You are starting a new business in which you have decided to sell two products, t-shirts and sweatshirts with your school logo. You have $500 available to start this business, meaning you cannot go over $500 when buying your products. The t-shirts will cost $2 each and the sweatshirts will cost $5 each. You plan on selling the t-shirts for $10 and the sweatshirts for $20. This is summarized in the chart below.

Question

t-shirts
sweatshirts
Cost
$2
$5
Sell Price
$10
$20

a) Write an inequality for the total cost of your t-shirts and sweatshirts keeping in mind that the cost cannot exceed $500.
b) Graph the inequality from part a). You can let the x-axis represent the number of t-shirts you make and the y-axis represent the number of sweatshirts you make. Limit your graph to just positive x and y values (i.e. there is no need to shade anything in the

Answer 1

a) Let $x$ represent the number of t-shirts and $y$ represent the number of sweatshirts. The total cost is given by the equation $2x + 5y \leq 500$.

b) To graph the inequality $2x + 5y \leq 500$, we first need to put it in slope-intercept form: $y \leq -\frac{2}{5}x + 100$.

The graph of this inequality would be a straight line with a slope of $-\frac{2}{5}$ and a $y$-intercept of 100, passing through the points (0,100) and (250,0) on the x-axis (found by setting $y=0$ and solving for $x$).

Since we are dealing with non-negative values, we only need to graph the portion of the line where $x \geq 0$ and $y \geq 0$, which is the first quadrant of the Cartesian plane.