Baseball fans can buy tickets for seats in the lower deck or upper deck of the stadium. Tickets for the lower deck cost $42 each. Ticket prices for the upper deck are 75% of the cost of tickets for the lower deck.

Which inequality represents all possible combinations of x, the number of tickets for the lower deck, and y, the number of tickets for the upper deck, that someone can buy for no more than $800?
A.42x + 56y ≤ 800
B.42x + 31.5y ≤ 800
C.42x + 56y > 800
D.42x + 31.5y > 800
5b. 
Which of the following can represent a possible combination of the upper deck and lower deck tickets?
A.22 lower deck and 5 upper deck
B.16 lower deck and 6 upper deck
C.8 lower deck and 14 upper deck
D.10 lower deck and 18 upper deck

Let's solve each part of the question.

### Part 1: Finding the Inequality

1. **Ticket Prices**:
   - **Lower Deck**: $42 each
   - **Upper Deck**: 75% of $42, which is $ 0.75 \times 42 = 31.5 $ dollars each.

2. **Cost Constraint**:
   - We want to find an inequality that represents spending no more than $800 on tickets. So, the total cost of $ x $ lower deck tickets and $ y $ upper deck tickets must satisfy:
     \[
     42x + 31.5y \leq 800
     \]

3. **Matching the Inequality with Answer Choices**:
   - The correct inequality is $ 42x + 31.5y \leq 800 $.
   - **Answer**: **B.** $ 42x + 31.5y \leq 800 $.

### Part 2: Checking Possible Combinations

Using the inequality $ 42x + 31.5y \leq 800 $, we substitute each combination of $ x $ (lower deck tickets) and $ y $ (upper deck tickets) to see if the cost is within the $800 budget.

1. **Option A**: 22 lower deck and 5 upper deck
   \[
   42(22) + 31.5(5) = 924 + 157.5 = 1081.5 \quad (\text{exceeds 800})
   \]

2. **Option B**: 16 lower deck and 6 upper deck
   \[
   42(16) + 31.5(6) = 672 + 189 = 861 \quad (\text{exceeds 800})
   \]

3. **Option C**: 8 lower deck and 14 upper deck
   \[
   42(8) + 31.5(14) = 336 + 441 = 777 \quad (\text{within 800})
   \]

4. **Option D**: 10 lower deck and 18 upper deck
   \[
   42(10) + 31.5(18) = 420 + 567 = 987 \quad (\text{exceeds 800})
   \]

**Answer** for possible combinations: **C.** 8 lower deck and 14 upper deck.

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Let me know if you would like further details or have any questions!

### Additional Practice Questions:
1. If the budget was $1,000, which combinations would satisfy the inequality $ 42x + 31.5y \leq 1000 $?
2. Calculate the maximum number of lower deck tickets possible within an $800 budget if no upper deck tickets are bought.
3. How would the inequality change if upper deck tickets were only 50% of the cost of lower deck tickets?
4. If someone buys 5 upper deck tickets, what is the maximum number of lower deck tickets they can buy while staying under $800?
5. Suppose each lower deck ticket also has a $5 service fee; how would this affect the inequality?

### Tip:
When working with inequalities, always check the boundary cases by substituting values that could maximize one variable while minimizing the other. This helps understand the limits fully.

Baseball fans can buy tickets for seats in the lower deck or upper deck of the stadium. Tickets for the lower deck cost $42 each. Ticket prices for the upper deck are 75% of the cost of tickets for the lower deck. Which inequality represents all possible combinations of x, the number of tickets for the lower deck, and y, the number of tickets for the upper deck, that someone can buy for no more than $800? A. 42x + 56y ≤ 800 B. 42x + 31.5y ≤ 800 C. 42x + 56y > 800 D. 42x + 31.5y > 800. Which of the following can represent a possible combination of the upper deck and lower deck tickets? A. 22 lower deck and 5 upper deck B. 16 lower deck and 6 upper deck C. 8 lower deck and 14 upper deck D. 10 lower deck and 18 upper deck

Learn how to solve inequalities involving ticket prices for lower and upper deck seats at a baseball stadium. This problem covers algebraic concepts like linear equations and properties of inequalities, suitable for students in Grades 9-11. Understand how to find combinations of ticket purchases that fit within a budget of $800.

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