https://file.aiquickdraw.com/mathbot/file/1728654023893zyq663cj.jpg

The number line in the image shows two blue dots located at \( x = -3 \) and \( x = 7 \). We are tasked with finding an absolute value equation that corresponds to these two points.

The general form of an absolute value equation is:

\[
|x - c| = d
\]

where \( c \) is the midpoint between the two values, and \( d \) is the distance from the midpoint to either point.

### Step 1: Find the midpoint \( c \)
To find the midpoint between \( x = -3 \) and \( x = 7 \), we use the formula for the midpoint:

\[
c = \frac{-3 + 7}{2} = \frac{4}{2} = 2
\]

So, the midpoint is \( x = 2 \).

### Step 2: Find the distance \( d \)
The distance between the midpoint and either of the points is:

\[
d = |7 - 2| = 5
\]

Thus, the absolute value equation that represents the solutions is:

\[
|x - 2| = 5
\]

This equation describes all points that are 5 units away from the midpoint \( x = 2 \), which includes \( x = -3 \) and \( x = 7 \).

Would you like further clarification on any steps, or need additional details?

Here are 5 related questions you might explore:
1. How would you solve the equation \( |x - 2| = 5 \) to get the solutions?
2. What is the geometric interpretation of an absolute value equation?
3. How would the equation change if the points on the number line were closer together?
4. Can absolute value equations have more than two solutions? If so, when?
5. How does the equation \( |x - 2| = 5 \) compare to \( |x + 3| = 5 \)?

**Tip:** The midpoint of two points on a number line can always be found by averaging the two values.

Write an equation that involves absolute value and has the solutions shown on the number line. Use x for your variable.

Learn how to write an absolute value equation based on a number line with solutions at x = -3 and x = 7. This example involves using the midpoint and distance to form the equation |x - 2| = 5. Suitable for students in Grades 7-10.

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