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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.

If a linear cost and revenue function are defined by C(x) = 15x + 100 and R(x) = 22x, how many items must be sold to break even?

Learn how to calculate the break-even point using linear cost and revenue functions. This problem covers key concepts such as linear equations and break-even analysis. Suitable for students in Grades 9-10.

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