The distance between x and 4 is less than 2.

Question

The distance between x and 4 is less than 2. Also, how would you graph it.

Answer 1

|x-4|< 2

 

The "absolute value" operation is actually a way to measure the distance between two points on a number line.   Since the absolute value turns negative numbers into positive numbers, we need to consider cases where x-4 may be negative as well as positive:

 

x-4 < 2

x < 6

 

x-4 > -2

x > 2

 

Hence the solution is 2 < x < 6.  The graph of the inequality is shown below.  The points 2 and 6 are represented with open circles to indicate they are NOT part of the solution (if they were, we'd use filled-in circles).  The orange section is the solution, excepting the points 2 and 6.

 

<---|-----|-----o-----|-----|-----|-----o-----|-----|--->

      0       1       2       3      4       5       6      7       8

 

All of the points within the orange section are within 2 units of the number 4.