Ai Mi is working two summer jobs, making $8 per hour walking dogs and making $10 per hour clearing tables. In a given week, she can work a maximum of 16 total hours and must earn a minimum of $140. If xx represents the number of hours walking dogs and yy represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.

Question

Ai Mi is working two summer jobs, making $8 per hour walking dogs and making $10 per hour clearing tables. In a given week, she can work a maximum of 16 total hours and must earn a minimum of $140. If xx represents the number of hours walking dogs and yy represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.

Answer 1

We can set up the following system of inequalities to represent the given information:

1. Ai Mi can work a maximum of 16 total hours:
x + y ≤ 16

2. Ai Mi must earn a minimum of $140:
8x + 10y ≥ 140

We can graph these two inequalities on a coordinate plane to find a solution:

First, let's graph the equation x + y ≤ 16:
Plotting the line x + y = 16:

x | y
------
0 | 16
16 | 0

Then, since it is less than or equal to, we will shade below the line.

Next, let's graph the equation 8x + 10y ≥ 140:
Plotting the line 8x + 10y = 140:
When x = 0, y = 14
When y = 0, x = 17.5

Due to the equation being greater than or equal to, we will shade above the line.

The shaded region that satisfies both inequalities is the feasible region. We can now find one possible solution by picking a point within that shaded region.

One possible solution is x = 6 hours walking dogs and y = 10 hours clearing tables. This satisfies both inequalities: 6 + 10 ≤ 16 and 8(6) + 10(10) ≥ 140.