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

- The Highville Resort costs $130 per night.
- The Coral Hotel costs $85 per night.

You want 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.

- You want to stay at least 6 nights: [tex] h + c \geq 6 [/tex]
- You want to stay at least 1 night at the Highville Resort: [tex] h \geq 1 [/tex]
- You want to spend no more than $700: [tex] 130h + 85c \leq 700 [/tex]

**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 and still stay within your budget? Explain how to use the solution set graphed above to name at least three options for your stay in Hawaii.

Answer :

Part A. 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, then h+c≥6.

2. You want to stay at least 1 night at the Highville Resort, then h≥1.

3. You want to spend no more than $700.

- The Highville Resort costs $130 per night, then it costs $130h per h nights;

- The Coral Hotel costs $85 per night, then it costs $85c per c nights.

In total, 130h+85c≤700.

You get the system of inequalities:

[tex]\left\{\begin{array}{l}h+c\ge 6\\h\ge 1\\130h+85c\le 700\end{array}\right.[/tex]

Part B. If h is on the y-axis and c is on the x-axis, then the system of inequalities is

[tex]\left\{\begin{array}{l}y+x\ge 6\\y\ge 1\\130y+85x\le 700\end{array}\right.[/tex]

See attached diagram for graphs.

Part C. You can stay:

  • 1 night at Highville Resort and 5 nights at Coral Hotel. The cost is $(130+5·85)=$555;
  • 1 night at Highville Resort and 6 nights at Coral Hotel. The cost is $(130+6·85)=$640;
  • 3 nights at Highville Resort and 3 nights at Coral Hotel. The cost is $(3·130+3·85)=$645;
  • 4 nights at Highville Resort and 2 nights at Coral Hotel. The cost is $(4·130+2·85)=$690.

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