ませんかwould you like to…?
ませんか is how you invite someone: "would you like to…?". It looks negative, but nothing negative is meant: asking this way leaves the other person room to say no.
Would you like to take a walk together during lunch break?
It's hot, isn't it? Why don't we take a short break?
How to form it
Verb ます-form: ます → ませんか
行きます → 行きませんか. With friends: 行かない？
Inviting someone
It is polite without being stiff, so it suits classmates, colleagues and people you have just met. People often check first that the other person is free.
Do you have time after the meeting?
Then, would you like to have lunch nearby?
Usually it is something to do together, and 一緒に often says so. It can also be something only the other person would do: come over, try something on.
Would you like to walk to the station together after this?
If you like, would you come over to my place sometime?
よかったら, "if you like", makes it gentler still.
in a shop
Would you like to try this hat on?
With friends
Among friends the same invitation is the plain negative, with the voice going up at the end: 行かない？, 見ない？.
Do you want to watch a movie together tonight?
ませんか or ましょう?
Would you like to go home together?
You are asking. They may not want to.
Let's go home together.
You both already expect to.
Saying yes and no
To say yes: いいですね, then ましょう when it is something you will do together. When only you would be doing it, the answer is thanks: ありがとうございます. To say no, you trail off after ちょっと; nobody expects the reason.
Would you like to go to the new restaurant this weekend?
That sounds good. Let's go.
Sounds odd: いいえ、
No, I'm not going.
Correct Japanese, but as an answer to an invitation it is a cold refusal.
Not たいですか
English invites with "Do you want to…?". In Japanese たいですか only asks about a wish, and it is too blunt for someone you speak politely to. The invitation is ませんか.
Sounds odd: 一緒に
Do you want to drink coffee together?
Would you like to have coffee together?
When it isn't an invitation
With ある, nobody is being invited to do anything. ありませんか is a soft way to ask for something: "do you have one?"
in a shop
Excuse me, do you have a slightly bigger one?
Asked about someone else, it is a plain question.
Won't Tanaka be coming today?
Close neighbours
ましょう is "let's", when doing it together is already settled or obvious. With か it asks: "shall we?".
Shall we be heading home?
ましょうかN5
Also an offer, "shall I?": you would do it for them, not with them.
Shall I carry your luggage?
After てもらえ or ていただけ, the same ending asks a favor: "could you…?".
Excuse me, could you take a photo?
In an invitation, でも after the thing is "or something". It keeps the invitation light.
Would you like to have some tea or something?
Check yourself
1What does this mean?
土曜日、
Choose an answer to see why.
2A classmate says this, but you can't go. What do you say?
明日、
Choose an answer to see why.
3You want to invite a colleague to a movie. Which do you say?
Choose an answer to see why.
More detail
With someone above you
ませんか is polite enough for a teacher or a boss. People often raise the verb as well: いらっしゃいませんか in place of 来ませんか or 行きませんか.
Would you like to come with us, Professor?
More sentences
Would you like to study at the library after school tomorrow?
Why don't we go to the venue a little early?
Why don't we wait for the next train?
Would you like to talk online for a little while tonight?