Let \[f(x) = \left\lfloor\frac{2 - 3x}{2x + 8}\right\rfloor.\] Evaluate $f(1)+f(2) + f(3) + \dots + f(999)+f(1000).$ (This sum has $1000$ terms, one for the result when we input each integer from $1$ to $1000$ into ${}f$.)

If x is between 1 and 10 then f(x) = -1

If x is between 11 and 1000 then f(x) = -2

Hence the sum is 10*(-1) + 990*(-2) =...