The problem asks to find how many items must be sold to break even given the cost function $C(x) = 15x + 100$ and the revenue function $R(x) = 22x$.

To break even, the cost must equal the revenue. Therefore, we set $C(x) = R(x)$:

$15x + 100 = 22x$

Now, solve for $x$:

$22x - 15x = 100$

$7x = 100$

$x = \frac{100}{7} \approx 14.29$

Since the question asks for the number of items sold (which must be a whole number), round up to the nearest whole number:

$x = 15$

Therefore, 15 items must be sold to break even.

Would you like more details on any step?

Here are 5 related questions you might find helpful:

1. How do we define the break-even point in business?
2. What does the slope of the cost function represent in this scenario?
3. How does the fixed cost of 100 affect the break-even point?
4. What would happen if the price per item (in revenue) changes?
5. How would we calculate profit if more than 15 items are sold?

Tip: Always check whether rounding is necessary when working with quantities of items, as fractional items usually don't make sense in practical terms.