1. **State the problem:** Simplify the expression $\frac{-x}{2x}$. 2. **Recall the rule:** When simplifying fractions, you can cancel common factors in the numerator and denominator as long as they are not zero. 3. **Simplify the expression:** $$\frac{-x}{2x} = \frac{-1 \cdot x}{2 \cdot x}$$ Since $x \neq 0$, cancel $x$ from numerator and denominator: $$= \frac{-1}{2}$$ 4. **Final answer:** The simplified form of $\frac{-x}{2x}$ is $\boxed{-\frac{1}{2}}$. This means the expression equals negative one half for all $x \neq 0$.