1. **State the problem:** Solve the rational equation $$\frac{9}{2} = \frac{7x - 8}{x + 1}$$ and check for extraneous solutions. 2. **Formula and rules:** To solve rational equations, multiply both sides by the least common denominator (LCD) to eliminate fractions. Then solve the resulting equation. Check for values that make any denominator zero, as these are excluded. 3. **Identify the LCD:** The denominators are 2 and $x+1$. The LCD is $2(x+1)$. 4. **Multiply both sides by the LCD:** $$\frac{9}{2} \times 2(x+1) = \frac{7x - 8}{x + 1} \times 2(x+1)$$ 5. **Simplify:** $$9(x+1) = 2(7x - 8)$$ 6. **Expand both sides:** $$9x + 9 = 14x - 16$$ 7. **Isolate variable terms:** $$9x + 9 = 14x - 16$$ $$9 + 16 = 14x - 9x$$ $$25 = 5x$$ 8. **Solve for $x$:** $$x = \frac{25}{5} = 5$$ 9. **Check for extraneous solutions:** Denominator $x+1 \neq 0 \Rightarrow x \neq -1$. Since $x=5$ is valid, it is the solution. **Final answer:** $$\boxed{5}$$