The inequality provided is $|x| > 2$. To solve and graph this inequality, we need to understand what the absolute value inequality means:

1. $|x| > 2$ implies that the distance of $x$ from 0 is greater than 2. In other words, $x$ is either less than -2 or greater than 2.

   • This splits into two inequalities: $x < -2 \quad \text{or} \quad x > 2$

In terms of a number line:

• The solution consists of two regions: one to the left of -2 and one to the right of 2.
• The values at $x = -2$ and $x = 2$ are not included in the solution, so we use open circles at -2 and 2.
• The graph should have arrows extending leftward from -2 and rightward from 2, representing all values less than -2 and greater than 2.

Correct Answer:

The correct graph will have open circles at -2 and 2, with shaded lines extending outwards from these points (to the left of -2 and to the right of 2).

From the image:

• The bottom-right graph correctly represents the inequality $|x| > 2$, with open circles at -2 and 2 and shading to the left of -2 and right of 2.

Would you like more details or have any questions?

Here are five related questions to expand on this concept:

1. How would you graph $|x| \geq 2$ instead of $|x| > 2$?
2. What does the graph of $|x| < 2$ look like?
3. How do you solve inequalities with absolute values inside a quadratic equation?
4. How do you represent compound inequalities on a number line?
5. What real-life scenarios can be modeled by absolute value inequalities?

Tip: When graphing absolute value inequalities, always remember to split the inequality into two cases and check whether to include or exclude the boundary points (using open or closed circles).