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.