Joe is responsible for reserving hotel rooms for a company trip. His company changes plans and increases how many people are going on the trip, so they need at least \[50\] total rooms.

Joe had already reserved and paid for \[16\] rooms, so he needs to reserve additional rooms. He can only reserve rooms in blocks, and each block contains \[8\] rooms and costs \[\$900\].

Let \[B\] represent the number of additional blocks that Joe reserves.

**1) Which inequality describes this scenario?**

Choose 1 answer:

Choose 1 answer:

(Choice A)

\[16+8B \leq 50\]

A

\[16+8B \leq 50\]

(Choice B)

\[16+8B \geq 50\]

B

\[16+8B \geq 50\]

(Choice C)

\[16+B \leq 50\]

C

\[16+B \leq 50\]

(Choice D)

\[16+B \geq 50\]

D

\[16+B \geq 50\]

**2) What is the least amount of additional money Joe can spend to get the rooms they need?**

dollars

### 1) Which inequality describes this scenario?

Joe already has 16 rooms reserved and needs at least 50 rooms in total. Each additional block of rooms contains 8 rooms, and he will reserve $ B $ blocks. The total number of rooms he will have after reserving additional blocks is:

\[
16 + 8B
\]

He needs this to be at least 50 rooms, so the inequality that describes this scenario is:

\[
16 + 8B \geq 50
\]

**Answer: (Choice B) \[16 + 8B \geq 50\]**

### 2) What is the least amount of additional money Joe can spend to get the rooms they need?

First, we solve the inequality $ 16 + 8B \geq 50 $ to find the minimum number of blocks $ B $ that Joe needs to reserve:

\[
16 + 8B \geq 50
\]

Subtract 16 from both sides:

\[
8B \geq 34
\]

Divide by 8:

\[
B \geq \frac{34}{8} = 4.25
\]

Since $ B $ must be a whole number (he can only reserve full blocks), we round up to the next whole number, so $ B = 5 $.

Now, each block costs \$900, so the cost for 5 blocks is:

\[
5 \times 900 = 4500 \text{ dollars}
\]

Therefore, the least amount of additional money Joe can spend is **\$4,500**.

### Follow-up Questions:
1. If Joe initially reserved 20 rooms, how many more blocks would he need to reserve?
2. How much would Joe spend in total if he needed 100 rooms and had initially reserved 16 rooms?
3. What is the minimum number of rooms Joe would have if he reserved 6 blocks in addition to the initial 16 rooms?
4. If the cost of each block increased to \$1,000, how much would Joe have to pay for the required additional rooms?
5. How would the inequality change if each block contained 10 rooms instead of 8?

### Tip:
When solving inequalities, always remember to check if the final value needs to be rounded up or down based on the context of the problem.

Joe is responsible for reserving hotel rooms for a company trip. His company changes plans and increases how many people are going on the trip, so they need at least 50 total rooms. Joe had already reserved and paid for 16 rooms, so he needs to reserve additional rooms. He can only reserve rooms in blocks, and each block contains 8 rooms and costs $900. Let B represent the number of additional blocks that Joe reserves.

1) Which inequality describes this scenario?

2) What is the least amount of additional money Joe can spend to get the rooms they need?

Explore how to solve an inequality problem involving hotel room reservations, where Joe needs at least 50 rooms, and has already reserved 16. Learn how to determine the minimum number of additional room blocks and the least amount Joe will spend. This problem is suitable for students in Grades 6-8.

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