https://file.aiquickdraw.com/mathbot/file/17254360702183zwjobkx.jpg

Let's analyze the problem based on the image.

### Given Information:
- **Room A**:
  - Cleaning fee: $50
  - Rate: $24 per hour

- **Room B**:
  - Cleaning fee: $80
  - Rate: $18 per hour

- **Total budget**: $250

Let $ t $ represent the time (in hours) that the band rents the room.

### Task A: Inequality for Room A
The total cost for Room A is given by:
\[
\text{Total Cost for Room A} = 50 + 24t
\]
Since the band has $250 to spend, the inequality for Room A is:
\[
50 + 24t \leq 250
\]

### Task B: Inequality for Room B
The total cost for Room B is given by:
\[
\text{Total Cost for Room B} = 80 + 18t
\]
Similarly, the inequality for Room B is:
\[
80 + 18t \leq 250
\]

### Task C: Determine Which Room the Band Should Rent
Now, let's solve both inequalities to find the maximum time $ t $ that the band can rent each room.

#### Solving for Room A:
\[
50 + 24t \leq 250
\]
Subtract 50 from both sides:
\[
24t \leq 200
\]
Divide both sides by 24:
\[
t \leq \frac{200}{24} \approx 8.33 \text{ hours}
\]

#### Solving for Room B:
\[
80 + 18t \leq 250
\]
Subtract 80 from both sides:
\[
18t \leq 170
\]
Divide both sides by 18:
\[
t \leq \frac{170}{18} \approx 9.44 \text{ hours}
\]

### Conclusion:
- **Room A**: The band can rent it for up to 8.33 hours.
- **Room B**: The band can rent it for up to 9.44 hours.

Given that Room B allows for more rental time within the budget, **the band should choose Room B**.

---

Would you like any more details or have any questions?

Here are five related questions to consider:
1. How would the inequalities change if the budget increased to $300?
2. If the rates for Room A and Room B were swapped, which room would be more cost-effective?
3. What would be the maximum rental time if the band only had $200?
4. How would the solution change if Room A's cleaning fee was reduced to $40?
5. Can you derive a formula for the maximum rental time for any room given a budget, rate, and cleaning fee?

**Tip:** When solving inequalities, remember to check if the result is logical in the context of the problem (e.g., time cannot be negative).

The band has $250 to spend. Let t be the time (in hours) the band rents the room. A. Write and solve an inequality to represent the time (in hours) the band could rent Room A. B. Write and solve an inequality to represent the time (in hours) the band could rent Room B. C. Which room should the band rent? Explain your answer.

Solve the band budget problem by writing and solving linear inequalities for Room A and Room B. Learn how to compare rental options using algebraic methods to determine the best choice within a $250 budget. Ideal for students in Grades 6-8.

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