If you divide an inequality by a negative number you have to flip the sign.

If you want to divide out the g before knowing if it’s positive or negative you need to make the case distinction.

If you make the case distinction and one branch results in a false statement (5 < 2 in this case), you have disproven the original inequality, or at least discovered a constraint on the original inequality. Then you arrive back at FooBarrington’s statement :-)