What does the absolute value of a real number represent?
Example Solution: |x+2| ≥ 0 is true for all real numbers. Example Solution: |x+2| < -5 has no solution.
Practice makes perfect
Recall that the absolute value of a real number |x| is the distance from 0 on the number line. We can use this to construct an absolute value inequality whose solutions are all real numbers. We can start with the absolute value of any mathematical expression involving the variable x.
|x+2|
Now, since this quantity represents a distance regardless of the value of x, it will always be grater or at least equal to 0.
|x+2| ≥ 0
The absolute value inequality shown above is, therefore, true for every real number x. Thinking in a similar way, since we know that there are no negative distances the absolute of any mathematical expression will never be less than 0. Therefore, the absolute value inequality shown below will have no solution.
|x+2| < -5
Notice that we could have used the absolute value of any mathematical expression and the results would have been the same. Therefore, there are infinitely many possible solutions, and these are just example solutions.
