Question

Before his parents anniversary party Lee is considering using one of two venues a hotel Clarksville will cost $500 for a reservation plus $8 per person a restaurant the same city will cost $6 per person and addition to a $600 for the reservation in order to make the best decision we figures out how many intendees it will take to have the Venus cost the same amount how many attendees would that be write a system of inequalities graph them and type the solution

Answer 1

To determine the break-even point between the two venues based on the number of attendees, let's define the costs for each venue as follows:

• Cost of the hotel (H): $$H=500+8x$$ where $x$ is the number of attendees.

• Cost of the restaurant (R): $$R=600+6x$$

Finding the Break-even Point

To find how many attendees it would take for both venues to cost the same, we can set the two equations equal to each other:

$$500+8x=600+6x$$

Solving for $x$

1. Rearranging the equation: $$8x-6x=600-500$$ $$2x=100$$ $$x=50$$

So, it takes 50 attendees for the costs of the hotel and restaurant to be the same.

Setting Up Inequalities

Next, to create a system of inequalities to determine which venue is more cost-effective for different numbers of attendees, we can express the inequalities based on comparative costs:

1. For the hotel to be cheaper than the restaurant: $$500+8x<600+6x$$

2. For the restaurant to be cheaper than the hotel: $$600+6x<500+8x$$

Graphing the Inequalities

1. Hotel cheaper than Restaurant: Rearrange the first inequality: $$8x-6x<600-500$$ $$2x<100⟹x<50$$

2. Restaurant cheaper than Hotel: Rearrange the second inequality: $$6x-8x<500-600$$ $$-2x<-100⟹x>50$$

Summary of the Inequalities

• The hotel is cheaper when $x<50$.
• The restaurant is cheaper when $x>50$.

Graphing the Inequalities

You can graph these two inequalities on a number line:

• From 0 to 50, you shade the region indicating the hotel is cheaper (below the line).
• After 50, you shade the region for the restaurant being cheaper (above the line).

Conclusion

• At $x=50$, both venues will cost the same.
• For fewer than 50 attendees, the hotel is the more cost-effective choice.
• For more than 50 attendees, the restaurant becomes the cheaper option.