Solve equations using the $\max$ function 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)


Check each of the possible cases. In your equations the "critical" points (i. e. the points where one of the max's switches) are $8$, $272$ and $-100$. For $x \le -100$ your equation reads 
\[ 8-x + 272 - x + (-100-x) = 180 \iff 180 - 3x = 180 \]
which doesn't have a solution in $(-\infty, -100]$. 
For $-100 \le x \le 8$, we have 
\[ 8-x + 272 - x = 180 \iff 280 - 2x = 180 \]
and the only solution $50\not\in [-100, 8]$.
For $8 \le x \le 272$ we have
\[ 272-x = 180 \iff x = 92 \]
so here we have a solution. 
And finally for $x \ge 272$ the equation reads 
\[ 0 = 180 \]
so no more solutions.
