Macro with Solver loop

Without more information, I will probably have to take your word for it that this cannot be solved algebraically. If AI2 is independent of AJ2, though, it really looks to me like the algebra should work out for a closed form solution of the equation/inequality. I think if I were solving this problem, I would pursue solving the above equation/inequality for AJ2. From that, I would learn how this function behaves and have a better idea how to solve it. If it ends up requiring a numeric Solver solution, I would expect to end up with an expression for AJ2 as a function of AP2. Then call Solver and tell it to set AJ2 to a minimum by changing AP2 subject to the constraint that AP2>=5 and any other constraints needed when the problem is rearranged in that way.