Question

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?(1 point)
Responses

(2,5)
left parenthesis 2 comma 5 right parenthesis

(10,5)
left parenthesis 10 comma 5 right parenthesis

(3,4.5)
left parenthesis 3 comma 4.5 right parenthesis

(−3,5)

Answer 1

To model the situation involving the Ramirez family's trip to the space museum, we can define two variables:

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

The cost of adult tickets is $15, and the cost of children's tickets is $12. The total amount they want to spend should not exceed $150. Therefore, we can create the following inequality:

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

Next, we can analyze the contextual constraints for this situation:

1. **Non-negativity constraints**: The number of tickets cannot be negative, so:
- \( x \geq 0 \)
- \( y \geq 0 \)

2. **Maximum budget constraint**: The total cost must not exceed $150.

Now, let's check which of the provided points (pair of integers representing adult and children's tickets) satisfies the inequality \( 15x + 12y \leq 150 \) and the non-negativity constraints.

1. **Point (2, 5)**:
\[
15(2) + 12(5) = 30 + 60 = 90 \quad \text{(valid, since } 90 \leq 150\text{)}
\]

2. **Point (10, 5)**:
\[
15(10) + 12(5) = 150 + 60 = 210 \quad \text{(not valid, since } 210 > 150\text{)}
\]

3. **Point (3, 4.5)**:
\[
15(3) + 12(4.5) = 45 + 54 = 99 \quad \text{(valid, since } 99 \leq 150\text{)}
\]
However, \( y = 4.5 \) is not a valid solution since the number of children's tickets must be a whole number.

4. **Point (-3, 5)**:
This point is invalid since \( x = -3 \) (negative tickets are not allowed).

After evaluating the options, the only viable solution (considering that both \( x \) and \( y \) must be non-negative integers) is **(2, 5)**.