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. Which inequality represents the situation described?

To express the situation in an inequality, we need to identify the variables and constants involved in the costs of the party.

Let:

• \( c \) = the number of children
• \( a \) = the number of adults

From the information given, the total cost of the party can be represented as:

\[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

According to her budget, the total cost must not exceed $300:

\[ 150 + 10.50c + 5.50a \leq 300 \]

Now, simplifying that inequality:

\[ 10.50c + 5.50a \leq 300 - 150 \]

\[ 10.50c + 5.50a \leq 150 \]

Additionally, the conditions state that there must be at least one adult and one child:

\[ c \geq 1 \] \[ a \geq 1 \]

Putting all of this together, the complete set of inequalities representing the situation is:

1. \( 10.50c + 5.50a \leq 150 \)
2. \( c \geq 1 \)
3. \( a \geq 1 \)

This system of inequalities describes her budget constraints and the requirement for adults and children at the party.