Exercise 46 Page 280

Let's suppose that the gift card we were given is loaded with $\$120.$ We can let $x$ be the cost of a t-shirt and $y$ be the cost of a sweatshirt. A situation in which we are able to buy $9$ t-shirts and $1$ sweatshirt can be written as an inequality. Notice that we can afford things that total to less than or equal to the amount on the gift card. Conversely, a situation where we are unable to purchase $3$ t-shirts and $8$ sweatshirts can be written as the following inequality. We cannot afford a purchase that totals to more than the value of our gift card. Combining these inequalities, we get the following system. To graph this system we need to first create boundary lines by writing the inequalities in slope-intercept form. The first inequality already has a coefficient of $1$ for $y,$ so we can simply subtract $9x$ from both sides. Now let's solve the second inequality for slope-intercept form. Having two of the inequalities in the slope-intercept form, we can write our system. Next, we can graph our boundary lines. Remember that the first inequality will have a solid line and the second will have a dashed line.

The overlapping area shows us the possible price combinations of t-shirts $x$ and sweatshirts $y$ such that we could purchase $9$ t-shirts and $1$ sweatshirt but we could not purchase $3$ t-shirts and $8$ sweatshirts. To view only the solution set for the system, we can cut away the non-overlapping area.

Keep in mind that this is just one possible solution to this problem.