Leyla is selling hot dogs and spicy sausages at the fair. She has only 40 buns, so she can sell no more than a total of 40 hot dogs and spicy sausages. Each hot dog sells for $2, and each sausage sells for $2.50. Leyla needs at least $90 in sales to meet her goal. Write and graph a system of inequalities that models this situation.

Let d represent the number of hot dogs, and let s represent the number of sausages.

The total number of buns Leyla has can be modeled by the inequality d + s ≤ 40.

The amount of money that Leyla needs to meet her goal can be modeled by 2d + 2.5s ≥ 90.

The system of inequalities is . d + s ≤ 40

2d + 2.5s ≥ 90

d 0s 0

Graph the solid boundary line d + s = 40, and shade below it.

Graph the solid boundary line 2d + 2.5s ≥ 90, and shade above it. The overlapping region is the solution region.

Check Test the point (5, 32) in both inequalities. This point represents selling 5 hot dogs and 32 sausages.

d + s ≤ 40 2d + 2.5s ≥ 90

5 + 32 ≤ 4037 ≤ 40

2(5) + 2.5(32) ≥ 90

90 ≥ 90

Graph the system of inequalities.Graphing Systems of Inequalities

y ≥ –x + 2

y < – 3

For y < – 3, graph the dashed boundary line y = – 3, and shade below it. For y ≥ –x + 2, graph the solid boundary line y = –x + 2, and shade above it.

The overlapping region is the solution region.

If you are unsure which direction to shade, use the origin as a test point.

Helpful Hint

Systems of inequalities may contain more than two inequalities.

Graph the system of inequalities, and classify the figure created by the solution region.

x ≥ –2

y ≥ –x + 1

x ≤ 3

y ≤ 4

Graph the solid boundary line x = –2 and shade to the right of it. Graph the solid boundary line x = 3, and shade to the left of it.

Graph the solid boundary line y = –x + 1, and shade above it. Graph the solid boundary line y = 4, and shade below it. The overlapping region is the solution region.

Graph the system of inequalities, and classify the figure created by the solution region.

x ≤ 6

y ≥ –2x + 4

y ≤ x + 1

Graph the solid boundary line x = 6 and shade to the left of it.Graph the solid boundary line, y ≤ x + 1 and shade below it. Graph the solid boundary line y ≥ –2x + 4, and shade below it. The overlapping region is the solution region. The solution is a triangle.