1. **State the problem:** We have a triangle with side lengths 11, 12, 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. This gives three inequalities: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the theorem to our sides:** - $11 + 12 > x$ which simplifies to $$23 > x$$ - $11 + x > 12$ which simplifies to $$x > 1$$ - $12 + x > 11$ which simplifies to $$x > -1$$ 4. **Combine the inequalities:** Since $x > 1$ is stronger than $x > -1$, the two lower bounds combine to $x > 1$. The upper bound is $x < 23$. 5. **Write the final inequality:** $$1 < x < 23$$ This means $x$ must be greater than 1 and less than 23 for the three lengths to form a triangle.