1. **State the problem:** Solve the equation $$\frac{6}{x+6} = \frac{4}{7}$$ for $x$. 2. **Formula and rules:** To solve equations involving fractions, we use cross-multiplication: if $$\frac{a}{b} = \frac{c}{d}$$ then $$a \times d = b \times c$$. 3. **Apply cross-multiplication:** $$6 \times 7 = (x+6) \times 4$$ 4. **Simplify both sides:** $$42 = 4(x+6)$$ 5. **Distribute 4 on the right side:** $$42 = 4x + 24$$ 6. **Isolate the variable term:** $$42 - 24 = 4x$$ $$18 = 4x$$ 7. **Divide both sides by 4 to solve for $x$:** $$x = \frac{18}{4}$$ 8. **Simplify the fraction:** $$x = \frac{\cancel{18}^{9}}{\cancel{4}^{2}} = \frac{9}{2}$$ **Final answer:** $$x = \frac{9}{2}$$