Exercise 48 Page 262

We are also told that we have saved $\$120$ to buy materials for our goods. While we cannot go over that budget, we can choose to spend less. We know that each necklace $x$ costs $\$2$ to make and each key chain $y$ costs $\$3$ to make. Let's now write the total amount of money spent preparing for the fair. The exercise asks us to create a system with four inequalities, so we can write inequalities to represent the restrictions on the domain and range. We cannot make a negative number of necklaces or key chains. This gives us the following as our final system of inequalities. We can graph this system by first creating boundary lines and then shading accordingly. Let's look at each inequality individually before we combine the graphs.

$0.5x+0.25y\le 20$

Before we can graph the inequality we need to find the boundary line. To do this, we need to rewrite our inequality in slope-intercept form. Now we can graph the boundary line, which will be solid because the symbol in our inequality is or equal to. To choose on which side of the line we should shade, we can choose an arbitrary point and see if it satisfies the inequality. For simplicity's sake, let's choose $(0,0).$ Because $(0,0)$ is a solution to the inequality, we should shade on the side of the line containing that point.

$2x+3y\le 120$

Just like with the previous inequality, we need to first find the boundary line by rewriting our inequality in the slope-intercept form. Once again the boundary line will be solid because the symbol in our inequality is or equal to. To choose on which side of the line we should shade, we can choose an arbitrary point and see if it satisfies the inequality. For simplicity's sake, let's choose $(0,0).$ Because $(0,0)$ is a solution to the inequality, we should shade on the side of the line containing that point.

$x\ge 0$

Because this is just a vertical line, we can immediately graph the boundary line. We also know that we can shade to the right of the line, as it is simply values of $x$ greater than or equal to $0.$

$y\ge 0$

Because this is just a horizontal line, we can immediately graph the boundary line. We also know that we can shade above the line, as it is simply values of $y$ greater than or equal to $0.$

Combining the Inequalities

To find the final solution set for the system, we can graph all four inequalities on the same coordinate plane.

The overlapping area is the solution set. Let's look at that without any of the extra information.