Question: Solve the following equation for x. Express your answer in the simplest form.
2 - 2(-x + 1) = -3 + \frac{1}{2} (4x + 6)
1. **State the problem:** Solve the equation $$2 - 2(-x + 1) = -3 + \frac{1}{2} (4x + 6)$$ for $$x$$.
2. **Distribute the terms:**
$$2 - 2(-x) - 2(1) = -3 + \frac{1}{2} (4x + 6)$$
which simplifies to
$$2 + 2x - 2 = -3 + \frac{1}{2} (4x + 6)$$
3. **Simplify the left side:**
$$2 - 2 = 0$$
so the equation becomes
$$2x = -3 + \frac{1}{2} (4x + 6)$$
4. **Distribute the fraction on the right side:**
$$2x = -3 + \frac{1}{2} \times 4x + \frac{1}{2} \times 6$$
which is
$$2x = -3 + 2x + 3$$
5. **Simplify the right side:**
$$-3 + 3 = 0$$
so
$$2x = 2x$$
6. **Subtract $$2x$$ from both sides:**
$$2x - \cancel{2x} = 2x - \cancel{2x}$$
which simplifies to
$$0 = 0$$
7. **Interpretation:** The equation is true for all values of $$x$$, so the solution is all real numbers.
**Final answer:** $$\boxed{\text{all real numbers}}$$