The Ramirez family is going to the space museum. Adult tickets are $15, and children's tickets are $12. They want to spend a maximum of $150. Write a two- variable inequality to model this situation and determine the contextual constraints. Which of the following points is a viable solution? A.(3,4.5) B. (2,5) C. (-3,5)

Answer 1

To model the situation with a two-variable inequality, let's define the variables:

• Let \( x \) be the number of adult tickets.
• Let \( y \) be the number of children's tickets.

The cost of tickets can be expressed by the equation:

\[ 15x + 12y \leq 150 \]

This represents the total cost of the tickets not exceeding $150.

Contextual Constraints

1. Non-negativity: The number of tickets must be zero or more, so:
   • \( x \geq 0 \) (there cannot be a negative number of adult tickets)
   • \( y \geq 0 \) (there cannot be a negative number of children's tickets)

Summary of Inequalities:

Putting it all together, the two-variable inequality along with the constraints is: \[ 15x + 12y \leq 150 \] \[ x \geq 0 \] \[ y \geq 0 \]

Viable Solution Evaluation

Now, let's evaluate the provided points to see which, if any, are viable solutions.

1. Point A: (3, 4.5) \[ 15(3) + 12(4.5) = 45 + 54 = 99 \] Since \( 99 \leq 150 \), this point is a viable solution.

2. Point B: (2, 5) \[ 15(2) + 12(5) = 30 + 60 = 90 \] Since \( 90 \leq 150 \), this point is also a viable solution.

3. Point C: (-3, 5) Since \( x = -3 \) is not non-negative, this point is not a viable solution.

Conclusion

Points A and B are viable solutions, while Point C is not. Thus, either (3, 4.5) or (2, 5) could be chosen as viable points in this context, but (3, 4.5) is preferable if fractional children's tickets are acceptable, otherwise (2, 5) would be ideal since both numbers of tickets are whole numbers.