Question: User: 2x=2*|x|-2
1. **State the problem:** Solve the equation $$2x = 2 \times |x| - 2$$ for $$x$$.
2. **Understand the absolute value:** Recall that $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$.
3. **Case 1: $$x \geq 0$$**
- Substitute $$|x| = x$$ into the equation:
$$2x = 2x - 2$$
- Simplify:
$$2x - 2x = -2$$
$$0 = -2$$
- This is a contradiction, so no solution for $$x \geq 0$$.
4. **Case 2: $$x < 0$$**
- Substitute $$|x| = -x$$ into the equation:
$$2x = 2(-x) - 2$$
$$2x = -2x - 2$$
- Add $$2x$$ to both sides:
$$2x + 2x = -2$$
$$4x = -2$$
- Divide both sides by 4:
$$x = \frac{-2}{4}$$
- Simplify:
$$x = \frac{\cancel{-2}}{\cancel{4}} = -\frac{1}{2}$$
- Check if $$x = -\frac{1}{2} < 0$$, which is true, so this solution is valid.
5. **Final answer:** $$x = -\frac{1}{2}$$.