Select the correct answer.

Darren is going to the state fair. Each ride costs $6 to ride, and each exhibit costs $3 to view. Darren can spend at most $84 at the fair.

The inequality and graph that represent this situation are shown, with x representing the number of rides Darren can ride and y representing the number of exhibits he can view.

6x + 3y ≤ 84

A linear graph of a solid boundary line intersects X-axis at the unit (15, 0) and Y-axis at the unit (0, 29)

Which solution is valid within the context of the situation?

 A. 
(9,10)
 B. 
(8.5,11)
 C. 
(11,-2)
 D. 
(-3,15)

To determine the correct solution within the context of this situation, let's evaluate each option based on the inequality $6x + 3y \leq 84$ and the context.

**Step 1: Check the inequality condition $6x + 3y \leq 84$:**

1. **Option A: (9,10)**
   \[
   6(9) + 3(10) = 54 + 30 = 84
   \]
   Since $84 \leq 84$, this satisfies the inequality.

2. **Option B: (8.5,11)**
   \[
   6(8.5) + 3(11) = 51 + 33 = 84
   \]
   Since $84 \leq 84$, this also satisfies the inequality.

3. **Option C: (11, -2)**
   \[
   6(11) + 3(-2) = 66 - 6 = 60
   \]
   While this does satisfy the inequality $60 \leq 84$, the context of the situation indicates that $y$ (the number of exhibits) cannot be negative. Therefore, this option is not valid in this context.

4. **Option D: (-3,15)**
   \[
   6(-3) + 3(15) = -18 + 45 = 27
   \]
   Although $27 \leq 84$, the context also requires that both $x$ (the number of rides) and $y$ be non-negative, so this option is also not valid.

**Conclusion:**

The valid options that satisfy both the inequality and the context of the problem are:
- **A. (9,10)**
- **B. (8.5,11)**

Since both A and B are valid solutions, the problem does not specify if we need integer values only. However, if integer solutions are preferred, **Option A** would be more appropriate.

### Final Answer:
**A. (9,10)**

Would you like a further breakdown on any part of this answer, or do you have any questions?

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Here are 5 additional related questions:
1. What are the possible integer solutions for $ x $ and $ y $ that satisfy the inequality $6x + 3y \leq 84$?
2. How would the inequality change if each ride cost $5 and each exhibit cost $4?
3. What is the maximum number of rides Darren can go on if he views 5 exhibits?
4. How would the graph of this inequality change if Darren’s budget increased to $100?
5. If Darren wanted to spend exactly $84, what are the possible combinations of rides and exhibits he could choose?

**Tip:** When working with inequalities, always consider the context to ensure solutions make practical sense (e.g., non-negative values in real-world contexts).

Darren is going to the state fair. Each ride costs $6 to ride, and each exhibit costs $3 to view. Darren can spend at most $84 at the fair. The inequality and graph that represent this situation are shown, with x representing the number of rides Darren can ride and y representing the number of exhibits he can view. 6x + 3y ≤ 84 A linear graph of a solid boundary line intersects X-axis at the unit (15, 0) and Y-axis at the unit (0, 29). Which solution is valid within the context of the situation? A. (9,10) B. (8.5,11) C. (11,-2) D. (-3,15)

Explore a linear inequality problem involving budget constraints for rides and exhibits at a fair. Learn how to find valid solutions within the context, graph inequalities, and apply non-negative constraints. Ideal for students in Grades 8-10.

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