1. **State the problem:** Solve for $x$ in the equation $$\frac{6}{x+2} = \frac{1}{4} + \frac{x-7}{x+2}.$$ 2. **Combine terms on the right side:** We want to combine the fractions on the right side. The common denominator is $4(x+2)$. Rewrite each term: $$\frac{1}{4} = \frac{x+2}{4(x+2)}, \quad \frac{x-7}{x+2} = \frac{4(x-7)}{4(x+2)}.$$ So, $$\frac{1}{4} + \frac{x-7}{x+2} = \frac{x+2}{4(x+2)} + \frac{4(x-7)}{4(x+2)} = \frac{x+2 + 4(x-7)}{4(x+2)}.$$ 3. **Simplify numerator:** $$x+2 + 4(x-7) = x + 2 + 4x - 28 = 5x - 26.$$ 4. **Rewrite the equation:** $$\frac{6}{x+2} = \frac{5x - 26}{4(x+2)}.$$ 5. **Multiply both sides by $4(x+2)$ to clear denominators:** $$4(x+2) \times \frac{6}{x+2} = 4(x+2) \times \frac{5x - 26}{4(x+2)}.$$ Simplify: $$4 \cancel{(x+2)} \times \frac{6}{\cancel{x+2}} = \cancel{4(x+2)} \times \frac{5x - 26}{\cancel{4(x+2)}}$$ $$4 \times 6 = 5x - 26$$ $$24 = 5x - 26.$$ 6. **Solve for $x$:** Add 26 to both sides: $$24 + 26 = 5x$$ $$50 = 5x.$$ Divide both sides by 5: $$\frac{\cancel{50}}{\cancel{5}} = \frac{5x}{5}$$ $$10 = x.$$ **Final answer:** $$x = 10.$$