Dr. Byrne
Fall 2010
Math 112
Worksheet 3.5: Solving Absolute Value Inequalities
Three Algebraic Properties if a>0
Recall:
x means ______________________________.
8 = sssssssssss
 11 =
sssssssssssss
x  a  x  a or x  a
x  a  a  x  a
x  a  x  a or x  a
Each of these properties can be understood geometrically:
Algebraic Property
x  a  x  a or x  a
x  a  a  x  a
Geometric Explanation
There are two ways on the real number line x can be distance
a from 0. Either x = -a or x = a.
This means the distance from x and 0 must be less than a.
This specifies the interval from –a to a on the real line.
This means the distance from x and 0 must be greater than a.
This specifies the interval outside of [-a, a] on the real line.
x  a  x  a or x  a
x  a  x  a or x  a
x  a  a  x  a
x  a  x  a or x  a
Solve the following for x.
x  7.5
x7 5
2 x  7  3.5
1
x  6  2  10
3
3x  24
8  4x  7
Exam Hint: If you forget the algebraic properties for ‘<’ and ‘>’, you can always solve the absolute
value as though it was an equality and then test points in each of the different intervals.
First solve as an equality:
Example:
x2  6
Then test points on the real line to find
the intervals where the inequality holds:
x2 6
Precalculus Tip:
xa b
means that the distance between x and a is equal to b.
xa b
means that the distance between x and a is less than b.