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?

Answer 1

To represent the constraints of Amy's purchasing situation as a system of inequalities, we need to consider her conditions regarding the number of pencils \( p \) and notebooks \( n \).

1. **Pencils need to be at least twice as many as notebooks**: This translates mathematically to the inequality:
\[
p \geq 2n
\]

2. **Total cost constraint**: The total cost for the pencils and notebooks must not exceed $100. Since pencils cost $0.25 each and notebooks cost $4 each, we can represent this as:
\[
0.25p + 4n \leq 100
\]

3. **Non-negativity**: Amy cannot buy a negative quantity of pencils or notebooks, which gives us the inequalities:
\[
p \geq 0
\]
\[
n \geq 0
\]

Therefore, the system of inequalities representing Amy's constraints is as follows:

\[
\begin{align*}
1) & \quad p \geq 2n \\
2) & \quad 0.25p + 4n \leq 100 \\
3) & \quad p \geq 0 \\
4) & \quad n \geq 0 \\
\end{align*}
\]