To solve this problem, we need to apply the triangle inequality theorem, which states that for any triangle with sides $a$, $b$, and $c$, the following must hold true:

$a + b > c, \quad a + c > b, \quad b + c > a$

Given two sides of the triangle are 6 and 9, let the third side be $x$. The conditions become:

$6 + 9 > x \quad \text{(i.e., } 15 > x\text{)}$ $6 + x > 9 \quad \text{(i.e., } x > 3\text{)}$ $9 + x > 6 \quad \text{(i.e., } x > -3 \text{, but this is automatically true for positive values of } x\text{)}$

Thus, $x$ must satisfy: $3 < x < 15$

Now, check the given options:

• $2$ does not satisfy $x > 3$ → Not valid.
• $7$ satisfies $3 < 7 < 15$ → Valid.
• $12$ satisfies $3 < 12 < 15$ → Valid.
• $4$ satisfies $3 < 4 < 15$ → Valid.
• $10$ satisfies $3 < 10 < 15$ → Valid.
• $15$ does not satisfy $x < 15$ → Not valid.

So, the correct answers are 7, 12, 4, and 10, which you have correctly selected.

Would you like further details or clarification?

Here are 5 related questions to expand your understanding:

1. What are the implications of the triangle inequality theorem?
2. How would the solution change if one of the given sides was larger?
3. Can the third side ever be equal to the sum of the other two sides? Why or why not?
4. What happens if the two given sides are the same length?
5. How do the triangle inequality conditions change for different types of triangles (e.g., right, obtuse)?

Tip: Always apply the triangle inequality theorem to verify if three given lengths can form a triangle.