Problem Analysis:
The problem involves setting up a system of inequalities based on the constraints provided. Let's break it down step by step:
#### Given Information:
1.
Total hours constraint: You can work no more than 41 hours per week at your two jobs.
- Let \( h \) be the number of hours spent on housecleaning.
- Let \( s \) be the number of hours spent on the sales job.
- Therefore, the total hours worked must satisfy:
\[
h + s \leq 41
\]
2.
Earnings constraint: You need to earn at least \$254 per week to pay your bills.
- Housecleaning pays \$5 per hour, so the earnings from housecleaning are \( 5h \).
- The sales job pays \$8 per hour, so the earnings from the sales job are \( 8s \).
- Therefore, the total earnings must satisfy:
\[
5h + 8s \geq 254
\]
3.
Non-negativity constraints: You cannot work negative hours, so:
\[
h \geq 0 \quad \text{and} \quad s \geq 0
\]
#### System of Inequalities:
Combining all the constraints, the system of inequalities is:
\[
\begin{cases}
h + s \leq 41 \\
5h + 8s \geq 254 \\
h \geq 0 \\
s \geq 0
\end{cases}
\]
Explanation of Each Inequality:
1.
Hours Constraint (\( h + s \leq 41 \)):
- This inequality ensures that the total number of hours worked at both jobs does not exceed 41 hours per week.
2.
Money Constraint (\( 5h + 8s \geq 254 \)):
- This inequality ensures that the total earnings from both jobs are at least \$254 per week. The earnings from housecleaning are \( 5h \), and the earnings from the sales job are \( 8s \).
3.
Non-Negativity Constraints (\( h \geq 0 \) and \( s \geq 0 \)):
- These inequalities ensure that the number of hours worked at each job is non-negative. It is not possible to work negative hours.
Final Answer:
The system of inequalities that represents the problem is:
\[
\boxed{
\begin{cases}
h + s \leq 41 \\
5h + 8s \geq 254 \\
h \geq 0 \\
s \geq 0
\end{cases}
}
\]
Parent Tip: Review the logic above to help your child master the concept of system of linear inequalities word problems worksheet.
