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

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

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

(−3,5)

Answer 1

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

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

The cost of adult tickets is $15, and the cost of children's tickets is $12. Therefore, the inequality that models their total spending would be:

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

Next, we need to consider the contextual constraints:

1. Non-negativity Constraint: The number of tickets can't be negative, so we have:
   • \( x \geq 0 \)
   • \( y \geq 0 \)

Now, we evaluate the given points to see if they satisfy both the inequality and the non-negativity constraints.

1. Point (2, 5):

   • \( 15(2) + 12(5) = 30 + 60 = 90 \leq 150 \) (satisfies the inequality)
   • \( 2 \geq 0 \) and \( 5 \geq 0 \) (satisfies the non-negativity constraints)

2. Point (10, 5):

   • \( 15(10) + 12(5) = 150 + 60 = 210 \not\leq 150 \) (does not satisfy the inequality)

3. Point (3, 4.5):

   • \( 15(3) + 12(4.5) = 45 + 54 = 99 \leq 150 \) (satisfies the inequality)
   • \( 3 \geq 0 \) (satisfies the non-negativity constraint, but \( 4.5 \) is not an integer, which may not be appropriate for number of tickets)

4. Point (−3, 5):

   • Negative value for \( x \); \( -3 < 0 \) (does not satisfy the non-negativity constraint)

From this evaluation, the viable solution that meets the conditions set out in the problem is:

• Point (2, 5).

So, the answer is (2, 5).