College Algebra Lecture Notes
Section 1.3
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Section 1.3: Absolute Value Equations and Inequalities
Big Idea: Absolute values can be thought of as stating the distance from a point. When
combined with inequalities, they result in compound inequalities.
Big Skill: You should be able to solve absolute value equations and inequalities using the
Property of Absolute Value Equations.
Examples of when you might want to use absolute value equations or inequalities:
 State the two x values that are 5 units away from x = 1
 if x is greater than 1, then x – 1 = 5
 if x is less than 1, then 1 – x = 5  x – 1 = -5
 x – 1 = 5 or x – 1 = -5
 |x – 1| = 5
 If you want to be avoid a parade route, you must be more than 5 blocks east or west of
block #22
 location > block #27 or location < block #17
 |location – block #22| > 5
A. SOLVING ABSOLUTE VALUE EQUATIONS
 Isolate the absolute value using the Properties of Equality, then finish the solution using
the Properties of Absolute Values:
Property of Absolute Value Equations
If X, represents an algebraic expression, and k is a positive real number,
then |X| = k
implies X = k or X = -k
Multiplicative Property of Absolute Value
If A, and B represent algebraic expressions,
then |A B| = |A||B|
Practice:
2
1. Solve 5  x  9  8 .
3
College Algebra Lecture Notes
Section 1.3
2. Solve 3x  12  13 .
B. SOLVING “LESS THAN” ABSOLUTE VALUE INEQUALITIES
 Use the below Property of Absolute Values Inequality:
Property 1 of Absolute Value Inequalities
If X, represents an algebraic expression, and k is a positive real number,
then |X| < k
implies –k < X < k
Practice:
3. Solve
3x  2
4
1  4 .
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College Algebra Lecture Notes
Section 1.3
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C. SOLVING “GREATER THAN” ABSOLUTE VALUE INEQUALITIES
 Use the below Property of Absolute Values Inequality:
Property 2 of Absolute Value Inequalities
If X, represents an algebraic expression, and k is a positive real number,
then |X| > k
implies X < -k or X > k
Practice:
4. Solve 
1
x
3   2 .
4
2
D. APPLICATIONS INVOLVING ABSOLUTE VALUE