Question

(\frac{22w + 11}{8w^2 - 6w}-\frac{3}{2w})

Explanation:

Step1: Factor denominators

Factor \(8w^2 - 6w\) as \(2w(4w - 3)\), and the second fraction has denominator \(2w\). So the expression becomes \(\frac{22w + 11}{2w(4w - 3)} - \frac{3}{2w}\).

Step2: Find common denominator

The common denominator is \(2w(4w - 3)\). Rewrite the second fraction: \(\frac{3(4w - 3)}{2w(4w - 3)}\).

Step3: Subtract the fractions

Subtract the numerators: \(\frac{22w + 11 - 3(4w - 3)}{2w(4w - 3)}\).

Step4: Simplify numerator

Expand and simplify the numerator: \(22w + 11 - 12w + 9 = 10w + 20 = 10(w + 2)\).

Step5: Simplify the fraction

The expression becomes \(\frac{10(w + 2)}{2w(4w - 3)}\), then simplify by dividing numerator and denominator by 2: \(\frac{5(w + 2)}{w(4w - 3)}\) or \(\frac{5w + 10}{4w^2 - 3w}\).

Answer:

\(\frac{5w + 10}{4w^2 - 3w}\) (or \(\frac{5(w + 2)}{w(4w - 3)}\))