1. **Problem:** Evaluate $\int_4^2 x \, dx$ given $\int_2^4 x \, dx = 6$. 2. **Recall the property of definite integrals:** $$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$$ 3. Applying this property: $$\int_4^2 x \, dx = -\int_2^4 x \, dx = -6$$ 4. **Answer:** $\boxed{-6}$ --- **Explanation:** When the limits of integration are reversed, the integral changes sign. Since $\int_2^4 x \, dx = 6$, reversing the limits from 4 to 2 gives $-6$.