To solve the problem, let's break it down step by step:

Problem Setup:

1. Myranda received a $100 gift card.
2. Matinee tickets cost $7.50 each.
3. Evening tickets cost $12.50 each.
4. Let:
   • $x$ = number of matinee tickets
   • $y$ = number of evening tickets
5. The goal is to determine the maximum number of matinee tickets ($x$) she can purchase using her gift card.

Step 1: Write the Inequality

The total cost of the tickets cannot exceed $100. This can be expressed as: $7.50x + 12.50y \leq 100$

Step 2: Analyze the Constraint

To maximize the number of matinee tickets ($x$), we need to assume $y$, the number of evening tickets, is as small as possible. Let's explore two cases:

• $y = 0$: This means Myranda only purchases matinee tickets.
• $y \neq 0$: This involves combinations of matinee and evening tickets.

Step 3: Solve for Maximum $x$ (When $y = 0$)

If $y = 0$, the inequality becomes: $7.50x \leq 100$ Solve for $x$: $x \leq \frac{100}{7.50} = 13.33$ Since $x$ must be a whole number, the maximum number of matinee tickets is 13.

Step 4: Graph the Inequality

To graph:

1. Rewrite the inequality $7.50x + 12.50y \leq 100$:
   • Solve for $y$: $y \leq \frac{100 - 7.50x}{12.50}$.
2. Plot points:
   • When $x = 0$, $y = 8$ (100 ÷ 12.50).
   • When $y = 0$, $x = 13$ (100 ÷ 7.50).
3. Shade the region below the line to represent all possible combinations of $x$ and $y$ that satisfy the inequality.

Final Answer:

The maximum number of matinee tickets Myranda can purchase is 13, assuming she doesn't buy any evening tickets.

Would you like help plotting the graph or further explanation of this solution?

Related Questions:

1. What would the maximum number of evening tickets be if no matinee tickets are purchased?
2. How does the graph change if the ticket prices increase?
3. What combinations of matinee and evening tickets sum exactly to $100?
4. How would the solution change if Myranda’s gift card was $150 instead of $100?
5. What is the equation of the line that represents the boundary of this inequality?

Tip:

Always simplify inequalities by considering extreme cases, such as one variable being zero, to find maximum or minimum values efficiently.