Absolute values can be interpreted as the distance from a center point.
|x-8| < 1
Practice makes perfect
Absolute values can be interpreted as the distance away from a midpoint. For one-variable absolute value inequalities, this distance can be represented by two points on a number line. These are the endpoints of the given compound inequality.
Because our inequality needs a distance from a midpoint, we should find the halfway point between the endpoints. We can do this by calculating their mean.
Mean=7+ 9/2= 8
Now we need to find the distance between this midpoint and each of the endpoints. To do this, we will find the difference between each of the endpoints and the midpoint.
distance= 8-7=9- 8=1
We see that both 7 and 9 are 1 unit away from 8. Notice that the given inequality is an and inequality, and the symbols used can be read as less than.
7< x < 9
7 is less than x
and
x is less than 9
To write the given compound inequality as an absolute value inequality, we can show that the difference between a number x and the midpoint is less than the distance we found above.
|x- midpoint| < distance
|x- 8| < 1
