1. **Problem statement:** We have a triangle with two sides of lengths 9 and 14, and we want to find the possible lengths of the third side $x$. 2. **Formula and rule:** The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. For sides $a$, $b$, and $c$, this means: $$a + b > c, \quad a + c > b, \quad b + c > a$$ 3. **Apply the triangle inequality:** Let the sides be 9, 14, and $x$. Then: - $9 + 14 > x \implies 23 > x$ - $9 + x > 14 \implies x > 5$ - $14 + x > 9 \implies x > -5$ (which is always true since side lengths are positive) 4. **Combine inequalities:** The possible lengths for $x$ satisfy: $$5 < x < 23$$ 5. **Explanation:** The third side must be longer than 5 but shorter than 23 to form a valid triangle with sides 9 and 14. **Final answer:** The compound inequality describing the possible lengths for the third side $x$ is: $$5 < x < 23$$