1. **State the problem:** Solve the equation $-2 \cdot 7^x = 3$ for $x$. 2. **Isolate the exponential term:** Divide both sides by $-2$ to isolate $7^x$. $$-2 \cdot 7^x = 3$$ $$\cancel{-2} \cdot 7^x = \frac{3}{\cancel{-2}}$$ $$7^x = -\frac{3}{2}$$ 3. **Analyze the result:** The expression $7^x$ represents an exponential function with base 7, which is always positive for any real $x$. 4. **Conclusion:** Since $7^x$ cannot be negative, there is no real solution to the equation $-2 \cdot 7^x = 3$. **Final answer:** No real solution exists.