Which system of linear inequalities represents all possible combinations of the number of hours babysitting, x, and the number of hours as a library assistant, y, Jeremy can work?
Understand the Problem
The question is asking for a system of linear inequalities that describes the constraints on the number of hours Jeremy can work at two different jobs, given his pay rates and weekly earnings goals.
Answer
The system of inequalities is: $$ \begin{align*} 7.25x + 14.50y > 100 \\ x + y \leq 20 \\ x \geq 0 \\ y \geq 0 \\ \end{align*} $$
Answer for screen readers
The system of linear inequalities that represents the situation is: $$ \begin{align*}
- & \quad 7.25x + 14.50y > 100 \
- & \quad x + y \leq 20 \
- & \quad x \geq 0 \
- & \quad y \geq 0 \ \end{align*} $$
Steps to Solve
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Define Variables Let ( x ) represent the number of hours Jeremy works babysitting, and ( y ) represent the number of hours he works as a library assistant.
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Determine Pay Rates The babysitting job pays $7.25 per hour, while the library assistant job pays twice as much: $$ 2 \times 7.25 = 14.50 $$ Therefore, ( y ) hours worked at the library assistant job yields ( 14.50y ).
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Set Up Weekly Earnings Inequality Jeremy wants to earn more than $100 a week: $$ 7.25x + 14.50y > 100 $$
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Set Up Total Hours Inequality Jeremy wants to work at most 20 hours a week: $$ x + y \leq 20 $$
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Include Non-negativity Constraints Since hours worked cannot be negative: $$ x \geq 0 $$ $$ y \geq 0 $$
By combining all of these inequalities, we define a system of linear inequalities.
The system of linear inequalities that represents the situation is: $$ \begin{align*}
- & \quad 7.25x + 14.50y > 100 \
- & \quad x + y \leq 20 \
- & \quad x \geq 0 \
- & \quad y \geq 0 \ \end{align*} $$
More Information
This set of inequalities outlines all the combinations of hours Jeremy can work at both jobs while meeting his income goals and time constraints. The inequality ( 7.25x + 14.50y > 100 ) ensures he earns more than $100, while ( x + y \leq 20 ) limits his total working hours.
Tips
- Omitting the non-negativity constraints ( x \geq 0 ) and ( y \geq 0 ), which are important because negative hours do not make sense in this context.
- Misinterpreting what "twice as much" means in calculating the library assistant's pay.
