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 represent the situation described, we first define variables for the number of children and adults at the party:

• Let \( c \) represent the number of children.
• Let \( a \) represent the number of adults.

The total cost of the party can be expressed as: \[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

According to the problem, the total cost should not exceed her budget of $300. Therefore, we can set up the following inequality: \[ 150 + 10.50c + 5.50a \leq 300 \]

Next, we also need to include the constraints regarding the minimum number of children and adults:

• There needs to be at least one child: \( c \geq 1 \)
• There needs to be at least one adult: \( a \geq 1 \)

Combining all of these elements, the inequalities representing the situation are:

1. \( 150 + 10.50c + 5.50a \leq 300 \)
2. \( c \geq 1 \)
3. \( a \geq 1 \)

These inequalities collectively represent the conditions of the party planning situation.