Solve the Absolute Value Equation: |2x + |x - 2|| = 1

To solve the equation $|2x + |x - 2|| = 1$, we need to consider different cases based on the expression within the absolute value signs.

Case 1: $x \geq 2$

• Here, $|x - 2| = x - 2$.
• The equation becomes $|2x + x - 2| = 1$ or $|3x - 2| = 1$.
• This gives two sub-cases to solve:
  • Sub-case 1.1: $3x - 2 = 1$ leads to $3x = 3$, so $x = 1$. However, $x = 1$ does not satisfy $x \geq 2$. Discard this solution.
  • Sub-case 1.2: $3x - 2 = -1$ leads to $3x = 1$, so $x = \frac{1}{3}$. But $x = \frac{1}{3}$ does not satisfy $x \geq 2$. Discard this solution.

Case 2: $x < 2$

• In this case, $|x - 2| = 2 - x$.
• The equation transforms into $|2x + 2 - x| = 1$, then simplifies to $|x + 2| = 1$.
• This leads to two sub-cases:
  • Sub-case 2.1: $x + 2 = 1$ leads to $x = -1$. Since $x = -1 < 2$, it is a valid solution.
  • Sub-case 2.2: $x + 2 = -1$ leads to $x = -3$. Again, $x = -3 < 2$ holds, so this is also a valid solution.

Thus, the solutions to the equation are $x = -1$ and $x = -3$.

Therefore, the correct answers are $x = -1$ and $x = -3$.