IndietroAbsolute Value Equations and Inequalities (Chapter 2.5) – College Algebra Study Notes
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Absolute Value Equations and Inequalities
Introduction
This section explores the properties, graphs, and solution methods for absolute value equations and inequalities. Mastery of these concepts is essential for understanding a wide range of algebraic and real-world problems.
Absolute Value Functions
Definition and Properties
Absolute Value: The absolute value of a real number x, denoted , is its distance from zero on the number line. It is always non-negative.
Mathematical Definition:
if
if
Graph of : The graph is a "V" shape with its vertex at the origin (0,0).
Graphing Absolute Value Functions
To graph , reflect the portion of that is below the x-axis across the x-axis.
For , the graph is the standard absolute value graph shifted right by 2 units.
For , the graph is reflected and stretched compared to .
Example: Graph and . The graph of is identical to for , but for , it is reflected above the x-axis.
Example: Graph and . The graph of is a reflection of above the x-axis wherever is negative.
Solving Absolute Value Equations
General Approach
An equation of the form (where ) has two solutions: or .
If , there is no solution because absolute value cannot be negative.
Example 1
Solve
Set up two equations:
Solving:
Solution set:
Example 2
Solve
Since the right side is negative, there is no solution.
Example 3
Solve
Add 3 to both sides:
Set up two equations:
Solution set:
Example 4
Solve graphically, numerically, and symbolically.
Graphical Solution: Plot and ; the intersection points give the solutions.
Numerical Solution: Use a table of values to find where .
Symbolic Solution:
Solution set:
Solving Absolute Value Inequalities
General Approach
For (where ):
For (where ): or
If , has no solution; is always true for all real numbers except if .
Example 5
The diameter in inches of a cylindrical can must satisfy . Find the maximum and minimum acceptable values for .
Set up:
Add 3 to all parts:
Maximum value: 3.02, Minimum value: 2.98
Example 6
Solve
Set up:
Subtract 4:
Multiply by -1 (reverse inequalities): or
Solution set:
Example 7
Solve
Set up: or
Solution set:
Example 8
The thickness of a phone is about 9.33 mm, but actual phones may vary by not more than 0.02 mm. Write and solve an absolute value statement for this situation.
Statement:
Set up:
Add 9.33:
Interpretation: The actual thickness must be between 9.31 mm and 9.35 mm.
Example 9
Solve
Set up:
Subtract 3:
Divide by -2 (reverse inequalities): or
Solution set:
Example 10
Solve
Set up: or
Solution set:
Summary Table: Absolute Value Equations and Inequalities
Form | Solution Method | Number of Solutions |
|---|---|---|
() | or | 2 (unless , then 1) |
() | No solution | 0 |
() | Infinite (interval) | |
() | or | Infinite (union of intervals) |
Additional info: The above notes expand on the examples and methods shown in the slides, providing full algebraic steps, definitions, and interval notation for solution sets.
