Ask a New Question
Menu

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.

Part 1: Write the two-step inequality that represents this situation. Remember, she can AT MOST, spend $500.

9 answers

To set up the inequality representing Maria's budget for renting the hall, we start with the following components:

  • The base fee for renting the hall is $200.
  • The hall charges an additional $25 per hour for the number of hours \( h \).

We need to ensure that the total cost does not exceed Maria's budget of $500. Therefore, we can represent the total cost with the inequality:

\[ 200 + 25h \leq 500 \]

This inequality states that the base fee plus the hourly charges must be less than or equal to $500.

Part 2: Solve the inequality to find the maximum number of hours \( h \) Maria can afford.

  1. Subtract 200 from both sides of the inequality:

\[ 25h \leq 500 - 200 \]

\[ 25h \leq 300 \]

  1. Divide both sides by 25:

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

\[ h \leq 12 \]

So, the maximum number of hours Maria can afford to rent the hall is \( \boxed{12} \).

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!

To solve the inequality \( 200 + 25h \leq 500 \) step by step, we'll use inverse operations to isolate the variable \( h \).

Step 1: Subtract the base fee from both sides

The first step is to eliminate the base fee of $200 from the left side of the inequality. We do this by subtracting 200 from both sides:

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

This simplifies to:

\[ 25h \leq 300 \]

Step 2: Divide by the hourly rate

Next, we divide both sides of the inequality by 25 in order to solve for \( h \):

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

This simplifies to:

\[ h \leq 12 \]

Conclusion of the Solution

The inequality \( h \leq 12 \) means that Maria can rent the hall for at most 12 hours without exceeding her budget of $500.

Context of the Situation

In the context of the situation, this solution indicates that if Maria wants to have her birthday party at the hall, she needs to plan for a rental time of 12 hours or fewer. If she intends to rent the hall for more than 12 hours, she will go over her budget and may need to adjust either her rental hours or her budget. This helps her make informed choices about the timing of her event and any possible extra costs she might have to consider.

Fifteen less than four times a number is more than 49. Write an inequality that could be used to solve this problem. The inequality has been started for you. Fill in the remaining blanks.

4x ____ _____ _____

To write an inequality representing the phrase "Fifteen less than four times a number is more than 49," we start by defining the variable:

Let \( x \) be the unknown number.

Now, we translate the statement step by step:

  • "Four times a number" is expressed as \( 4x \).
  • "Fifteen less than four times a number" translates to \( 4x - 15 \).
  • "is more than 49" means we set up the inequality \( (4x - 15) > 49 \).

So, the complete inequality is:

\[ 4x - 15 > 49 \]

In the form you provided, filling in the blanks gives us:

\[ 4x ; - ; 15 ; > ; 49 \]

This inequality can now be solved to find the possible values of \( x \).

Lily is saving up to buy a cellphone. She needs to save at least $300 before she is able to buy the phone. Her grandfather gives her $40, and she earns $65 tutoring after school each week. Write an inequality for the number of weeks Lily will need to save to have at least $300.

_____ x + ______ ≥ _______, where x is the number of weeks Lily will need to save.

To create the inequality for the number of weeks Lily needs to save to have at least $300, we can break down her savings:

  1. She starts with $40 from her grandfather.
  2. She earns $65 each week from tutoring.

Let \( x \) represent the number of weeks Lily saves.

Now, we can express her total savings after \( x \) weeks:

  • Her total savings after \( x \) weeks will be the initial $40 plus \( 65x \), where \( 65x \) is the amount she earns from tutoring over \( x \) weeks.

The inequality stating that she needs to save at least $300 is:

\[ 40 + 65x \geq 300 \]

So, in the form you provided, it would be:

\[ 65x + 40 \geq 300 \]

This expresses Lily's requirement to save at least $300.

how many weeks does she need.

To find out how many weeks \( x \) Lily needs to save in order to have at least $300, we will solve the inequality we formed:

\[ 40 + 65x \geq 300 \]

Step 1: Subtract 40 from both sides

\[ 65x \geq 300 - 40 \]

\[ 65x \geq 260 \]

Step 2: Divide both sides by 65

\[ x \geq \frac{260}{65} \]

\[ x \geq 4 \]

Conclusion

Lily needs to save for at least 4 weeks to have enough money to buy the cellphone. If she saves for 4 weeks or more, she will reach her savings goal of at least $300.