1. **State the problem:** Solve the linear equation $4x + 3 = 1 - 2x$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. We can do this by adding or subtracting terms on both sides and then dividing if necessary. 3. **Add $2x$ to both sides to get all $x$ terms on one side:** $$4x + 2x + 3 = 1 - 2x + 2x$$ which simplifies to $$6x + 3 = 1$$ 4. **Subtract 3 from both sides to isolate the $x$ term:** $$6x + 3 - 3 = 1 - 3$$ which simplifies to $$6x = -2$$ 5. **Divide both sides by 6 to solve for $x$:** $$\frac{\cancel{6}x}{\cancel{6}} = \frac{-2}{6}$$ which simplifies to $$x = -\frac{1}{3}$$ 6. **Final answer:** $$x = -\frac{1}{3}$$