Purchasing triplex- what do do about one potential problem tenant

[Nicholas W.]

I am a somewhat new investor, see my intro post: https://www.biggerpockets.com/forums/55/topics/440...

I am in the process of purchasing my first multi-family building. It is a triplex in a lower-income area (but not a slum). The building is currently rented at $675, $750, and $800. All of the units are substantially the same, the $800 unit is slightly nicer.

The tenant who is renting at $675 has been in for a while (2014) and is on a month-to-month basis. 

I did the property inspection yesterday and two of the units were relatively clean and well kept. However, the $675 tenant's unit was extremely dirty, unkempt, had a broken range, had trash inside and outside, and smelled like pot. I also think $675 is below market in that area.

The current owner said that she's been a good tenant and always paid on time. However, I don't want to be a slumlord and have tenants like her. I'm considering raising her rent when I close on the building to $750 in order to bring her up to the rest other tenants and help cashflow. I know that it may drive her out of the building which would cost me time that the unit isn't rented, cost me listing fees, cleaning fees, repair time and expense, etc.

Thoughts or advice? Is this a good strategy long term? Should I maybe be more incremental with rent increases? Like raise it to $700 first? Or is it typically better to just let sleeping dogs lie?