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*} \]