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To solve this problem, we need to compare the costs of two different payment options for the team.

### Option 1: Pay $12 per person
The cost for the first option is $ 12n $, where $ n $ represents the number of people on the team.

### Option 2: $40 fixed fee plus $8 per person
The cost for the second option is $ 40 + 8n $, which includes a fixed fee of $40 and $8 per person.

### Problem:
We need to determine when the first option is cheaper than the second option, which can be expressed as the inequality:
\[
12n < 40 + 8n
\]

### Solving the inequality:
1. Start with the inequality:
   \[
   12n < 40 + 8n
   \]
2. Subtract $ 8n $ from both sides:
   \[
   12n - 8n < 40
   \]
   \[
   4n < 40
   \]
3. Divide both sides by 4:
   \[
   n < 10
   \]

### Conclusion:
The inequality that determines when the first option is cheaper than the second option is $ n < 10 $. This corresponds to option **D** from the answer choices.

### Would you like further explanation or details on any part of the solution?

### Five related questions to expand your understanding:
1. How would the solution change if the fixed fee in the second option increased to $50?
2. What happens to the inequality if the per-person cost in option 1 is reduced to $10?
3. Can we determine a situation where both options cost the same for a specific number of people?
4. How would this problem change if there was a discount for larger groups in the second option?
5. What if there was an additional fee added in option 1 for groups larger than 10 people?

### Tip: When comparing two pricing options, always set up an inequality to find the range of values where one option is cheaper than the other.

A cheerleading team wants to attend a competition. They have two options for paying the entrance fee. The first option is for each person on the team to pay a $12 fee. The second option is to pay a $40 fixed fee plus $8 per person. If the team has n number of people, determine when the first option is the cheaper one using an inequality.

In this math problem, learn how to compare two pricing options using inequalities. One option involves a fixed fee plus a per-person cost, and the other is based on a single per-person fee. Follow the step-by-step solution, suitable for students in Grades 7-9.

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