To find the range of the possible third side of a triangle given two sides, we apply the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given sides $a = 6$ and $b = 9$, let the third side be $x$. According to the Triangle Inequality Theorem, we have the following conditions:

1. $a + b > x$ → $6 + 9 > x$ → $15 > x$
2. $a + x > b$ → $6 + x > 9$ → $x > 3$
3. $b + x > a$ → $9 + x > 6$ → $x > -3$ (this inequality is redundant since $x > 3$ already covers this condition)

From these conditions, we conclude that the third side $x$ must satisfy: $3 < x < 15$

Thus, the range for the possible third side is between 3 and 15.

Would you like further details or clarifications? Here are some related questions:

1. How is the Triangle Inequality Theorem used in different triangle types?
2. What happens if the triangle inequality is violated?
3. Can we form a triangle with any three side lengths?
4. What are some real-world applications of the Triangle Inequality Theorem?
5. How does this theorem extend to other polygons beyond triangles?

Tip: Always check all three conditions of the Triangle Inequality Theorem when solving for possible side lengths.