1. **State the problem:** Solve the equation $$\frac{x + 9}{7} = \frac{5x - 8}{6}$$ for $x$. 2. **Use the cross-multiplication formula:** When two fractions are equal, their cross products are equal: $$ (x + 9) \times 6 = (5x - 8) \times 7 $$ 3. **Multiply both sides:** $$ 6(x + 9) = 7(5x - 8) $$ 4. **Distribute the multiplication:** $$ 6x + 54 = 35x - 56 $$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$ 6x - 35x = -56 - 54 $$ 6. **Simplify both sides:** $$ -29x = -110 $$ 7. **Divide both sides by $-29$ to solve for $x$:** $$ x = \frac{-110}{\cancel{-29}} \cancel{-1} = \frac{110}{29} $$ **Final answer:** $$ x = \frac{110}{29} $$