Question

eal World Situation: Planning a Party Maria wants to rent a hall for a birthday party, and she has a budget of $500 for the rental. The hall charges a base fee of $200 plus an additional $25 per hour. Write an inequality to represent this situation, where h is the number of hours Maria can rent the hall without exceeding her budget. Then, solve the inequality to find the maximum number of hours Maria can afford to rent the hall. Question 1 Part 1: Write the two-step inequality that represents this situation. Remember, she can AT MOST, spend $500.

Answer 1

To represent the situation with an inequality, we can define the total cost of renting the hall. The total cost \( C \) is composed of a base fee of $200 plus an hourly rate of $25 times the number of hours \( h \) that Maria rents the hall.

This can be written as:

\[ C = 200 + 25h \]

Since Maria wants to stay within her budget of $500, we can set up the inequality:

\[ 200 + 25h \leq 500 \]

This inequality represents the maximum amount Maria is willing to spend on the rental.

Question 1 Part 1:

The two-step inequality that represents this situation is:

\[ 200 + 25h \leq 500 \]

Next, let's solve this inequality.

Step 1: Subtract 200 from both sides

\[ 25h \leq 500 - 200 \]

\[ 25h \leq 300 \]

Step 2: Divide both sides by 25

\[ h \leq \frac{300}{25} \]

\[ h \leq 12 \]

Thus, the maximum number of hours Maria can afford to rent the hall without exceeding her budget is 12 hours.

Answer 2

Part 2: Solve the inequality you wrote in part 1. Show/explain all steps including your inverse operations to solve for the variable. EXPLAIN what your solution means in the context of the situation

Answer 3

To solve the inequality from Part 1, we have:

\[ 200 + 25h \leq 500 \]

Step 1: Isolate the term with the variable

To start solving for \( h \), we want to isolate the term that contains \( h \). We do this by subtracting 200 from both sides of the inequality:

\[ 200 + 25h - 200 \leq 500 - 200 \]

This simplifies to:

\[ 25h \leq 300 \]

Step 2: Solve for the variable

Next, we want to get \( h \) by itself. Since \( h \) is multiplied by 25, we can do this by dividing both sides of the inequality by 25. Remember, when dividing or multiplying both sides of an inequality by a positive number, the direction of the inequality does not change:

\[ \frac{25h}{25} \leq \frac{300}{25} \]

This simplifies to:

\[ h \leq 12 \]

Final Answer

The solution to the inequality is:

\[ h \leq 12 \]

Explanation of the Solution in Context

In the context of the situation, the solution \( h \leq 12 \) means that Maria can rent the hall for a maximum of 12 hours without exceeding her budget of $500. If she rents the hall for exactly 12 hours, her total cost will be:

• Base fee: $200
• Additionally, for 12 hours at $25 per hour: \( 25 \times 12 = 300 \)

So, the total cost will be:

\[ 200 + 300 = 500 \]

This means Maria can exactly afford to rent the hall for 12 hours or any amount of time less than that, provided she does not exceed her budget. This gives her the flexibility to choose how many hours she wants based on her preferences, as long as it does not go over the 12-hour mark.