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The problem involves a fast-food restaurant's production costs and revenue. We're given:

- The cost function \( C(x) = 6x + 120 \), where \( x \) represents the number of units produced.
- The revenue function \( R(x) = 13x \).

The company starts to turn a profit when its revenue exceeds its cost. This occurs when:

\[
R(x) > C(x)
\]

Substituting the given functions:

\[
13x > 6x + 120
\]

Now, let's solve for \( x \):

\[
13x - 6x > 120
\]
\[
7x > 120
\]
\[
x > \frac{120}{7}
\]
\[
x > 17.14
\]

Since the company starts turning a profit after producing a whole number of units, we round \( x \) up to the nearest whole number, which is **18**.

Thus, the company starts to turn a profit when the production level reaches at least **18 units**.

Would you like more details, or have any questions about this solution?

### Relative Questions:
1. How do fixed costs affect a company's break-even point?
2. What would happen if the cost function had a different slope?
3. How is profit maximization different from the break-even analysis?
4. How do changes in revenue affect the production threshold for profit?
5. What if the cost function were non-linear—how would the calculation change?

### Tip:
Always check if the profit function involves rounding or additional conditions, as small details can affect the final solution.

A fast-food restaurant has a cost of production C(x) = 6x + 120 and a revenue function R(x) = 13x. When does the company start to turn a profit?

Learn how to calculate when a company starts to turn a profit using cost and revenue functions. This problem involves solving a linear inequality and rounding to find the production level where revenue exceeds cost. Suitable for Grades 9-12.

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