If the distance between the points (3, x) and (-2, -6) is 13 units, then find the value of x.

Answer

To solve the problem, we need to find the value of \( x \) given that the distance between the points \( (3, x) \) and \( (-2, -6) \) is 13 units. We will use the distance formula to set up our equation.

### Step-by-step Solution:

1. **Write the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. **Identify the Points**:
Here, the points are \( (3, x) \) and \( (-2, -6) \). Thus:
- \( x_1 = 3 \), \( y_1 = x \)
- \( x_2 = -2 \), \( y_2 = -6 \)

3. **Substitute the Points into the Distance Formula**:
We know the distance is 13 units, so we set up the equation:
\[ 13 = \sqrt{((-2) - 3)^2 + ((-6) - x)^2} \]

4. **Simplify the Equation**:
Calculate \( (-2) - 3 \):
\[ -2 - 3 = -5 \]
Therefore, \( (-2 - 3)^2 = (-5)^2 = 25 \).

Now, substitute this back into the equation:
\[ 13 = \sqrt{25 + (-6 - x)^2} \]

5. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ 13^2 = 25 + (-6 - x)^2 \]
\[ 169 = 25 + (-6 - x)^2 \]

6. **Isolate the Squared Term**:
Subtract 25 from both sides:
\[ 169 - 25 = (-6 - x)^2 \]
\[ 144 = (-6 - x)^2 \]

7. **Take the Square Root**:
Taking the square root of both sides gives:
\[ \sqrt{144} = | -6 - x | \]
\[ 12 = | -6 - x | \]

8. **Set Up Two Equations**:
This absolute value equation leads to two cases:
- Case 1: \( -6 - x = 12 \)
- Case 2: \( -6 - x = -12 \)

9. **Solve Each Case**:
- **Case 1**:
\[ -6 - x = 12 \implies -x = 12 + 6 \implies -x = 18 \implies x = -18 \]
- **Case 2**:
\[ -6 - x = -12 \implies -x = -12 + 6 \implies -x = -6 \implies x = 6 \]

10. **Final Values**:
The values of \( x \) are:
\[ x = -18 \quad \text{or} \quad x = 6 \]