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

(−3,5)
left parenthesis negative 3 comma 5 right parenthesis

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

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

(3,4.5)

Answer 1

To model the situation with a two-variable inequality, we can define the variables as follows:

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

The cost of adult tickets is $15 per ticket, and the cost of children's tickets is $12 per ticket. The total amount they want to spend should not exceed $150. Thus, we can write an inequality to represent this situation:

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

Next, we should consider the contextual constraints. Since the number of tickets cannot be negative, we have:

\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

Thus, the full set of inequalities that models the situation is:

\[
\begin{align*}
15x + 12y & \leq 150 \\
x & \geq 0 \\
y & \geq 0
\end{align*}
\]

Now, let’s evaluate the potential solutions provided:

1. **For the point (−3, 5)**:
- \( x = -3 \), which is not allowed since you cannot purchase a negative number of tickets. This point is **not viable**.

2. **For the point (10, 5)**:
- Calculate the total cost:
\[
15(10) + 12(5) = 150 + 60 = 210
\]
- This exceeds the budget of $150. This point is **not viable**.

3. **For the point (2, 5)**:
- Calculate the total cost:
\[
15(2) + 12(5) = 30 + 60 = 90
\]
- This is within the budget of $150. This point is **viable**.

4. **For the point (3, 4.5)**:
- Since the number of children’s tickets (4.5) must be a whole number (you can't have half a ticket), this point is **not viable**.

Thus, the only viable solution from the given options is:

**(2, 5)**.