HERO button [100 bb] AK
SB [50bb]
BB [75 BB]
LJ [100 BB]
CO [100 BB]
HJ [30BB]

Preflop:
LJ 3x, FOLD, FOLD, HERO 9x, FOLD, FOLD, CALL

Flop (18.5 BB)
J7Q

Checks to HERO and I check back. Without any backdoor flush draws, how should I be playing this flop? I find myself in this position a lot and am not sure what to do. I am confused b/c if I had a good hand on this board, I would bet large b/c it is draw heavy. However, I am reluctant to bet large on this flop with just a gut show and no backdoor flush draws. Also, considering we 3-bet and villain called, he has a lot more Q's and J's in his range than us.

Can someone please enlighten me on how I should be playing this flop? Thanks in advance.

It's a tricky board...you can bet it and plan when bet is to go all the way on brick run outs, you block AQs KQs KJs AJs and ublcoks flush draws and BDFD so you will have some extra fold equity in dry turns and rivers. Ofc if you play against calling station don't bluff.
Checking is fine too, problem is it's s hart to play vs turn probe unless we hit.

thanks for the reply, you triple barrel this more than checking and playing passively?

Depends on opponent and i wouldn't barrel every run out.

Can you explicitly show some run outs where you would barrel off turns and rivers? As well as run outs where you would avoid barreling off? While I have a good idea, I want to make sure I make the right decisions, as my default is to usually play passive and want to begin to try your barreling strategy.

Many thanks in advance.

Two cards lower then 7 without flush for example would be good to barrel.
If 8c is turn probably give up.

if you bet, you're going small
there are merits to both betting and checking, depends on how you want to construct your range. solver will mix for various reasons, mostly board coverage.
more importantly, you have to be aware of exactly what your range is going to look like on future streets so think about what you want to do in advance otf

if you want to bet range small, that's likely fine as well because your opponents will tend to not xr enough. either way, the prime candidates to x are going to be close underpairs to the J w/o a club and weaker tp to put some bluff catchers in your x/ range, as well as top 2, which block too much of his continuing range

Betting and checking are both fine and each one has equal pros and cons. Otf we have range advantage but nut combos are equivalent so dont bet too large.

Postflop we can barrel with impunity since we block villains top pair combos and unlock flushdraws.

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bet 1/3