1. **Problem:** Resolve the fraction $$\frac{3x - 2}{2x^2 - x}$$ into partial fractions. 2. **Step 1: Factor the denominator.** $$2x^2 - x = x(2x - 1)$$ 3. **Step 2: Set up the partial fractions.** $$\frac{3x - 2}{x(2x - 1)} = \frac{A}{x} + \frac{B}{2x - 1}$$ 4. **Step 3: Multiply both sides by the denominator to clear fractions.** $$3x - 2 = A(2x - 1) + Bx$$ 5. **Step 4: Expand and group terms.** $$3x - 2 = 2Ax - A + Bx = (2A + B)x - A$$ 6. **Step 5: Equate coefficients of like terms.** - Coefficient of $$x$$: $$3 = 2A + B$$ - Constant term: $$-2 = -A$$ 7. **Step 6: Solve for $$A$$ and $$B$$.** From constant term: $$A = 2$$ Substitute into $$3 = 2A + B$$: $$3 = 2(2) + B \Rightarrow 3 = 4 + B \Rightarrow B = -1$$ 8. **Final answer:** $$\frac{3x - 2}{2x^2 - x} = \frac{2}{x} - \frac{1}{2x - 1}$$