1. **State the problem:** Simplify the expression $\frac{1}{2} \times (3x+1) \times (2x-3)$. 2. **Recall the distributive property:** To multiply expressions, multiply each term in one expression by each term in the other. 3. **Multiply the binomials:** $$ (3x+1)(2x-3) = 3x \times 2x + 3x \times (-3) + 1 \times 2x + 1 \times (-3) $$ $$ = 6x^2 - 9x + 2x - 3 $$ $$ = 6x^2 - 7x - 3 $$ 4. **Multiply by $\frac{1}{2}$:** $$ \frac{1}{2} \times (6x^2 - 7x - 3) = \frac{1}{2} \times 6x^2 - \frac{1}{2} \times 7x - \frac{1}{2} \times 3 $$ $$ = 3x^2 - \frac{7}{2}x - \frac{3}{2} $$ 5. **Final simplified expression:** $$ 3x^2 - \frac{7}{2}x - \frac{3}{2} $$ This is the simplified form of the original expression.