Final answer:
Mary's budget constraint for snacks at the movie theater is represented by the inequality 3x + 5y ≤ 30, while Samantha's is given by 3x + 5y ≤ 35. These inequalities can be graphed by plotting the corresponding lines and shading the area that represents all combinations they can afford without going over their budgets.
Step-by-step explanation:
The problem given is related to budget constraints and the purchase of items within that constraint, which is a typical question in the field of mathematics, particularly dealing with systems of inequalities in algebra. To determine the combinations of sodas and popcorn that Mary and Samantha can afford, we set up two separate inequalities with variables x and y to represent the quantity of sodas and popcorn, respectively. Mary has $30 to spend, and the inequality that represents her budget constraint is 3x + 5y ≤ 30, where x is the number of sodas and y is the number of popcorns. Samantha has $35 to spend, so her constraint is 3x + 5y ≤ 35. The inequalities represent that they can spend up to but not exceed their respective budgets on these items.
To graph these inequalities, we would plot the lines of the equations setting each equal to the budget amount (e.g., 3x + 5y = 30 for Mary), and shade the area below and to the left of each line. This shaded area represents all possible combinations of sodas and popcorn they can purchase without exceeding their budget. The inequalities a) and c) are the correct ones for Mary and Samantha, respectively. Option b) and d) suggest that they are required to spend at least the given amounts, which contradicts the context of the question.