1. **State the problem:** Solve the equation $$\frac{4}{x+6} = \frac{5}{4x-8}$$ for $x$ and simplify the answer fully. 2. **Use the cross-multiplication formula:** For equations of the form $$\frac{a}{b} = \frac{c}{d}$$, cross-multiply to get $$a \cdot d = b \cdot c$$. 3. **Apply cross-multiplication:** $$4 \cdot (4x - 8) = 5 \cdot (x + 6)$$ 4. **Expand both sides:** $$16x - 32 = 5x + 30$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$16x - 5x = 30 + 32$$ 6. **Simplify both sides:** $$11x = 62$$ 7. **Solve for $x$ by dividing both sides by 11:** $$x = \frac{62}{11}$$ 8. **Show cancellation step:** $$x = \frac{\cancel{62}}{\cancel{11}}$$ (no common factors to cancel, so fraction is simplified) 9. **Final answer:** $$x = \frac{62}{11}$$ This is the fully simplified solution for $x$.