1. **State the problem:** Solve the inequality $2 + 2x \geq 3x - 1$ for $x$. 2. **Write down the inequality:** $$2 + 2x \geq 3x - 1$$ 3. **Isolate the variable terms on one side:** Subtract $2x$ from both sides: $$2 + \cancel{2x} \geq 3x - 1 - \cancel{2x}$$ which simplifies to $$2 \geq x - 1$$ 4. **Isolate $x$:** Add $1$ to both sides: $$2 + 1 \geq x - 1 + 1$$ which simplifies to $$3 \geq x$$ 5. **Rewrite the inequality:** $$x \leq 3$$ 6. **Interpretation:** The solution to the inequality is all real numbers $x$ such that $x$ is less than or equal to $3$. **Final answer:** $$\boxed{x \leq 3}$$