The equation is \( |x| = |x - 2| \). To solve it, we need to break this into cases based on the properties of absolute values.

### Case 1: \( x \geq 2 \)
In this case, both \( |x| \) and \( |x - 2| \) simplify to:
\[
|x| = x \quad \text{and} \quad |x - 2| = x - 2
\]
Thus, the equation becomes:
\[
x = x - 2
\]
Subtracting \( x \) from both sides, we get:
\[
0 = -2
\]
This is a contradiction, so there is **no solution** in this case.

### Case 2: \( 0 \leq x < 2 \)
In this case, we have:
\[
|x| = x \quad \text{and} \quad |x - 2| = 2 - x
\]
Thus, the equation becomes:
\[
x = 2 - x
\]
Adding \( x \) to both sides, we get:
\[
2x = 2
\]
Dividing by 2, we get:
\[
x = 1
\]
Since \( 1 \) is within the range \( 0 \leq x < 2 \), this is a valid solution.

### Case 3: \( x < 0 \)
In this case, we have:
\[
|x| = -x \quad \text{and} \quad |x - 2| = 2 - x
\]
Thus, the equation becomes:
\[
-x = 2 - x
\]
Adding \( x \) to both sides, we get:
\[
0 = 2
\]
This is another contradiction, so there is **no solution** in this case.

### Final Solution:
The only solution is \( x = 1 \).

Would you like more details or have any questions? Here are some related questions to think about:

1. How do absolute values affect equations?
2. What are some methods to graph absolute value functions?
3. Can equations with absolute values have no solution?
4. What are other types of piecewise functions besides absolute value?
5. How does inequality change the approach to absolute value equations?

**Tip:** When solving absolute value equations, always consider breaking the problem into cases to handle different ranges of the variables.

Learn how to solve the absolute value equation |x| = |x - 2| with a detailed step-by-step solution. Explore key concepts like piecewise functions, absolute value properties, and algebraic manipulation. This problem is suitable for students in Grades 8-10.

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