1. **State the problem:** Solve the equation $3x2^x=1$ for $x$. 2. **Rewrite the equation:** The equation is $3x2^x=1$. 3. **Isolate the exponential term:** Divide both sides by 3 to get $$x2^x=\frac{1}{3}$$ 4. **Analyze the equation:** This is a transcendental equation involving both $x$ and $2^x$. It cannot be solved algebraically using elementary functions. 5. **Use numerical methods or graphing:** To find $x$ such that $x2^x=\frac{1}{3}$, we can try values or use a graph. 6. **Check some values:** - For $x=0$, $0\cdot2^0=0$ (too small) - For $x=0.2$, $0.2\cdot2^{0.2}\approx0.2\cdot1.1487=0.2297$ (still less than $1/3$) - For $x=0.3$, $0.3\cdot2^{0.3}\approx0.3\cdot1.231=0.3693$ (slightly greater than $1/3$) 7. **By interpolation, the solution is near $x\approx0.28$**. **Final answer:** $x\approx0.28$