If the distance between the points (x, 0) and (-7, 0) be 10 units, then the possible values of x are

Options

A. 3 and 17
B. `-3` and 17
C. 3 and `-17`
D. `-3` and `-17`

Answer

To solve the problem, we need to find the possible values of \( x \) given that the distance between the points \( (x, 0) \) and \( (-7, 0) \) is 10 units.

### Step-by-step Solution:

1. **Understand the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, the points are \( (x, 0) \) and \( (-7, 0) \).

2. **Set Up the Distance Equation**:
Here, we can substitute \( (x_1, y_1) = (x, 0) \) and \( (x_2, y_2) = (-7, 0) \):
\[ d = \sqrt{((-7) - x)^2 + (0 - 0)^2} \]
This simplifies to:
\[ d = \sqrt{(-7 - x)^2} \]

3. **Simplify the Distance**:
Since the square root of a square is the absolute value, we have:
\[ d = | -7 - x | = | x + 7 | \]

4. **Set the Distance Equal to 10**:
We know that the distance is 10 units, so we set up the equation:
\[ | x + 7 | = 10 \]

5. **Solve the Absolute Value Equation**:
The absolute value equation \( |A| = B \) gives us two cases:
- Case 1: \( x + 7 = 10 \)
- Case 2: \( x + 7 = -10 \)

**For Case 1**:
\[ x + 7 = 10 \implies x = 10 - 7 = 3 \]

**For Case 2**:
\[ x + 7 = -10 \implies x = -10 - 7 = -17 \]

6. **Conclusion**:
The possible values of \( x \) are \( 3 \) and \( -17 \).

### Final Answer:
The possible values of \( x \) are \( 3 \) and \( -17 \).

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