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Distilling down some of the good information posted: The loudest source at the mic wins. If the loudest sound at the mic is a copy of itself from the PA system, you will have feedback. You will likely have some ability to affect this with EQ, but the most complete solution is to either decrease the distance from the mic to the source, and/or increase the distance between the mic and the speakers.

General things:
– make sure you’re not sending these mics to any monitors.
– Lapels suck, and are borderline useless for live sound reinforcement. Move to headset mics to get the element closer to the mouth.
– Train your speaker to project. Many people are afraid of hearing themselves in the system, and as you turn them up, they speak even more softly. Reassure them that they need to speak loudly and clearly, and that you will control the volume.