1. **Problem Statement:** We have a triangle with two sides of lengths 17.9 and 13.9, and we want to find the possible lengths of the third side $x$. 2. **Triangle Inequality Theorem:** For any triangle with sides $a$, $b$, and $c$, the length of any side must be less than the sum of the other two sides and greater than the difference of the other two sides. This gives us the inequalities: $$|a - b| < c < a + b$$ 3. **Apply to our sides:** Let the two known sides be $a = 17.9$ and $b = 13.9$, and the unknown side be $x$. Then: $$|17.9 - 13.9| < x < 17.9 + 13.9$$ 4. **Calculate the bounds:** $$|4.0| < x < 31.8$$ 5. **Final compound inequality:** $$4.0 < x < 31.8$$ This means the third side $x$ must be greater than 4.0 and less than 31.8 to form a valid triangle.