Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and at least one child at the party. She has budgeted $300 for the party and that is the maximum amount she wants to spend.

Question

Question 1
A.) Write a two-variable inequality to represent this situation. Use x for the number of children and y for the number of adults. (2 points)

Answer 1

To represent the situation with a two-variable inequality, we can start by calculating the total cost of the party based on the number of children (x) and adults (y) invited.

The total cost of the party includes a fixed cost of $150, a variable cost of $10.50 per child, and a variable cost of $5.50 per adult. Therefore, the total cost \(C\) can be expressed as:

\[ C = 150 + 10.50x + 5.50y \]

Given that Meha's aunt has budgeted a maximum amount of $300 for the party, we need to express this as an inequality:

\[ 150 + 10.50x + 5.50y \leq 300 \]

To simplify, we can subtract 150 from both sides:

\[ 10.50x + 5.50y \leq 150 \]

Now, we also need to consider the constraints that there needs to be at least one adult and at least one child at the party, so we can write:

\[ x \geq 1 \quad (at least , 1 , child) \] \[ y \geq 1 \quad (at least , 1 , adult) \]

Putting this together, the complete representation of the situation would be:

1. \(10.50x + 5.50y \leq 150\)
2. \(x \geq 1\)
3. \(y \geq 1\)

These inequalities represent the constraints and budget limits for the party planning.