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 solve the problem regarding Eula’s purchasing situation, we begin by establishing the variables and the inequality representing her constraints. Let ( x ) represent the number of binders and ( y ) represent the number of notebooks she wants to buy. The costs associated with these items lead us to the inequality:
[
4x + 2y \leq 20
]
Finding Maximum Binders and Notebooks
- Maximum Binders:
To determine the maximum number of binders ( x ) that Eula can buy, we set ( y = 0 ) in the inequality: [
4x + 2(0) \leq 20 \implies 4x \leq 20 \implies x \leq 5
] Hence, the greatest number of binders Eula can buy is 5 binders. - Maximum Notebooks:
Similarly, to find the maximum number of notebooks ( y ) she can buy, we set ( x = 0 ): [
4(0) + 2y \leq 20 \implies 2y \leq 20 \implies y \leq 10
] Therefore, the greatest number of notebooks Eula can buy is 10 notebooks.
Binders with 7 Notebooks
If Eula decides to purchase 7 notebooks, we need to substitute ( y = 7 ) back into the inequality to find out how many binders she can still buy:
[
4x + 2(7) \leq 20 \implies 4x + 14 \leq 20 \implies 4x \leq 6 \implies x \leq 1.5
]
Since ( x ) must be a whole number, the greatest number of binders Eula can buy in this scenario is 1 binder.
Conclusion
In summary, Eula can buy a maximum of 5 binders or 10 notebooks. If she buys 7 notebooks, she can afford to purchase only 1 binder. This analysis highlights the constraints imposed by her budget, allowing her to make informed purchasing decisions while adhering to her financial limitations. The graph of the inequality ( 4x + 2y \leq 20 ) visually represents the feasible combinations of binders and notebooks Eula can afford within her budget.