To reach the numbers 1 through 10 using the digits 5, 6, 9, and 8 with the operations allowed, let's explore possible solutions for each target number. I’ll try to find expressions for each result from 1 to 10.

Using 5, 6, 9, and 8 to make numbers 1 through 10:

1. 1:
   $(6 - 5) \times (9 - 8)$

2. 2:
   $(9 - 8) + (6 - 5)$

3. 3:
   $(9 - 6) - (8 - 5)$

4. 4:
   $8 - 6 + (9 - 5)$

5. 5:
   $9 - (8 - (6 - 5))$

6. 6:
   $5 + (9 - 8) + (6 - 5)$

7. 7:
   $9 - 8 + (6 - 5) + 5$

8. 8:
   $(9 - 5) + (6 - 8)$

9. 9:
   $9 + (6 - 5) - 8$

10. 10:
    $(8 + 6 - 5) + 1$

These are a few examples of how we might structure each expression. Let me know if you'd like any specific explanation on these solutions or if you'd like further examples with alternative solutions.

---

Would you like further clarification on any of these solutions?

Here are five related questions to extend this exercise:

1. Can you create alternative solutions for each number using different combinations?
2. How would you solve if we limited the operations to only addition and subtraction?
3. What happens if you add constraints like only using each number and operation once?
4. How might you adjust if decimals were allowed in your expressions?
5. What strategies could help systematically approach this kind of number puzzle?

Tip: When working on these kinds of puzzles, breaking down each target number into possible expressions and experimenting with operations is key.