You work as a cashier for a grocery store and earn $5 per hour. You also mow lawns and earn $10 per hour. You want to earn at least $50 per week, but would like to work no more than 10 hours per week. Which system of inequalities, along with [tex]$y \geq 0$[/tex] and [tex]$x \geq 0$[/tex], would you use to solve the real-world problem?
[tex]\left\{\begin{array}{l}
y \leq-x+10 \\
y \geq \frac{1}{2} x+5
\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}
y \geq-x+10 \\
y \leq-\frac{1}{2} x+5
\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}
y \leq-x+10 \\
y \geq-\frac{1}{2} x+5
\end{array}\right.[/tex]
Answer (1)
Define variables: Let x be the number of hours working as a cashier and y be the number of hours mowing lawns.
Write inequalities: 5 x + 10 y ≥ 50 and x + y ≤ 10 .
Isolate y : y ≥ − 2 1 x + 5 and y ≤ − x + 10 .
The system of inequalities is { y ≥ − 2 1 x + 5 y ≤ − x + 10 , along with x ≥ 0 and y ≥ 0 , so the answer is { y ≤ − x + 10 y ≥ − 2 1 x + 5 .
Explanation
Define variables and write inequalities Let x be the number of hours you work as a cashier and y be the number of hours you mow lawns. You earn $5 per hour as a cashier and $10 per hour mowing lawns. You want to earn at least $50 per week, so the inequality representing your earnings is 5 x + 10 y ≥ 50 . You also want to work no more than 10 hours per week, so the inequality representing the total hours you work is x + y ≤ 10 . We also have the constraints x ≥ 0 and y ≥ 0 since you cannot work a negative number of hours.
Isolate y in the inequalities We want to rewrite the inequalities to isolate y . For the first inequality, 5 x + 10 y ≥ 50 , we subtract 5 x from both sides to get 10 y ≥ − 5 x + 50 . Then, we divide both sides by 10 to get y ≥ − 2 1 x + 5 . For the second inequality, x + y ≤ 10 , we subtract x from both sides to get y ≤ − x + 10 .
Write the system of inequalities Therefore, the system of inequalities is
{ y ≥ − 2 1 x + 5 y ≤ − x + 10
along with x ≥ 0 and y ≥ 0 . This corresponds to the third option.
Examples
Understanding systems of inequalities can help you manage your work hours and earnings to meet your financial goals. For example, if you want to save money for a new bike, you can use inequalities to determine how many hours you need to work at different jobs to reach your savings target while staying within your available time. By setting up and solving these inequalities, you can make informed decisions about your work schedule and achieve your financial objectives.
