To monitor only the X coordinate of the object's position, you could do this:

Vector3 lastPos;
Transform obj; // drag the object to monitor here
float threshold = 0.0f; // minimum displacement to recognize a 

void Start(){
  lastPos = obj.position;
}

void Update(){
  Vector3 offset = obj.position - lastPos;
  if (offset.x > threshold){
    lastPos = obj.position; // update lastPos
    // code to execute when X is getting bigger
  }
  else
  if (offset.x < -threshold){
    lastPos = obj.position; // update lastPos
    // code to execute when X is getting smaller 
  }
}

This will execute the code once when the X distance from the lastPos exceeds threshold. This also updates lastPos when the code is executed, thus the distance from this point must again exceed threshold for the code to be executed again.
This script is intended to be attached to another object; if you plan to use it in the monitored object, replace obj. in the text by transform. (or assign obj = transform; at Start).