Let's analyze the options to find the correct equation for the given situation.

The general form of an absolute value equation for distance:

If the midpoint is $m$ and the distance from the midpoint to either point is $d$, the equation can be written as: $|x - m| = d$ This represents the distance from $x$ to the midpoint $m$, which equals $d$.

Analyzing the Options:

• Top Left:

  • Midpoint = 6, Distance = 3.
  • Equation given: $|x + 6| = 3$.
  • This equation is incorrect because it should be $|x - 6| = 3$, not $|x + 6| = 3$, based on the general form.

• Top Right:

  • Midpoint = 3, Distance = 6.
  • Equation given: $|x - 3| = 6$.
  • This equation is correct because it follows the correct form $|x - m| = d$, where $m = 3$ and $d = 6$.

• Bottom Left (Selected):

  • Midpoint = 6, Distance = 3.
  • Equation given: $|x - 6| = 3$.
  • This equation is correct because it matches the form $|x - m| = d$ with $m = 6$ and $d = 3$.

• Bottom Right:

  • Midpoint = 3, Distance = 6.
  • Equation given: $|x + 3| = 6$.
  • This equation is incorrect because it should be $|x - 3| = 6$, not $|x + 3| = 6$.

Conclusion:

The correct answer is the Bottom Left option, where the equation is $|x - 6| = 3$.

Would you like further details on this, or do you have any other questions? Here are five related questions you might consider:

1. How do we derive absolute value equations for intervals?
2. What is the geometric interpretation of an absolute value equation?
3. How do we solve absolute value equations algebraically?
4. What are some common mistakes to avoid when forming absolute value equations?
5. How does the concept of midpoint apply to coordinate geometry?

Tip: Always remember to check the sign in absolute value equations. The correct form is $|x - \text{midpoint}| = \text{distance}$.