In the studio there is one overriding rule of EQ – make it sound good!
Of course there are subtleties, compromises, technical requirements and generally lots to learn, but at the end of the day if it sounds good, then it is good.
Live sound however has certain different requirements. Obviously the end result has to sound good, but there are things to think about.
Firstly and foremostly, there is the requirement for freedom from feedback (the FFF movement!)
There is nothing in live sound that sounds worse than feedback, sometimes called howlround. So it wouldn't matter how good the sound was, due to well-judged use of EQ. If feedback happens more than once in a concert, that is the worst thing that could happen other than the artist falling off the stage.
Why should feedback happen once during the concert? That's a good question, for another day…
Anyway, back to the point. Feedback always occurs at the frequency where the gain in the system, including the reverberation of the auditorium, is highest. This is something therefore that EQ can deal with.
Where there is a peak in the response, use an equalizer – probably a graphic equalizer on the mixing console's main outputs – to bring it down. This will leave a secondary peak, so EQ that away too. Keep on going as far as practical to flatten out the overall response of the system and the room.
But when you have EQ'ed the system for FFF (freedom from feedback, remember), the overall sound might be a little dull. And you will notice this particularly on vocals – feedback often occurs right in the middle of the vocal range, and therefore to correct the feedback you have made changes that will affect the vocal quality significantly.
So there has to be a compromise. Feedback is best handled by keeping microphones as close as possible to the sound sources. EQ is a secondary tool.
Always in PA, there is the consideration of the room. In recording, you have to consider the requirements of the track, and the likely speakers people will use at home. But hardly ever is the listening environment considered (except some radio stations, who fine tune their output for in-car listening).
In PA however, the room is an important part of the overall system and its EQ characteristics will have a significant impact on the sound.