Madelyn is working two summer jobs, making $20 per hour lifeguarding and making $12 per hour clearing tables. In a given week, she can work no more than 13 total hours and must earn no less than $180. If Madelyn worked 8 hours lifeguarding, determine all possible values for the number of whole hours clearing tables that she must work to meet her requirements. Your answer should be a comma separated list of values. If there are no possible solutions, submit an empty answer.

The possible values of the number of hours clearing tables are 2, 3, 4, 5 hours

Step-by-step explanation:

* Lets explain how to solve the problem

- Madelyn is making $20 per hour life guarding

- She is making $12 per hour clearing tables

- She can work no more than 13 total hours

∴ The greatest time she can work is 13 hours (≤)

- She must earn no less than $180

∴ The least money she wants to earn is $180 (≥)

- If Madelyn worked 8 hours life guarding and assume that she

worked x hours clearing tables

∵ She can work no more than 13 hours

∴ 8 + x ≤ 13

- Subtract 8 from both sides

∴ x ≤ 5 ⇒ (1)

∵ She must earn no less than $180

∴ 8 * 20 + x * 12 ≥ 180

∴ 160 + 12x ≥ 180

- Subtract 60 from both sides

∴ 12x ≥ 20 ⇒ (2)

- From equation (1) ⇒ x = 1, 2, 3, 4, 5

- Substitute x in equation (2) by these values to find the right values

of x

# If x = 1

∴ 12 (1) ≥ 20

- But 12 is less than 20

∴ x can not be 1

# x = 2

∴ 12 (2) ≥ 20

∴ 24 ≥ 20

- 24 is greater than 20 then x can be 2

∵ 3, 4, 5 greater than 2

∴ All of them give answers greater than 20

∴ x can be 3, 4, 5

∵ x represents the number of hours for clearing tables

∴ The possible values of the number of hours clearing tables are

2, 3, 4, 5 hours