To solve this, we need to analyze the graph and the options provided.

The graph shows two green dots at $-12$ and $-4$ on a number line, which indicates that these values are solutions of the absolute value equation.

The general form of an absolute value equation is:

$|x - a| = b$

Where $a$ is the central point (midpoint between the two solutions) and $b$ is the distance from $a$ to one of the solutions.

1. Find the midpoint:

   • The two points are $-12$ and $-4$.
   • The midpoint is $\frac{-12 + (-4)}{2} = \frac{-16}{2} = -8$.

2. Find the distance:

   • The distance between $-12$ and $-8$ is $4$, which is the value of $b$.

Thus, the absolute value equation is:

$|x + 8| = 4$

The correct answer is the option $|x + 8| = 4$.

Would you like more details or have any questions?

Related questions:

1. What are other examples of absolute value equations with different midpoints?
2. How do we determine the solutions to an absolute value equation like $|x + 8| = 4$?
3. What happens if the distance between the two points is greater than 4?
4. Can you explain the geometric meaning of an absolute value function on a number line?
5. How do we solve an absolute value equation when the absolute value is equal to zero?

Tip:

Always remember that the absolute value represents the distance from a central point, so finding the midpoint is a key step to constructing the equation.