1. **State the problem:** We have a rectangle with sides $2x$ and $4x$, and the area $A$ satisfies the inequality $32 < A < 392$. 2. **Formula for area of a rectangle:** $$A = \text{length} \times \text{width}$$ Here, length = $4x$ and width = $2x$. 3. **Express the area in terms of $x$:** $$A = 4x \times 2x = 8x^2$$ 4. **Set up the inequality with the area:** $$32 < 8x^2 < 392$$ 5. **Divide the entire inequality by 8 to isolate $x^2$:** $$\frac{32}{8} < \cancel{8}x^2 / \cancel{8} < \frac{392}{8}$$ $$4 < x^2 < 49$$ 6. **Take the square root of all parts, remembering to consider both positive and negative roots:** $$-\sqrt{49} < x < -\sqrt{4} \quad \text{or} \quad \sqrt{4} < x < \sqrt{49}$$ $$-7 < x < -2 \quad \text{or} \quad 2 < x < 7$$ 7. **Interpretation:** Since $x$ represents a length scale, we consider only positive values. **Final answer:** $$2 < x < 7$$