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:

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!