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Inequalities
With Absolute Value
The
absoluate value of a, denoted by IaI,
measures the distance from the origin to a
on a real number line.
For example, |-3| =3=|3|, the distance from the origin to either
-3 or 3 is 3.
Now if |x| = 3, the only solutions to this equation are -3 or 3.
If |x| < 3, this means that the distance from 0 to x is within
3 units and its solution set is the set of all real numbers x that
lies between -3 and 3.
If |x| >3, this means that the distance from0 to x is greater
than 3 units and its solution set is the set of all real numbers
x that is either less than -3 or or greater then 3. (See figure
below for illustration.)
Properties
of equations and inequalities involving the absolute value |x|:
Ex
1: Solve the inequality: 
Sol:
 
Ex
2: Solve the inequality: 
Sol:
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