1. The problem is to solve the inequality $$\omega - 6(4x + 6) < \sqrt{-111}$$. 2. Note that the square root of a negative number, such as $$\sqrt{-111}$$, is not a real number. Since we are working in the real number system, $$\sqrt{-111}$$ is undefined. 3. Therefore, the inequality involves a term that is not real, making the inequality impossible to satisfy for any real $$x$$ or $$\omega$$. 4. Hence, there is no solution in the real numbers for the inequality $$\omega - 6(4x + 6) < \sqrt{-111}$$. 5. If complex numbers are considered, $$\sqrt{-111} = i\sqrt{111}$$, but the inequality would then be in the complex domain, which is not typically ordered and cannot be compared with $$<$$. Final answer: No real solution exists for the inequality.