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

(2,5)

left parenthesis 2 comma 5 right parenthesis
(−3,5)
left parenthesis negative 3 comma 5 right parenthesis
(3,4.5)
left parenthesis 3 comma 4.5 right parenthesis
(10,5)

Answer 1

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 adult tickets is $15 each, and the cost of children's tickets is $12 each. The total expenditure must not exceed $150. This can be represented by the inequality:

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

Next, we need to identify the contextual constraints. Since you cannot have a negative number of tickets, both \( x \) and \( y \) must be non-negative integers:

\[ x \geq 0 \] \[ y \geq 0 \]

So, the complete system of inequalities that represents the situation is:

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

Now, let's evaluate the point options given in the responses:

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

2. (−3, 5): \[ 15(-3) + 12(5) = -45 + 60 = 15 \quad \text{✗ Invalid (negative tickets)} \]

3. (3, 4.5): \[ 3 \text{ is valid, but } 4.5 \text{ is not an integer (children’s tickets must be a whole number).} \quad \text{✗ Invalid} \]

4. (10, 5): \[ 15(10) + 12(5) = 150 + 60 = 210 \quad \text{✗ Invalid (exceeds budget)} \]

Among the provided options, only \( (2, 5) \) is a viable solution to the ticket purchasing scenario, while the other options are either invalid due to negative values, non-integer values, or exceeding the budget.