Question

Michael has $15 and wants to buy a combination of school lunches to feed at least three classmates. A sandwich costs $2, and hot lunch costs $3.

This system of inequalities models the scenario:

2x + 3y ≤ 15
x + y ≥ 3

Part A: Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)

Part B: Is the point (5, 1) included in the solution area for the system? Justify your answer mathematically. (3 points)

Part C: Choose a point in the solution set and interpret what it means in terms of the real-world context. (3 points)

Answer 1

for part A) you should probably graph the systems. since the inequality signs are ≤ and ≥ both lines should be solid lines. the direction of the shading will be done according to the sign of the inequality. for "description of the solution set" they're looking for something like "the area under _____ and above _____" where the appropriate inequalities go into the blanks

Answer 2

for part B), to determine whether (5,1) is in the solution area, simply let x = 5 and y = 1 and see whether both inequalities are satisfied or not. alternatively just look at the graph you made from part A) and see if (5,1)'s in the shaded area or not

Answer 3

for part C) you just need to pick a point within the solution set (since x and y are the numbers of sandwich lunches/hot dog lunches, you want to select integer solutions only)

then state something like "the point (__,__) within the solution set means he can buy ___ sandwich lunches and ___ hot dog lunches without going over his budget"