On the number line above, the tick marks are equally spaced. If a,c,d,s,v, and x are the coordinates of the indicated points, which of the following must be true?

|x−v|
The relationship |x−v|

A (|a−v|<d) YOUR ANSWER

While this expression suggests that the distance between points a and v is less than d, it may not hold true universally. The positions of a and v can vary significantly depending on their actual coordinates on the number line, making this statement potentially false.

B (|x−v|<d) CORRECT

This statement accurately reflects the condition set by the coordinates on the number line. Given the equal spacing of tick marks, the distance between x and v must always be less than d, as x is positioned within the limits defined by point d relative to v.

C (|c−v|<d) YOUR ANSWER

Similar to choice A, this statement cannot be universally true. The point c may be located anywhere on the number line, and its distance from v could either be less than or greater than d depending on their respective positions.

D (|d−v|<d) YOUR ANSWER

This statement suggests that the distance between d and v is less than d itself, which is a contradiction. Since the absolute difference cannot be less than itself, this choice is inherently false and cannot be true.

Conclusion

The correct assertion is that the distance |x−v| must be less than d, a reflection of the consistent spacing of tick marks on the number line. Other options either rely on variable positions of points or contradict mathematical principles, underscoring the importance of understanding spatial relationships in this context.