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Firstly apologies at the extremely basic level of my questions but I'm a total beginner.

The BTN seemed to me to be somewhat of a weak-tight player but certainly more competant than UTG, who seems to really have no idea at all, drawing rough and patting early, checking behind pat hands with people drawing 2 etc. Obviously from the chat the button wasn't very impressed by my play when I 3-bet their iso and then draw 2. Is it that out of line to do that? I did it because I had a premium draw in a situation they might potentially fold and leave me heads up with the UTG all-in.

Would it be better to just flat and play the hand multiway out of position?

Finally, once the BTN is out and the UTG is all in, given that there aren't any reverse implied odds, does that change the decision about whether to ever keep the nine and just draw one?

Poker Stars $0.10/$0.20 Limit 2-7 Triple Draw Lowball - 5 players - View hand 1692026
DeucesCracked Poker Videos Hand History Converter

Pre Draw: (1.5 SB) Hero is SB with 9 7 3 3 2
UTG calls, CO folds, BTN raises, Hero 3-bets, BB folds, UTG calls all in, BTN folds

First Draw: (6.9 SB) (2 players - 1 is all in)
Hero draws 2, UTG draws 2
Hand: A 7 5 3 2

Second Draw: (3.45 BB) (2 players - 1 is all in)
Hero draws 1, UTG draws 1
Hand: Q 7 5 3 2
BTN says "wot"

Third Draw: (3.45 BB) (2 players - 1 is all in)
Hero draws 1, UTG stands pat
Hand: J 7 5 3 2
BTN says "hmm"

Final Pot: 3.45 BB
Hero shows J 7 5 3 2 (Lo: J,7,5,3,2)
UTG shows 9 7 5 4 3 (Lo: 9,7,5,4,3)
Hero wins 0.1 BB
UTG wins 3.2 BB
(Rake: $0.03)

I got the feeling that everyone else thought I was the fish in the game because as soon as I quit (up a few BBs as usual) the table broke in less than a second.

3betting is just unbalanced, so unless you actually think BTN will fold anything (which is super unlikely and happens <1% IME) its not great. its not terrible either, but most people just flat their entire 2cd range so that people cant figure out when you have a good 2cd and a notsogood 2cd

OK thanks.

Edit: Quick question if you don't mind. What about if I'm on the btn and they are in the CO. Do you 3-bet any 2cd pre draw there?

under some circumstances yes. for example, bad player in CO opens and the 2 blinds are very good tough opponents. there il just 3bet everything as the really good players will find it difficult to call 2 cold with rough 2cds and 3cds.

OK thanks very much.

To answer your other question, I definitely don't like hanging on to the nine.
I'm pretty sure that your range of hands is stronger (i.e. more likely to win at showdown) by drawing two.