1. Stating the problem: Solve the equation $$|\frac{3 - 2x}{x + 6}| = 4$$ for $x$. 2. Breaking down the absolute value equation, we get two cases: - Case 1: $$\frac{3 - 2x}{x + 6} = 4$$ - Case 2: $$\frac{3 - 2x}{x + 6} = -4$$ 3. Case 1: Solve $$\frac{3 - 2x}{x + 6} = 4$$ Multiply both sides by $x + 6$ (noting that $x \neq -6$ to avoid division by zero): $$3 - 2x = 4(x + 6)$$ Simplify the right side: $$3 - 2x = 4x + 24$$ Bring all terms to one side: $$3 - 2x - 4x - 24 = 0$$ $$-6x - 21 = 0$$ Solve for $x$: $$-6x = 21$$ $$x = -\frac{21}{6} = -\frac{7}{2} = -3.5$$ 4. Case 2: Solve $$\frac{3 - 2x}{x + 6} = -4$$ Multiply both sides by $x + 6$ (again $x \neq -6$): $$3 - 2x = -4(x + 6)$$ Simplify the right side: $$3 - 2x = -4x - 24$$ Bring all terms to one side: $$3 - 2x + 4x + 24 = 0$$ $$2x + 27 = 0$$ Solve for $x$: $$2x = -27$$ $$x = -\frac{27}{2} = -13.5$$ 5. Check that neither solution makes the denominator zero: - For $x = -3.5$, denominator $-3.5 + 6 = 2.5 \neq 0$ - For $x = -13.5$, denominator $-13.5 + 6 = -7.5 \neq 0$ 6. Therefore, the solutions are $$\boxed{x = -3.5 \text{ or } x = -13.5}$$.