To find which combinations of hours James can work, let's evaluate each option against both constraints:

1. Total Hours Constraint:
   $x + y < 10$ — James can work less than 10 hours in total.

2. Earnings Constraint:
   $5x + 8y \geq 56$ — James needs to earn at least $56 per week.

Now, we will test each option:

Option A: $x = 3$, $y = 5$

• Total Hours Check: $x + y = 3 + 5 = 8$, which is less than 10 (satisfies the first constraint).
• Earnings Check: $5x + 8y = 5(3) + 8(5) = 15 + 40 = 55$, which is less than 56 (does not satisfy the second constraint).
• Conclusion: Not a valid option.

Option B: $x = 5$, $y = 3$

• Total Hours Check: $x + y = 5 + 3 = 8$, which is less than 10 (satisfies the first constraint).
• Earnings Check: $5x + 8y = 5(5) + 8(3) = 25 + 24 = 49$, which is less than 56 (does not satisfy the second constraint).
• Conclusion: Not a valid option.

Option C: $x = 2$, $y = 7$

• Total Hours Check: $x + y = 2 + 7 = 9$, which is less than 10 (satisfies the first constraint).
• Earnings Check: $5x + 8y = 5(2) + 8(7) = 10 + 56 = 66$, which is greater than 56 (satisfies the second constraint).
• Conclusion: Valid option.

Option D: $x = 4$, $y = 5$

• Total Hours Check: $x + y = 4 + 5 = 9$, which is less than 10 (satisfies the first constraint).
• Earnings Check: $5x + 8y = 5(4) + 8(5) = 20 + 40 = 60$, which is greater than 56 (satisfies the second constraint).
• Conclusion: Valid option.

Option E: $x = 8$, $y = 2$

• Total Hours Check: $x + y = 8 + 2 = 10$, which is not less than 10 (does not satisfy the first constraint).
• Earnings Check: Not needed, as it already fails the total hours constraint.
• Conclusion: Not a valid option.

Answer:

The two correct responses are:

• C. James can work 2 hours mowing lawns and 7 hours at the clothing store.
• D. James can work 4 hours mowing lawns and 5 hours at the clothing store.

Would you like further details or have any questions?

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Follow-up questions:

1. How would the answer change if James’s earnings goal increased to $60 per week?
2. If James could work up to 12 hours per week, would any additional options become valid?
3. What if James's clothing store job started paying $10 per hour? How would this affect possible combinations?
4. How would the scenario change if James could only work a maximum of 4 hours at each job?
5. Could James work only at the clothing store to meet his earnings goal?

Tip: When working with inequalities, testing each condition separately can simplify finding valid solutions.