How do you solve equations that involve the $\max$ function? For example: $$\max(8-x, 0) + \max(272-x, 0) + \max(-100-x, 0) = 180$$
In this case, I can work out in my head that $x = 92.$ But what is the general procedure to use when the number of $\max$ terms are arbitrary? Thanks for the help, here is a Python solution for the problem if anyone is interested.
def solve_max(y, a): y = sorted(y) for idx, y1 in enumerate(y): y_left = y[idx:] y_sum = sum(y_left) x = (y_sum - a) / len(y_left) if x <= y1: return x print solve_max([8, 272, -100], 180)