To solve the equation \( 2y = x + |x| \), we need to consider the two possible cases for the absolute value function \( |x| \), depending on whether \( x \) is positive or negative:

### Case 1: \( x \geq 0 \)
When \( x \geq 0 \), the absolute value function simplifies as \( |x| = x \). So, the equation becomes:
\[
2y = x + x = 2x
\]
Dividing both sides by 2:
\[
y = x
\]
Thus, for \( x \geq 0 \), the solution is \( y = x \).

### Case 2: \( x < 0 \)
When \( x < 0 \), the absolute value function becomes \( |x| = -x \). So, the equation becomes:
\[
2y = x + (-x) = 0
\]
Thus, \( y = 0 \) when \( x < 0 \).

### Conclusion:
The piecewise solution to the equation is:
\[
y = \begin{cases} 
x & \text{if } x \geq 0 \\
0 & \text{if } x < 0
\end{cases}
\]
This defines a function that is linear for non-negative values of \( x \), and constant (equal to 0) for negative values of \( x \).

Would you like further details or have any questions about this? Here are five related questions to deepen your understanding:

1. How does the absolute value function affect different types of equations?
2. Can you sketch the graph of \( y = x \) for \( x \geq 0 \) and \( y = 0 \) for \( x < 0 \)?
3. How would this equation change if \( 2y = x - |x| \)?
4. How can you express this solution as a single equation using step functions?
5. What are the implications of piecewise functions in real-world applications?

**Tip**: When dealing with absolute values in equations, break the problem into cases depending on whether the expression inside the absolute value is positive or negative!

Explore how to solve the equation 2y = x + |x| by breaking it into cases using the absolute value function. This problem highlights key concepts in algebra and piecewise functions, providing a detailed step-by-step solution. Ideal for high school students in Grades 9-12.

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