Question
Francesca is helping with the decorations for her uncle's retirement party. She fills up 70 mini balloons and 10 jumbo balloons. Unfortunately, some of the mini balloons pop. Francesca mixes the remaining mini balloons with the jumbo balloons to make a giant balloon arch.
Let p represent the number of mini balloons that might have popped. Which inequality models the story?
Graph the inequality that models the story.
To draw a ray, plot an endpoint and select an arrow. Select an endpoint to change it from closed to open. Select the middle of the ray to delete it.
Let p represent the number of mini balloons that might have popped. Which inequality models the story?
Graph the inequality that models the story.
To draw a ray, plot an endpoint and select an arrow. Select an endpoint to change it from closed to open. Select the middle of the ray to delete it.
Answers
GPT-4o mini
To model the situation with an inequality, let's first establish the relationships in the problem:
1. Francesca originally has 70 mini balloons.
2. She has 10 jumbo balloons.
3. Let \( p \) represent the number of mini balloons that popped, so the number of remaining mini balloons is \( 70 - p \).
The conditions for the situation are:
- The number of popped balloons \( p \) cannot be greater than the total number of mini balloons she originally had. Therefore, \( p \) must satisfy the inequality:
\[
p \leq 70
\]
Additionally, since popping can't yield a negative number of balloons, we also need:
\[
p \geq 0
\]
From combining both conditions, we can write the inequality as:
\[
0 \leq p \leq 70
\]
### Graphing the Inequality
To graph this inequality:
1. On a number line, represent the range of possible values for \( p \).
2. The closed endpoint at 0 indicates that \( p \) can be equal to 0 (some balloons may not have popped), and the closed endpoint at 70 indicates that \( p \) can be equal to 70 (all mini balloons have popped).
3. So, we draw a number line:
- Place a filled dot (closed dot) at 0 and a filled dot (closed dot) at 70.
- Shade the area between 0 and 70 to indicate all possible values of \( p \) in that range.
This graphical representation signals that \( p \) can take any value from 0 to 70, inclusive.
1. Francesca originally has 70 mini balloons.
2. She has 10 jumbo balloons.
3. Let \( p \) represent the number of mini balloons that popped, so the number of remaining mini balloons is \( 70 - p \).
The conditions for the situation are:
- The number of popped balloons \( p \) cannot be greater than the total number of mini balloons she originally had. Therefore, \( p \) must satisfy the inequality:
\[
p \leq 70
\]
Additionally, since popping can't yield a negative number of balloons, we also need:
\[
p \geq 0
\]
From combining both conditions, we can write the inequality as:
\[
0 \leq p \leq 70
\]
### Graphing the Inequality
To graph this inequality:
1. On a number line, represent the range of possible values for \( p \).
2. The closed endpoint at 0 indicates that \( p \) can be equal to 0 (some balloons may not have popped), and the closed endpoint at 70 indicates that \( p \) can be equal to 70 (all mini balloons have popped).
3. So, we draw a number line:
- Place a filled dot (closed dot) at 0 and a filled dot (closed dot) at 70.
- Shade the area between 0 and 70 to indicate all possible values of \( p \) in that range.
This graphical representation signals that \( p \) can take any value from 0 to 70, inclusive.