Final answer:
The student's mathematics question involves creating and graphically solving a system of inequalities to find feasible solutions for the number of hours Caroline works at two jobs. Multiple scenarios meet all the criteria provided, leading to multiple feasible solutions. A specific solution example is 8 hours lifeguarding and 7 hours car washing.
Step-by-step explanation:
The student is asked to write and solve a system of inequalities graphically to find a possible solution for Caroline's work hours as a lifeguard and car washer. Given the criteria that Caroline makes $19 per hour lifeguarding (x hours) and $10 per hour washing cars (y hours), the system of inequalities based on the conditions provided is as follows:
- x + y ≤ 15 (Total hours worked in a week cannot exceed 15)
- 19x + 10y ≥ 200 (Total earnings in a week must be at least $200)
- x ≥ 8 (Minimum of 8 hours lifeguarding)
- y ≥ 2 (Minimum of 2 hours washing cars)
To solve graphically, you plot each inequality on a coordinate plane and find the region of intersection that satisfies all inequalities. By doing this, you may find multiple feasible solutions for x and y within this region that satisfy all the given conditions.
One possible solution could be where Caroline works 8 hours lifeguarding and 7 hours washing cars. This meets all requirements since:
- 8 + 7 = 15 (which is the maximum total hours)
- 19(8) + 10(7) = $152 + $70 = $222 (which is above the minimum earning requirement)
- Caroline meets the minimum hours for both jobs
The system has multiple feasible solutions because there are several combinations of hours for x and y that meet all conditions within the intersection region.