1. **State the problem:** Solve the equation $$\frac{3}{N-1} + \frac{2N}{N+1} = 3$$ for $N$. 2. **Identify the formula and rules:** To solve this rational equation, find a common denominator and combine terms. Remember to check for values that make denominators zero: $N \neq 1$ and $N \neq -1$. 3. **Find the common denominator:** The denominators are $N-1$ and $N+1$, so the common denominator is $(N-1)(N+1)$. 4. **Rewrite each fraction with the common denominator:** $$\frac{3}{N-1} = \frac{3(N+1)}{(N-1)(N+1)}$$ $$\frac{2N}{N+1} = \frac{2N(N-1)}{(N+1)(N-1)}$$ 5. **Rewrite the equation:** $$\frac{3(N+1)}{(N-1)(N+1)} + \frac{2N(N-1)}{(N+1)(N-1)} = 3$$ 6. **Combine the left side:** $$\frac{3(N+1) + 2N(N-1)}{(N-1)(N+1)} = 3$$ 7. **Multiply both sides by $(N-1)(N+1)$ to clear denominators:** $$3(N+1) + 2N(N-1) = 3 \times (N-1)(N+1)$$ 8. **Expand each term:** $$3N + 3 + 2N^2 - 2N = 3(N^2 - 1)$$ 9. **Simplify left side:** $$2N^2 + (3N - 2N) + 3 = 3N^2 - 3$$ $$2N^2 + N + 3 = 3N^2 - 3$$ 10. **Bring all terms to one side:** $$2N^2 + N + 3 - 3N^2 + 3 = 0$$ 11. **Simplify:** $$-N^2 + N + 6 = 0$$ 12. **Multiply both sides by $-1$ to make the leading coefficient positive:** $$\cancel{-}N^2 + N + 6 = 0 \Rightarrow N^2 - N - 6 = 0$$ 13. **Factor the quadratic:** $$N^2 - N - 6 = (N - 3)(N + 2) = 0$$ 14. **Solve for $N$:** $$N - 3 = 0 \Rightarrow N = 3$$ $$N + 2 = 0 \Rightarrow N = -2$$ 15. **Check for excluded values:** $N \neq 1$, $N \neq -1$. Both $3$ and $-2$ are valid. **Final answer:** $$N = 3 \text{ or } N = -2$$