1. **State the problem:** Simplify the expression $\frac{x}{2} \div \frac{1}{2}$.\n\n2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $a \div b = a \times \frac{1}{b}$.\n\n3. **Apply the rule:**\n$$\frac{x}{2} \div \frac{1}{2} = \frac{x}{2} \times \frac{2}{1}$$\n\n4. **Multiply the fractions:**\n$$= \frac{x \times 2}{2 \times 1} = \frac{2x}{2}$$\n\n5. **Simplify the fraction by canceling common factors:**\n$$= \frac{\cancel{2}x}{\cancel{2}} = x$$\n\n6. **Final answer:** The expression simplifies to $x$.