Question

Mason is working two summer jobs, making $11 per hour babysitting and making $7 per hour walking dogs. In a given week, he can work at most 16 total hours and must earn a minimum of $140. If xx represents the number of hours babysitting and yy represents the number of hours walking dogs, write and solve a system of inequalities graphically and determine one possible solution.
Inequality 1: yy
≥


Inequality 2: yy
≥

Answer 1

Let x = hours babysitting, y = hours walking dogs.

Constraints:
- At most 16 hours: x + y ≤ 16 → y ≤ 16 − x.
- At least $140: 11x + 7y ≥ 140 → 7y ≥ 140 − 11x → y ≥ 20 − (11/7)x.
- Also x ≥ 0, y ≥ 0.

So the system (in slope form) is:
y ≤ 16 − x
y ≥ 20 − (11/7)x
x ≥ 0, y ≥ 0

Graphically the feasible region is the set of points between the two lines and in the first quadrant. The two boundary lines meet when
16 − x = 20 − (11/7)x → (4/7)x = 4 → x = 7,
so y = 16 − 7 = 9.

Thus one possible solution is (x,y) = (7, 9). Check: 7 + 9 = 16 hours and 11(7) + 7(9) = 77 + 63 = $140. (Any point in the feasible region, e.g. (8,8), would also work.)