1. **State the problem:** Simplify the expression $$\frac{3\sin x + 6\cos x}{5\sin x - 2\cos x}$$ and evaluate given that $$\tan x = 7 - 1 = 6$$. 2. **Recall the formula and rules:** We know that $$\tan x = \frac{\sin x}{\cos x}$$. 3. **Express numerator and denominator in terms of $$\cos x$$:** $$\frac{3\sin x + 6\cos x}{5\sin x - 2\cos x} = \frac{3(\sin x) + 6(\cos x)}{5(\sin x) - 2(\cos x)} = \frac{3\tan x \cos x + 6\cos x}{5\tan x \cos x - 2\cos x}$$ 4. **Factor out $$\cos x$$:** $$= \frac{\cos x (3\tan x + 6)}{\cos x (5\tan x - 2)}$$ 5. **Cancel $$\cos x$$:** $$= \frac{\cancel{\cos x} (3\tan x + 6)}{\cancel{\cos x} (5\tan x - 2)} = \frac{3\tan x + 6}{5\tan x - 2}$$ 6. **Substitute $$\tan x = 6$$:** $$= \frac{3(6) + 6}{5(6) - 2} = \frac{18 + 6}{30 - 2} = \frac{24}{28}$$ 7. **Simplify the fraction:** $$= \frac{\cancel{24}^{6}}{\cancel{28}^{7}} = \frac{6}{7}$$ **Final answer:** $$\frac{6}{7}$$