1. **Problem statement:** We have a triangle with two sides of lengths 4.3 and 0.6, 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 4.3, 0.6, and $x$. Then: - $4.3 + 0.6 > x \implies 4.9 > x$ - $4.3 + x > 0.6 \implies x > 0.6 - 4.3 \implies x > -3.7$ - $0.6 + x > 4.3 \implies x > 4.3 - 0.6 \implies x > 3.7$ 4. **Simplify inequalities:** Since $x$ must be positive (side length), and $x > 3.7$ is stronger than $x > -3.7$, the compound inequality is: $$3.7 < x < 4.9$$ 5. **Interpretation:** The third side $x$ must be greater than 3.7 and less than 4.9 to form a valid triangle with the other two sides. **Final answer:** $$3.7 < x < 4.9$$