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

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

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

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

(2,5)

Answer 1

Let's define the variables in the context of the problem:

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

The cost of the adult tickets is $15 each, and the cost of the children's tickets is $12 each. The Ramirez family wants to spend a maximum of $150 on tickets. Therefore, the inequality representing this situation can be written as:

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

Contextual Constraints

1. \( x \geq 0 \): The number of adult tickets must be a non-negative integer.
2. \( y \geq 0 \): The number of children's tickets must be a non-negative integer.

Now we need to check which of the provided points is a viable solution to the inequality \( 15x + 12y \leq 150 \) while also satisfying the non-negativity constraints.

1. Check point (3, 4.5):

   • \( x = 3 \), \( y = 4.5 \)
   • \( 15(3) + 12(4.5) = 45 + 54 = 99 \) (This satisfies \( 99 \leq 150 \))
   • However, \( y = 4.5 \) is not a valid number of children's tickets since it must be a whole number.

2. Check point (10, 5):

   • \( x = 10 \), \( y = 5 \)
   • \( 15(10) + 12(5) = 150 + 60 = 210 \) (This does not satisfy \( 210 \leq 150 \))

3. Check point (-3, 5):

   • \( x = -3 \), \( y = 5 \)
   • Point is invalid because \( x \) cannot be negative.

4. Check point (2, 5):

   • \( x = 2 \), \( y = 5 \)
   • \( 15(2) + 12(5) = 30 + 60 = 90 \) (This satisfies \( 90 \leq 150 \))
   • Both \( x \) and \( y \) are non-negative.

Viable Solution

The only viable solution among the options provided is (2, 5).