To solve this problem, we apply the triangle inequality theorem, which states:

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given:

• Two sides are $8 \, \text{cm}$ and $14 \, \text{cm}$.
• The third side has a length of $x \, \text{cm}$.

We write the inequalities for the triangle inequality as:

1. $8 + 14 > x$
2. $8 + x > 14$
3. $14 + x > 8$

Solve each inequality:

1. $8 + 14 > x$ gives $x < 22$.
2. $8 + x > 14$ gives $x > 6$.
3. $14 + x > 8$ is always true because $x > 6$ already satisfies it.

Conclusion:

The possible values of $x$ must satisfy: $6 < x < 22$

Answer:

• Write $6$ in the first box.
• Write $22$ in the second box.

Would you like me to provide more explanation or solve a similar problem? Here are 5 related questions for practice:

1. What happens if the side lengths of a triangle violate the triangle inequality?
2. How can you verify if three given side lengths can form a triangle?
3. Can a triangle exist if two of its sides are equal in length? What are the restrictions for the third side?
4. How does the triangle inequality theorem apply to right triangles?
5. How do the triangle inequalities change if one side length is a variable expression?

Tip: When solving triangle inequality problems, always check each of the three inequality conditions carefully!