1. **State the problem:** Simplify the expression $$\frac{22w + 11}{8w^2 - 6w} - \frac{3}{2w}$$. 2. **Factor where possible:** - Numerator of first fraction: $$22w + 11 = 11(2w + 1)$$. - Denominator of first fraction: $$8w^2 - 6w = 2w(4w - 3)$$. 3. **Rewrite the expression:** $$\frac{11(2w + 1)}{2w(4w - 3)} - \frac{3}{2w}$$. 4. **Find common denominator:** The denominators are $$2w(4w - 3)$$ and $$2w$$. The least common denominator (LCD) is $$2w(4w - 3)$$. 5. **Rewrite second fraction with LCD:** $$\frac{3}{2w} = \frac{3(4w - 3)}{2w(4w - 3)}$$. 6. **Combine the fractions:** $$\frac{11(2w + 1) - 3(4w - 3)}{2w(4w - 3)}$$. 7. **Expand numerator:** $$11(2w + 1) = 22w + 11$$ $$3(4w - 3) = 12w - 9$$ 8. **Subtract in numerator:** $$22w + 11 - (12w - 9) = 22w + 11 - 12w + 9 = (22w - 12w) + (11 + 9) = 10w + 20$$. 9. **Factor numerator:** $$10w + 20 = 10(w + 2)$$. 10. **Final simplified expression:** $$\frac{10(w + 2)}{2w(4w - 3)}$$. 11. **Simplify fraction by dividing numerator and denominator by 2:** $$\frac{\cancel{10}(w + 2)}{\cancel{2}w(4w - 3)} = \frac{5(w + 2)}{w(4w - 3)}$$. **Answer:** $$\boxed{\frac{5(w + 2)}{w(4w - 3)}}$$