1. **State the problem:** Solve the equation $$\frac{7w}{w+2} - 8 = \frac{3w}{w+2}$$ for $w$. 2. **Identify the formula and rules:** This is a rational equation where terms have denominators involving $w+2$. To solve, we first eliminate the denominators by multiplying both sides by $w+2$, noting that $w \neq -2$ to avoid division by zero. 3. **Multiply both sides by $w+2$ to clear denominators:** $$\cancel{(w+2)}\left(\frac{7w}{\cancel{w+2}} - 8\right) = \cancel{(w+2)}\frac{3w}{\cancel{w+2}}$$ which simplifies to $$7w - 8(w+2) = 3w$$ 4. **Expand and simplify:** $$7w - 8w - 16 = 3w$$ 5. **Combine like terms:** $$-w - 16 = 3w$$ 6. **Add $w$ to both sides:** $$-w + w - 16 = 3w + w$$ $$-16 = 4w$$ 7. **Divide both sides by 4:** $$\frac{-16}{\cancel{4}} = \frac{4w}{\cancel{4}}$$ $$-4 = w$$ 8. **Check for restrictions:** Since $w \neq -2$, $w = -4$ is valid. **Final answer:** $$w = -4$$