Solve Equation 30 A425E5

1. **State the problem:** Solve the equation $$-\frac{6}{2}\left(5 + \frac{x}{3}\right) = \frac{1}{3}\left(4 - \frac{4x}{2}\right) + \frac{3x}{2}$$ for $x$. 2. **Rewrite the equation:** Simplify the coefficients: $$-3\left(5 + \frac{x}{3}\right) = \frac{1}{3}\left(4 - 2x\right) + \frac{3x}{2}$$ 3. **Distribute terms:** $$-3 \times 5 - 3 \times \frac{x}{3} = \frac{1}{3} \times 4 - \frac{1}{3} \times 2x + \frac{3x}{2}$$ $$-15 - x = \frac{4}{3} - \frac{2x}{3} + \frac{3x}{2}$$ 4. **Combine like terms on the right:** Find common denominator for $-\frac{2x}{3} + \frac{3x}{2}$ which is 6: $$-\frac{2x}{3} = -\frac{4x}{6}, \quad \frac{3x}{2} = \frac{9x}{6}$$ So, $$-\frac{4x}{6} + \frac{9x}{6} = \frac{5x}{6}$$ 5. **Rewrite equation:** $$-15 - x = \frac{4}{3} + \frac{5x}{6}$$ 6. **Move all $x$ terms to one side and constants to the other:** Add $x$ to both sides: $$-15 = \frac{4}{3} + \frac{5x}{6} + x$$ Rewrite $x$ as $\frac{6x}{6}$: $$-15 = \frac{4}{3} + \frac{5x}{6} + \frac{6x}{6} = \frac{4}{3} + \frac{11x}{6}$$ 7. **Subtract $\frac{4}{3}$ from both sides:** $$-15 - \frac{4}{3} = \frac{11x}{6}$$ Convert $-15$ to thirds: $$-15 = -\frac{45}{3}$$ So, $$-\frac{45}{3} - \frac{4}{3} = -\frac{49}{3} = \frac{11x}{6}$$ 8. **Solve for $x$:** Multiply both sides by $\frac{6}{11}$: $$x = -\frac{49}{3} \times \frac{6}{11} = -\frac{49 \times 6}{3 \times 11}$$ Simplify numerator and denominator: $$-\frac{49 \times \cancel{6}}{\cancel{3} \times 11} = -\frac{49 \times 2}{11} = -\frac{98}{11}$$ 9. **Final answer:** $$x = -\frac{98}{11}$$ Note: The user provided answer is $x = -\frac{38}{11}$, but the correct solution from the steps is $x = -\frac{98}{11}$.