1. **State the problem:** We have a triangle with side lengths 10, 14, 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: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the triangle inequality to our sides:** - $10 + 14 > x$ which simplifies to $$24 > x$$ - $10 + x > 14$ which simplifies to $$x > 4$$ - $14 + x > 10$ which simplifies to $$x > -4$$ (this is always true since side lengths are positive) 4. **Combine the inequalities:** The meaningful inequalities are: $$4 < x < 24$$ 5. **Write the final answer as a single inequality using $x$ only once:** $$4 < x < 24$$ This means $x$ must be greater than 4 and less than 24 to form a valid triangle.