Is it possible for a business to be solvent but not profitable? Why or why not?

It could be possible because if you have enough to pay the bills and can keep on running the business. If you are able to pay the bills and such things, it would be an equal sign to both sides (profit and debt). However, sometimes being solvent may be a problem, sometimes there's thing you need to pay a bit more cash than the usual. The cost can go up and you may not have enough money anymore, so it's better to have profit than solvent.