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.
To write the system of inequalities, we first need to define our variables:
Let xx be the number of hours walking dogs.
Let yy be the number of hours clearing tables.
We can now write the system of inequalities based on the given information:
1. Ai Mi can work a maximum of 16 hours: xx + yy ≤ 16
2. Ai Mi must earn a minimum of $140: 8xx + 10yy ≥ 140
To graph this system of inequalities, we can start by graphing the first inequality:
xx + yy ≤ 16
We can rewrite this inequality as:
yy ≤ -xx + 16
Next, let's graph the boundary line yy = -xx + 16 as a dotted line:
Graph: https://www.desmos.com/calculator/fwrgposvxm
Now, let's shade the region below this line to satisfy the inequality.
Next, let's graph the second inequality:
8xx + 10yy ≥ 140
We can rewrite this inequality as:
10yy ≥ -8xx + 140
yy ≥ (-4/5)xx + 14
Now, let's graph the boundary line yy = (-4/5)xx + 14 as a dotted line:
Graph: https://www.desmos.com/calculator/83idlnyqry
Finally, let's shade the region above this line to satisfy the inequality.
The region that is shaded in both graphs (below yy = -xx + 16 and above yy = (-4/5)xx + 14) represents the feasible region where both inequalities are satisfied.
We can now find one possible solution within this feasible region by finding the intersection point. From the graph, one possible solution is (4, 12), which means Ai Mi can walk dogs for 4 hours and clear tables for 12 hours in a given week to meet the requirements.