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)

Question

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).