[Verified] Rob works part-time at the Fallbrook Riding Stable. He makes $5 an hour exercising horses and $10 an hour cleaning stalls. Because Rob is a full-time student, he can work no more than 12 hours per week. However, he must make at least $60 per week. Which of the following is a possible solution for this system of inequalities?

Rob works part-time at the Fallbrook Riding Stable. He makes $5 an hour exercising horses and $10 an hour cleaning stalls. Because Rob is a full-time student, he can work no more than 12 hours per week. However, he must make at least $60 per week.
Which of the following is a possible solution for this system of inequalities?
7 hours of exercising and 8 hours of cleaning
1 hour exercising and 1 hour cleaning
2 hours exercising and 8 hours cleaning
2 hours exercising and 5 hours of cleaning

Setting up the Inequalities

Let: x = number of hours Rob exercises y = number of hours Rob cleans stalls

From the information given:

Hours Worked:

Rob can work no more than 12 hours. �+�≤12x+y≤12 … (i)

Income:

Rob needs to make at least $60. For exercising, he earns $5 per hour, so he earns 5x. For cleaning stalls, he earns $10 per hour, so he earns 10y.

Total earnings: 5�+10�≥605x+10y≥60 … (ii)

2. Evaluating the Options

Option 1: 7 hours of exercising and 8 hours of cleaning

Plug in x = 7 and y = 8:

For (i) Hours Worked: 7+8=157+8=15 15 is NOT less than or equal to 12.

For (ii) Income: 5(7)+10(8)=35+80=1155(7)+10(8)=35+80=115 115 is greater than 60.

Conclusion for Option 1: Does not satisfy the hours condition.

Option 2: 1 hour exercising and 1 hour cleaning

Plug in x = 1 and y = 1:

For (i) Hours Worked: 1+1=21+1=2 2 is less than or equal to 12.

For (ii) Income: 5(1)+10(1)=5+10=155(1)+10(1)=5+10=15 15 is NOT greater than or equal to 60.

Conclusion for Option 2: Does not satisfy the income condition.

Option 3: 2 hours exercising and 8 hours cleaning

Plug in x = 2 and y = 8:

For (i) Hours Worked: 2+8=102+8=10 10 is less than or equal to 12.

For (ii) Income: 5(2)+10(8)=10+80=905(2)+10(8)=10+80=90 90 is greater than 60.

Conclusion for Option 3: Satisfies both the hours and income conditions.

Option 4: 2 hours exercising and 5 hours cleaning

Plug in x = 2 and y = 5:

For (i) Hours Worked: 2+5=72+5=7 7 is less than or equal to 12.

For (ii) Income: 5(2)+10(5)=10+50=605(2)+10(5)=10+50=60 60 is equal to 60.

Conclusion for Option 4: Satisfies both the hours and income conditions.

3. Final Answer

Options 3 and 4 both satisfy the conditions set by the inequalities. However, since the question asks for a possible solution, Option 3 (2 hours exercising and 8 hours cleaning) is a valid choice.

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