Question

Natalie uses a prepaid card to buy movie tickets. The initial value of the card is $180, and each movie ticket costs $12. Let m(x) represent the amount of money, m, in dollars that remains on the card after purchasing x movie tickets. what are the constraints?

Answer 1

To represent the situation with the prepaid card and movie ticket purchases, we can derive the function \( m(x) \) that describes the remaining balance on the card after purchasing \( x \) movie tickets.

Given:
- The initial amount on the card is $180.
- The cost of each movie ticket is $12.

The equation representing the remaining balance on the card after purchasing \( x \) tickets is:

\[
m(x) = 180 - 12x
\]

### Constraints
To establish the constraints for this scenario, we consider the following factors:

1. **Non-negative Remaining Balance**: The balance on the card cannot go below zero. This means:

\[
m(x) \geq 0
\]
Plugging in the equation:
\[
180 - 12x \geq 0
\]
Solving for \( x \):
\[
180 \geq 12x \implies x \leq \frac{180}{12} \implies x \leq 15
\]

This means that \( x \) (the number of movie tickets) must be less than or equal to 15.

2. **Non-negative Number of Tickets**: The number of movie tickets purchased cannot be negative, which gives us the constraint:

\[
x \geq 0
\]

### Summary of Constraints
Combining both constraints, we have:

\[
0 \leq x \leq 15
\]

These constraints imply that Natalie can purchase anywhere from 0 to a maximum of 15 movie tickets.