To solve this, let's test each option to see if it satisfies both inequalities:

1. James needs to work fewer than 10 hours in total: $x + y < 10$
2. James needs to earn at least $56: $5x + 8y \geq 56$

We'll check each option.

Option A: 3 hours mowing lawns and 5 hours at the clothing store

• Total hours: $3 + 5 = 8$ (satisfies $x + y < 10$)
• Earnings: $5(3) + 8(5) = 15 + 40 = 55$ (does not satisfy $5x + 8y \geq 56$)
• Option A does not work.

Option B: 5 hours mowing lawns and 3 hours at the clothing store

• Total hours: $5 + 3 = 8$ (satisfies $x + y < 10$)
• Earnings: $5(5) + 8(3) = 25 + 24 = 49$ (does not satisfy $5x + 8y \geq 56$)
• Option B does not work.

Option C: 2 hours mowing lawns and 7 hours at the clothing store

• Total hours: $2 + 7 = 9$ (satisfies $x + y < 10$)
• Earnings: $5(2) + 8(7) = 10 + 56 = 66$ (satisfies $5x + 8y \geq 56$)
• Option C works.

Option D: 4 hours mowing lawns and 5 hours at the clothing store

• Total hours: $4 + 5 = 9$ (satisfies $x + y < 10$)
• Earnings: $5(4) + 8(5) = 20 + 40 = 60$ (satisfies $5x + 8y \geq 56$)
• Option D works.

Option E: 8 hours mowing lawns and 2 hours at the clothing store

• Total hours: $8 + 2 = 10$ (does not satisfy $x + y < 10$)
• Option E does not work.

Final Answer:

The 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 clarification on any of these steps?

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Here are some related questions to consider:

1. If James worked at another job that paid $10 per hour, how would the inequalities change?
2. What would the solution be if James needed to earn at least $70 per week instead of $56?
3. How would the result change if he could work exactly 10 hours?
4. What if the mowing job paid $7 per hour and the clothing job $6 per hour?
5. Can we graph the system of inequalities to visualize the solution?

Tip: When solving inequalities with multiple constraints, test each possible solution against all given conditions to confirm it fits.