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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.

Two sides of a triangle are given: 6 and 9. Find the range of the possible third side.

Learn how to find the range of the third side in a triangle with two given sides (6 and 9) using the Triangle Inequality Theorem. This problem is suitable for students in Grades 6-8.

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