Use the graph to solve –|x – 2| + 4 ≤ 0.

In the xy plane the range of the both axes is negative 8 to 8 by increment of two. The curve is two lines starting from the point (2,4). The first line is passing through the point (6,0). The second line is passing through the point (-8,-6).

x ≤

x ≥

Given the problem \( -|x - 2| + 4 \leq 0 \), the goal is to find the values of \( x \) that satisfy this inequality.

The graph is described as two lines starting from the point \( (2, 4) \). This indicates that the graph corresponds to the function \( y = -|x - 2| + 4 \). Let's break down the graph and inequality-solving process:

### Step 1: Interpretation of the Graph
- The absolute value function is \( |x - 2| \), which creates two linear pieces starting at the vertex \( (2, 4) \).
- One line passes through \( (6, 0) \), implying that for \( x \geq 2 \), the slope is negative.
- Another line passes through \( (-8, -6) \), implying that for \( x < 2 \), the slope is positive.

### Step 2: Solve the Inequality
We need to solve the inequality \( -|x - 2| + 4 \leq 0 \).

First, rewrite the inequality:

\[
-|x - 2| \leq -4
\]
\[
|x - 2| \geq 4
\]

This means that the distance between \( x \) and 2 must be at least 4. Therefore, we have two cases:

1. \( x - 2 \geq 4 \) which simplifies to \( x \geq 6 \).
2. \( -(x - 2) \geq 4 \) which simplifies to \( x \leq -2 \).

### Final Answer:
- \( x \geq 6 \)
- \( x \leq -2 \)

### Conclusion:
The solution to \( -|x - 2| + 4 \leq 0 \) is \( x \geq 6 \) or \( x \leq -2 \).

Would you like more details on any part of this? Here are 5 related questions:

1. How do we derive inequalities from absolute value equations?
2. Can you explain how the graph of \( |x - 2| \) is constructed?
3. What does the solution set \( x \geq 6 \) or \( x \leq -2 \) look like on a number line?
4. How does shifting and flipping affect the graph of absolute value functions?
5. How would the inequality change if we replaced 0 with a positive number in \( -|x - 2| + 4 \leq c \)?

**Tip:** When solving absolute value inequalities, always consider both cases (positive and negative) for the expression inside the absolute value.

Use the graph to solve –|x – 2| + 4 ≤ 0. In the xy plane the range of the both axes is negative 8 to 8 by increment of two. The curve is two lines starting from the point (2,4). The first line is passing through the point (6,0). The second line is passing through the point (-8,-6).

Learn how to solve the inequality –|x – 2| + 4 ≤ 0 by analyzing the graph of two lines starting at (2, 4). Understand how absolute value functions and inequalities work through a step-by-step solution. Suitable for students in Grades 9-11.

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