Question

Amy needs to purchase pencils and notebooks for her students. The number of pencils needed is at least twice as many as the number of notebooks. The pencils cost $0.25 each and the notebooks cost $4. Amy’s budget for these supplies allows for a maximum cost of $100. How do you represent the constraints of this scenario as a system of inequalities, where p is the number of pencils and n is the number of notebooks Amy will buy?(1 point)
Responses

p<2n
𝑝
<
2
𝑛

0.25p+4n≤100
0.25
𝑝
+
4
𝑛
≤
100

p less than 2 n 0.25 p plus 4 n less-than-or-equal-to 100

p≥n+2
𝑝
≥
𝑛
+
2

0.25p+4n≤100
0.25
𝑝
+
4
𝑛
≤
100

p greater-than-or-equal-to n plus 2 0.25 p plus 4 n less-than-or-equal-to 100

p≥2n
𝑝
≥
2
𝑛

0.25p+4n≥100
0.25
𝑝
+
4
𝑛
≥
100

p greater-than-or-equal-to 2 n 0.25 p plus 4 n greater-than-or-equal-to 100

p≥2n
𝑝
≥
2
𝑛

0.25p+4n≤100
0.25
𝑝
+
4
𝑛
≤
100

p greater-than-or-equal-to 2 n 0.25 p plus 4 n less-than-o

Answer 1

To represent the constraints of Amy's purchasing scenario as a system of inequalities, we need to derive the inequalities based on the information given:

1. **The number of pencils needed is at least twice as many as the number of notebooks.** This can be represented as:
\[
p \geq 2n
\]

2. **The total cost of the pencils and notebooks must not exceed Amy's budget of $100.** The cost for pencils is $0.25 each and notebooks are $4 each. Thus, this constraint can be expressed as:
\[
0.25p + 4n \leq 100
\]

Therefore, the system of inequalities that represents the constraints is:
\[
\begin{cases}
p \geq 2n \\
0.25p + 4n \leq 100
\end{cases}
\]

This corresponds to the response:
- \( p \geq 2n \)
- \( 0.25p + 4n \leq 100 \)

So the correct selection is:
1. \( p \geq 2n \)
2. \( 0.25p + 4n \leq 100 \)