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

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.

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 negative areas since  negatives do not apply to this problem). 
c) Choose 3 points on your graph: one point within the shaded region, one point on directly on the  line, and one point outside the shaded region. For each point explain what the x and y  coordinates represent, and the significance in terms of total cost (i.e. are you under $500, at  $500, or over $500?).

GPT 3.5 · Answer

a) Let $x$ be the number of t-shirts and $y$ be the number of sweatshirts. The total cost can be calculated by the following inequality: $$2x + 5y \leq 500$$ b) The graph of the inequality $2x + 5y \leq 500$ will be a straight line with a shaded region below the line. c) - Point within the shaded region: Let's say the point is (50, 50). This means you made 50 t-shirts and 50 sweatshirts. The total cost would be $2(50) + 5(50) = 100 + 250 = $350$, which is within the $500 budget. - Point on the line: Let's say the point is (80, 60). This means you made 80 t-shirts and 60 sweatshirts. The total cost would be $2(80) + 5(60) = 160 + 300 = $460$, which is exactly at the $500 budget. - Point outside the shaded region: Let's say the point is (100, 80). This means you made 100 t-shirts and 80 sweatshirts. The total cost would be $2(100) + 5(80) = 200 + 400 = $600$, which is over the $500 budget.