1. **State the problem:** We have a triangle with side lengths 13, 16, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall the triangle inequality theorem:** For any triangle with sides $a$, $b$, and $c$, the sum of the lengths of any two sides must be greater than the length of the remaining side. This gives three inequalities: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the theorem to our sides:** Let the sides be 13, 16, and $x$. Then: - $13 + 16 > x$ which simplifies to $29 > x$ - $13 + x > 16$ - $16 + x > 13$ 4. **Simplify the inequalities involving $x$:** - From $13 + x > 16$, subtract 13 from both sides: $$\cancel{13} + x > \cancel{16}$$ $$x > 3$$ - From $16 + x > 13$, subtract 16 from both sides: $$\cancel{16} + x > \cancel{13}$$ $$x > -3$$ Since side lengths must be positive, $x > 0$ is stricter than $x > -3$. 5. **Combine all inequalities:** $$3 < x < 29$$ 6. **Final answer:** The possible values of $x$ satisfy the inequality $$\boxed{3 < x < 29}$$ This means $x$ must be greater than 3 and less than 29 for the three lengths to form a triangle.