The problem is to analyze the function $y = |x+2| - |2x-4|$. The solution involves finding the critical points where the expressions inside the absolute values are zero and then analyzing the function in the intervals defined by these critical points. The critical points are found by solving $x+2 = 0$ and $2x-4 = 0$.

1. Problem Description

The problem is to analyze the function $y = |x+2| - |2x-4|$. The solution involves finding the critical points where the expressions inside the absolute values are zero and then analyzing the function in the intervals defined by these critical points. The critical points are found by solving $x+2 = 0$ and $2x-4 = 0$.

2. Solution Steps

First, we find the critical points.

$x+2 = 0 \implies x = -2$

$2x-4 = 0 \implies 2x = 4 \implies x = 2$

Now, we analyze the function in the three intervals defined by the critical points $x = -2$ and $x = 2$:

Interval 1: $x < -2$

In this interval, $x+2 < 0$ and $2x-4 < 0$. Therefore, $|x+2| = -(x+2)$ and $|2x-4| = -(2x-4)$.

$y = -(x+2) - (-(2x-4)) = -x-2 + 2x-4 = x - 6$

Interval 2: $-2 \le x < 2$

In this interval, $x+2 \ge 0$ and $2x-4 < 0$. Therefore, $|x+2| = x+2$ and $|2x-4| = -(2x-4)$.

$y = (x+2) - (-(2x-4)) = x+2 + 2x-4 = 3x - 2$

Interval 3: $x \ge 2$

In this interval, $x+2 > 0$ and $2x-4 \ge 0$. Therefore, $|x+2| = x+2$ and $|2x-4| = 2x-4$.

$y = (x+2) - (2x-4) = x+2 - 2x+4 = -x + 6$

So the function can be defined 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

The function is defined as:

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