To represent the constraints of Amy's situation using a system of inequalities, we can break down the information provided:
- The number of pencils needed is at least twice the number of notebooks.
- The cost of pencils is 25 cents each, and the cost of notebooks is $4 each.
- Amy has a maximum budget of $100.
Let's define:
- \( p \) = number of pencils
- \( n \) = number of notebooks
1. Inequality from the relationship between pencils and notebooks:
The requirement that the number of pencils is at least twice the number of notebooks can be expressed as: \[ p \geq 2n \]
2. Inequality from the budget constraint:
The total cost of the pencils and notebooks must not exceed $100. The cost equation based on the number of items is:
- Cost of pencils: \( 0.25p \) (since each pencil costs 25 cents)
- Cost of notebooks: \( 4n \) (since each notebook costs $4)
The budget constraint can be written as: \[ 0.25p + 4n \leq 100 \]
3. Non-negativity constraints:
Since the number of pencils and notebooks cannot be negative, we also have: \[ p \geq 0 \] \[ n \geq 0 \]
Complete System of Inequalities:
Combining all of the above, the system of inequalities representing the scenario is: \[ \begin{align*} p & \geq 2n \quad \text{(1)} \ 0.25p + 4n & \leq 100 \quad \text{(2)} \ p & \geq 0 \quad \text{(3)} \ n & \geq 0 \quad \text{(4)} \end{align*} \]
This system summarizes all the constraints Amy needs to consider when purchasing pencils and notebooks for her students within her budget.