A family knows two babysitters they can hire. One charges $6 an hour and the other charges $7. The family can't afford more than $42 a week for babysitting. Write the system of inequalities.

$\begin{array}{l}
7 x-6 y \leq 42 \\
x \geq 0 \\
y \geq 0
\end{array}$

$\begin{array}{l}
7 x+6 y \geq 42 \\
x \geq 0 \\
y \geq 0
\end{array}$

$7 x+6 y \leq 42$

$\begin{array}{l}
x \geq 0 \\
y \geq 0
\end{array}$

Define variables: Let x be the hours for the first babysitter and y be the hours for the second babysitter.
Express the total cost: The total cost is 6 x + 7 y , which must be less than or equal to $42.
Non-negativity constraints: The hours worked cannot be negative, so xg e q 0 and y g e q 0 .
The system of inequalities is: 6 x + 7 y ≤ 42 , xg e q 0 , and y g e q 0 . The correct answer is 6 x + 7 y ≤ 42 , xg e q 0 , and y g e q 0 . Therefore, the answer is: 6 x + 7 y ≤ 42 , xg e q 0 , y g e q 0 ​

Explanation

Understanding the Problem Let's break down this problem. We need to create a system of inequalities that represents the family's babysitting budget. The key is to define our variables and then translate the given information into mathematical expressions.

Defining Variables Let's define our variables:



Let x be the number of hours the family hires the first babysitter (who charges $6 per hour).
Let y be the number of hours the family hires the second babysitter (who charges $7 per hour).


Expressing the Costs Now, let's express the cost of hiring each babysitter:


The cost of hiring the first babysitter is 6 x dollars.
The cost of hiring the second babysitter is 7 y dollars.


Formulating the Main Inequality The total cost of babysitting is the sum of the costs for each babysitter, which is 6 x + 7 y . The family can't afford more than $42 per week, so the total cost must be less than or equal to $42. This gives us the inequality: 6 x + 7 y ≤ 42

Non-Negativity Constraints Since the number of hours worked cannot be negative, we also have the following constraints:



xg e q 0 (The family cannot hire the first babysitter for a negative number of hours).
y g e q 0 (The family cannot hire the second babysitter for a negative number of hours).


The Final System of Inequalities Therefore, the system of inequalities that represents this situation is:

⎩ ⎨ ⎧ ​ 6 x + 7 y ≤ 42 xg e q 0 y g e q 0 ​
Comparing this to the options provided, the correct system of inequalities is the one that includes 6 x + 7 y ≤ 42 , xg e q 0 , and y g e q 0 .
Examples
Imagine you're planning a birthday party and have a limited budget for decorations and food. You can use a system of inequalities to figure out how much you can spend on each item without exceeding your budget. For example, if decorations cost $5 per item and food costs $8 per item, and you have a total budget of $100, you can set up an inequality like 5 x + 8 y ≤ 100 , where x is the number of decoration items and y is the number of food items. This helps you make informed decisions and stay within your financial constraints.

Answered by GinnyAnswer | 2025-07-03

To represent the family's babysitting budget as inequalities, we define x as hours for the first babysitter and y as hours for the second babysitter. The inequalities form a system: 6 x + 7 y ≤ 42 , x ≥ 0 , and y ≥ 0 . Thus, the answer is 6 x + 7 y ≤ 42 , x ≥ 0 , y ≥ 0 ​ .
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Answered by Anonymous | 2025-07-04