1. **State the problem:** Solve the equation $$\frac{5}{x-1} + \frac{1}{4-3x} = \frac{3}{6x-8}$$ for $x$. 2. **Rewrite the denominators:** Notice that $$4-3x = -(3x-4)$$ and $$6x-8 = 2(3x-4)$$. 3. **Find a common denominator:** The denominators are $x-1$, $4-3x$, and $6x-8$. Since $4-3x = -(3x-4)$ and $6x-8=2(3x-4)$, the common denominator can be taken as $2(x-1)(3x-4)$. 4. **Multiply both sides by the common denominator to clear fractions:** $$2(x-1)(3x-4) \times \left( \frac{5}{x-1} + \frac{1}{4-3x} \right) = 2(x-1)(3x-4) \times \frac{3}{6x-8}$$ 5. **Simplify each term:** - First term: $$2(x-1)(3x-4) \times \frac{5}{x-1} = 2(3x-4) \times 5 = 10(3x-4)$$ - Second term: $$2(x-1)(3x-4) \times \frac{1}{4-3x} = 2(x-1)(3x-4) \times \frac{1}{-(3x-4)} = -2(x-1)$$ - Right side: $$2(x-1)(3x-4) \times \frac{3}{2(3x-4)} = 3(x-1)$$ 6. **Write the equation after clearing denominators:** $$10(3x-4) - 2(x-1) = 3(x-1)$$ 7. **Expand terms:** $$30x - 40 - 2x + 2 = 3x - 3$$ 8. **Combine like terms:** $$30x - 2x = 28x$$ $$-40 + 2 = -38$$ So the equation becomes: $$28x - 38 = 3x - 3$$ 9. **Bring all terms to one side:** $$28x - 3x = -3 + 38$$ $$25x = 35$$ 10. **Solve for $x$:** $$x = \frac{35}{25} = \frac{7}{5} = 1.4$$ 11. **Check for restrictions:** Denominators cannot be zero: - $x-1 \neq 0 \Rightarrow x \neq 1$ - $4-3x \neq 0 \Rightarrow x \neq \frac{4}{3} \approx 1.333$ - $6x-8 \neq 0 \Rightarrow x \neq \frac{8}{6} = \frac{4}{3} \approx 1.333$ Since $x=1.4$ is not equal to any restricted value, it is valid. **Final answer:** $$\boxed{x = \frac{7}{5}}$$