Let's represent the information as a system of inequalities using the variables x and y, where:
x represents the number of hours lifeguarding.
y represents the number of hours walking dogs.
Based on the given information, we can establish the following constraints:
Isabella can work at most 11 total hours:
x + y ≤ 11
She must earn a minimum of $140, considering her hourly rates:
17x + 10y ≥ 140
She must work a minimum of 7 hours lifeguarding:
x ≥ 7
Now, let's graph these inequalities:
The first inequality, x + y ≤ 11, represents a boundary line on the graph. To graph it, you can consider the following points:
When x = 0, y = 11
When y = 0, x = 11
So, you can draw a line connecting these two points.
The second inequality, 17x + 10y ≥ 140, represents another boundary line on the graph. To graph it, you can consider the following points:
When x = 0, y = 14 (since 10y ≥ 140, y ≥ 14)
When y = 0, x = 8.24 (approximately, since 17x ≥ 140, x ≥ 140/17 ≈ 8.24)
So, you can draw a line connecting these two points.
The third inequality, x ≥ 7, represents a vertical line at x = 7.
Now, shade the regions on the graph that satisfy all three inequalities. The solution to the system of inequalities is the shaded area where all three conditions are met.
One possible solution within this shaded area is where the lines representing the first two inequalities intersect. You can find this point by solving the system of equations formed by the two boundary lines:
x + y = 11 (from the first inequality)
17x + 10y = 140 (from the second inequality)
You can solve this system of equations to find the values of x and y that satisfy both equations. This will give you a specific solution within the shaded area.