Question

Alex knits hats and scarves to sell at an art fair. He can make at most 20 hats and 30 scarves, but no more than 40 items altogether, in time for the art fair. Write and graph a system of inequalities that show possible numbers of hats and scarves Alex can bring to the art fair if he wants to bring at least 25 items.

Answer 1

Let h be the number of hats and s be the number of scarves that Alex can bring to the art fair. Then, the system of inequalities can be written as:

h ≤ 20 (since he can make at most 20 hats)

s ≤ 30 (since he can make at most 30 scarves)

h + s ≤ 40 (since he can make no more than 40 items altogether)

h + s ≥ 25 (since he wants to bring at least 25 items)

To graph this system of inequalities, we can plot the lines h = 20, s = 30, h + s = 40, and h + s = 25, and shade the region that satisfies all the inequalities. The resulting graph will be a quadrilateral with vertices at (20, 0), (0, 30), (10, 15), and (20, 5). The shaded region will be the area inside this quadrilateral that is above the line h + s = 25.