか。
  Would you like to take a little break?

In this case, ませんか feels more like a direct invitation because its negative question form gently asks for the listener's preference. This structure makes it easy for them to decline, which is why it's often perceived as more considerate and polite than ましょうか.