Eula needs to buy binders that cost $4 each and notebooks that cost $2 each. She has $20. The graph of the inequality 4x + 2y ≤ 20, which represents the situation, is shown. What is the greatest number of binders Eula can buy? What is the greatest number of notebooks Eula can buy? If Eula buys 7 notebooks, what is the greatest number of binders she can buy?
The Correct Answer and Explanation is :
To determine how many binders (x) and notebooks (y) Eula can buy with her $20 budget, we start with the inequality representing her spending:
[
4x + 2y \leq 20
]
Step 1: Analyzing the Inequality
- Understanding the Costs:
- Each binder costs $4.
- Each notebook costs $2.
- Eula has $20 total.
- Setting Up the Inequality:
The inequality states that the total cost of binders and notebooks cannot exceed $20.
Step 2: Finding Maximum Quantities of Binders and Notebooks
- Maximize Binders (x):
To find the maximum number of binders Eula can buy when she spends all her money on binders, set (y = 0):
[
4x \leq 20 \implies x \leq 5
]
Therefore, the greatest number of binders Eula can buy is 5 binders. - Maximize Notebooks (y):
To find the maximum number of notebooks she can buy when she spends all her money on notebooks, set (x = 0):
[
2y \leq 20 \implies y \leq 10
]
Thus, the greatest number of notebooks Eula can buy is 10 notebooks.
Step 3: If Eula Buys 7 Notebooks
Now, if Eula decides to buy 7 notebooks, we can substitute (y = 7) into the inequality to find how many binders she can still buy:
[
4x + 2(7) \leq 20 \implies 4x + 14 \leq 20
]
Subtract 14 from both sides:
[
4x \leq 6 \implies x \leq \frac{6}{4} = 1.5
]
Since Eula can only buy a whole number of binders, she can buy at most 1 binder if she purchases 7 notebooks.
Summary:
- Greatest number of binders Eula can buy: 5 binders
- Greatest number of notebooks Eula can buy: 10 notebooks
- If she buys 7 notebooks, the greatest number of binders she can buy: 1 binder.
This analysis shows the relationship between Eula’s budget and her purchasing options for binders and notebooks, helping her make informed decisions based on her needs.