To determine how many items must be produced to keep the average cost below a specific value, we can model the scenario with a rational inequality. For instance, if the total cost function is $C(x) = 10x + 3000$, the average cost function is found by dividing by $x$, giving $c(x) = \frac{10x + 3000}{x}$. To find when the average cost is less than $40$ dollars, set up the inequality $\frac{10x + 3000}{x} < 40$, where $x \neq 0$. First, subtract $40$ to get $0$ on the right: $\frac{10x + 3000}{x} - 40 < 0$. Next, rewrite the left side as a single quotient by finding the common denominator: $\frac{10x + 3000}{x} - \frac{40x}{x} < 0$, which simplifies to $\frac{-30x + 3000}{x} < 0$. Factoring the numerator gives $\frac{-30(x - 100)}{x} < 0$. The zero partition numbers are found by setting the numerator and denominator to zero: $-30(x - 100) = 0 \implies x = 100$ and $x = 0$. These partition numbers divide the number line, and testing intervals reveals the inequality is satisfied when $x > 100$. Thus, more than $100$ items must be produced to keep the average cost below $40$ dollars per item.