The problem presents the function $y = |x+2| - |2x-4|$. We want to understand this function, perhaps by simplifying it or finding its values for different ranges of $x$.

1. Problem Description

The problem presents the function $y = |x+2| - |2x-4|$. We want to understand this function, perhaps by simplifying it or finding its values for different ranges of $x$.

2. Solution Steps

To analyze this function, we need to consider the cases when the expressions inside the absolute values are positive or negative.

Case 1: $x+2 \ge 0$ and $2x-4 \ge 0$. This means $x \ge -2$ and $x \ge 2$. Thus, $x \ge 2$.

In this case, $|x+2| = x+2$ and $|2x-4| = 2x-4$.

So, $y = (x+2) - (2x-4) = x+2 - 2x + 4 = -x + 6$.

Case 2: $x+2 \ge 0$ and $2x-4 < 0$. This means $x \ge -2$ and $x < 2$. Thus, $-2 \le x < 2$.

In this case, $|x+2| = x+2$ and $|2x-4| = -(2x-4) = -2x+4$.

So, $y = (x+2) - (-2x+4) = x+2 + 2x - 4 = 3x - 2$.

Case 3: $x+2 < 0$ and $2x-4 \ge 0$. This means $x < -2$ and $x \ge 2$. This case is impossible.

Case 4: $x+2 < 0$ and $2x-4 < 0$. This means $x < -2$ and $x < 2$. Thus, $x < -2$.

In this case, $|x+2| = -(x+2) = -x-2$ and $|2x-4| = -(2x-4) = -2x+4$.

So, $y = (-x-2) - (-2x+4) = -x-2 + 2x - 4 = x - 6$.

Therefore, the function can be written piecewise as:

$y = \begin{cases} x-6 & \text{if } x < -2 \\ 3x-2 & \text{if } -2 \le x < 2 \\ -x+6 & \text{if } x \ge 2 \end{cases}$

3. Final Answer

$y = \begin{cases} x-6 & \text{if } x < -2 \\ 3x-2 & \text{if } -2 \le x < 2 \\ -x+6 & \text{if } x \ge 2 \end{cases}$