1. **State the problem:** Solve the equation $$\frac{2x - 4}{3} - \frac{3x + 2}{4} = \frac{x - 5}{6} - 3$$ given $x = 12$. 2. **Substitute $x = 12$ into the equation:** $$\frac{2(12) - 4}{3} - \frac{3(12) + 2}{4} = \frac{12 - 5}{6} - 3$$ 3. **Simplify each term:** - Left numerator 1: $2(12) - 4 = 24 - 4 = 20$ - Left numerator 2: $3(12) + 2 = 36 + 2 = 38$ - Right numerator: $12 - 5 = 7$ 4. **Rewrite the equation with simplified numerators:** $$\frac{20}{3} - \frac{38}{4} = \frac{7}{6} - 3$$ 5. **Convert all terms to have a common denominator to simplify:** - Find the least common denominator (LCD) of 3, 4, and 6, which is 12. - Convert each fraction: - $\frac{20}{3} = \frac{20 \times 4}{12} = \frac{80}{12}$ - $\frac{38}{4} = \frac{38 \times 3}{12} = \frac{114}{12}$ - $\frac{7}{6} = \frac{7 \times 2}{12} = \frac{14}{12}$ 6. **Rewrite the equation with common denominators:** $$\frac{80}{12} - \frac{114}{12} = \frac{14}{12} - 3$$ 7. **Combine the left side:** $$\frac{80 - 114}{12} = \frac{-34}{12} = -\frac{17}{6}$$ 8. **Rewrite the right side:** $$\frac{14}{12} - 3 = \frac{14}{12} - \frac{36}{12} = -\frac{22}{12} = -\frac{11}{6}$$ 9. **Compare both sides:** Left side: $-\frac{17}{6}$ Right side: $-\frac{11}{6}$ 10. **Conclusion:** Since $-\frac{17}{6} \neq -\frac{11}{6}$, the equation is not true for $x = 12$. **Final answer:** $x = 12$ is not a solution to the equation.