たらどう・
A suggestion put as a question: "what if you did this?" With friends it shrinks to たら？.
polite
You seem a little tired. Why don't you take a break here?
to a friend
Why don't you talk to the teacher about that problem?
shortest
Does it still hurt? Why don't you go to the doctor?
How to form it
Verb in the た-form + ら + どうですか
行った → 行ったらどうですか. Word for word: "if you went, how would that be?"
To a friend
…たらどう？ …たら？
行ったらどう？, or just 行ったら？ with the voice going up.
More polite
…たらいかがですか
いかが is the polite どう.
Putting an idea forward
Someone is tired, stuck or worried, and you have an idea for them. A reason, before or after, makes it sound kind.
I haven't been sleeping well lately.
Why don't you try not looking at your phone before bed? It might make it a little easier to sleep.
If you can't decide, why don't you sleep on it?
With friends: たら？
In casual talk どう often drops. てみたら？, "why not try…?", is softer still.
If you're sleepy, why don't you go to bed?
Why don't you try asking the station staff?
Why don't you quit smoking already?
To suggest not doing something, people pick a verb for stopping or resting: やめたら, 休んだら.
How it sounds
You don't look well. You should go home early today.
Going home is the better choice, and staying is the worse one.
You don't look well. Why don't you go home early today?
Just an idea. Neither choice is called worse.
It is still advice. Said unasked to a teacher or a boss, even the polite たらいかがですか can sound forward.
When it is a complaint
If the idea is something the other person plainly ought to be doing already, the same words mean "why don't you just…?". You hear it in the voice, and often in a そんなに…なら or 少しは at the front.
If you have so many complaints, why don't you do it yourself?
Why don't you at least help a little?
Advice, not an invitation
English "why don't we…?" and "how about…?" also invite. たらどう doesn't: it is advice about what the other person does. An invitation is ませんか. With a noun, "how about…?" is コーヒーはどうですか.
Why don't you two go together?
Advice to two other people. You aren't going.
Why don't we go together?
An invitation. You are going too.
Close neighbours
たほうがいいN5
Advice that says which choice is the better one.
If you have a fever, you should go to bed early today.
Asks for advice. たらどうですか gives it.
My computer suddenly stopped working. What should I do?
たらいい・ばいいN3
The same idea as a statement: "you could…", "you can just…".
If you're unsure, you could ask the teacher.
たらN4
The たら form and its main job: "if", "when".
If it rains tomorrow, the game will probably be canceled.
Check yourself
1A friend says her head hurts, and you say this. What are you doing?
薬を
Choose an answer to see why.
2You'd like a colleague to see a film with you. Which one invites her?
Choose an answer to see why.
More detail
たらどうか
With plain か it mostly sits inside a longer sentence. Said straight to someone, it is stiff, the way a boss might talk.
The doctor suggested I get a little exercise.
I think it might be worth talking to the teacher.
More sentences
Why don't you have a doctor look at it?
If you're tired, why don't you go home early today?
If you have too much work, why don't you talk to your supervisor?
If you say it's that easy, why don't you explain it?