A drugstore is selling two types of notebooks for $3 and $4. If a person wants to spend between $10 and $15 on notebooks, write the system of inequalities that represents this situation.
[tex]
\begin{array}{l}
3 x+4 y \geq 15 \\
3 x+4 y \leq 10
\end{array}
[/tex]
[tex]
\begin{array}{l}
3 x+4 y \leq 15 \\
3 x+4 y \geq 10
\end{array}
[/tex]
[tex]
\begin{array}{l}
x \geq 0, y \geq 0 \\
3 x+4 y \leq 15
\end{array}
[/tex]
[tex]
\begin{array}{l}
x \geq 0, y \geq 0 \\
3 x+4 y \geq 10
\end{array}
[/tex]
Answer (2)
Define x as the number of $3 notebooks and y as the number of $4 notebooks.
The total cost is represented by 3 x + 4 y .
The cost must be between $10 and $15 , leading to the inequalities 3 x + 4 y ≥ 10 and 3 x + 4 y ≤ 15 .
The number of notebooks must be non-negative, so x ≥ 0 and y ≥ 0 . The system of inequalities is: 3 x + 4 y ≤ 15 and 3 x + 4 y ≥ 10 .
Explanation
Problem Analysis Let x be the number of notebooks that cost $3 and y be the number of notebooks that cost $4 . The total cost of the notebooks is 3 x + 4 y . Since the person wants to spend between $10 and $15 , the total cost must be greater than or equal to $10 and less than or equal to $15 . This can be written as 10 ≤ 3 x + 4 y ≤ 15 . Also, the number of notebooks must be non-negative, so x ≥ 0 and y ≥ 0 .
System of Inequalities We can split the inequality 10 ≤ 3 x + 4 y ≤ 15 into two inequalities: 3 x + 4 y ≥ 10 and 3 x + 4 y ≤ 15 . Combining these with the non-negativity constraints, we get the following system of inequalities:
Final System 3 x + 4 y ≥ 10
3 x + 4 y ≤ 15
x ≥ 0
y ≥ 0
Final Answer The system of inequalities that represents the situation is:
3 x + 4 y ≥ 10 3 x + 4 y ≤ 15 x ≥ 0 y ≥ 0
This corresponds to the second option provided.
Examples
Imagine you're planning a bake sale. You want to make cookies and brownies. Cookies cost $3 to make per batch, and brownies cost $4 per batch. You have between $10 and $15 to spend on ingredients. The system of inequalities helps you determine how many batches of cookies and brownies you can make without exceeding your budget. This kind of problem arises in resource allocation and budgeting scenarios.
The system of inequalities representing the situation is that the total cost of notebooks, given by 3 x + 4 y , must satisfy 3 x + 4 y ≥ 10 and 3 x + 4 y ≤ 15 . Additionally, x ≥ 0 and y ≥ 0 to ensure non-negative quantities of notebooks. This system is crucial for determining an appropriate quantity of each type of notebook while staying within budget.
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