1. **State the problem:** We have a triangle with two sides of lengths 4.6 and 8.3, and we want to find the possible lengths of the third side $x$. 2. **Formula and rule:** The triangle inequality states that the length of any side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides. 3. **Apply the triangle inequality:** - The third side $x$ must be greater than the difference of the two given sides: $$x > |8.3 - 4.6|$$ - The third side $x$ must be less than the sum of the two given sides: $$x < 8.3 + 4.6$$ 4. **Calculate the values:** - Difference: $$|8.3 - 4.6| = 3.7$$ - Sum: $$8.3 + 4.6 = 12.9$$ 5. **Write the compound inequality:** $$3.7 < x < 12.9$$ 6. **Explanation:** This means the third side must be longer than 3.7 units but shorter than 12.9 units to form a valid triangle. **Final answer:** $$3.7 < x < 12.9$$