Middle School

You want to spend no more than $700 on hotels while vacationing in Hawaii for at least 6 days. A travel agent has provided the names and prices of two hotels in Hawaii.

- The Highville Resort (four stars) costs $130 per night.
- The Coral Hotel (two stars) costs $85 per night.

You plan to stay at the Highville Resort for at least one night and at the Coral Hotel for the remainder of your stay.

**Part A:** Represent the following conditions of your vacation as a system of inequalities. Let \( h \) represent the number of nights you stay at the Highville Resort, and \( c \) represent the number of nights you stay at the Coral Hotel.

1. You want to stay at least 6 nights:
\[ h + c \geq 6 \]

2. You want to stay at least 1 night at the Highville Resort:
\[ h \geq 1 \]

3. You want to spend no more than $700:
\[ 130h + 85c \leq 700 \]

**Part B:** Graph the system of inequalities from Part A with \( c \) on the x-axis and \( h \) on the y-axis.

**Part C:** If you want to stay in Hawaii for at least 6 nights, how many nights could you spend at each hotel while staying within your budget? Explain how to use the solution set graphed above to name at least three options for your stay in Hawaii.

Answer :

Answer:

Part A)

[tex]c+h \geq 6[/tex]

[tex]h\geq 1[/tex]

[tex]85c+130h \leq 700[/tex]

Part B) The graph in the attached figure

Part C) [tex](5,1)[/tex], [tex](2,4)[/tex], [tex](5,2)[/tex]

Step-by-step explanation:

Part A)

Let

c------> the number of nights you stay at the Coral Hotel.

h-----> the number of nights you stay at the Highville Resort

we know that

The system of a inequalities is equal to

[tex]c+h \geq 6[/tex] ------> inequality A

[tex]h\geq 1[/tex] -------> inequality B

[tex]85c+130h \leq 700[/tex] -----> inequality C

Part B)

Using a graphing tool

see the attached figure

The solution is the triangular shaded region

Part C)

the maximum number of nights you stay at the Coral Hotel is for [tex]6[/tex] nights

the maximum number of nights you stay at the Highville Resorts is for [tex]4[/tex] nights

we know that

To name at least three options for your stay in Hawaii , find three pair of whole numbers that are in the shaded area of the graph.

Option N [tex]1[/tex]

point [tex](5,1)[/tex]

[tex]5[/tex] nights at the Coral Hotel.

[tex]1[/tex] nights at the Highville Resort

Option N [tex]2[/tex]

point [tex](2,4)[/tex]

[tex]2[/tex] nights at the Coral Hotel.

[tex]4[/tex] nights at the Highville Resort

Option N [tex]3[/tex]

point [tex](5,2)[/tex]

[tex]5[/tex] nights at the Coral Hotel.

[tex]2[/tex] nights at the Highville Resort

see the attached figure N [tex]2[/tex] to better understand the problem

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