Absolute Value Inequalities

Types of Absolute Value Inequalities

1. Absolute Value Less Than:

   • Example: ( |x| < 5 )
   • Represents a distance that is no more than five units from zero.
   • Algebraic interpretation: ( x ) is between (-5) and (5).
   • Verification:
     • For ( x = -3 ), ( |x| = 3 < 5 )
     • For ( x = 4 ), ( |x| = 4 < 5 )
   • Rule: For absolute value less than or less than or equal to, it's a "between" scenario.
   • Interval Notation: ((-5, 5))

2. Absolute Value Greater Than:

   • Example: ( |x| > 5 )
   • Represents a distance that is more than five units from zero.
   • Algebraic interpretation: ( x ) can be greater than (5) or less than (-5).
   • Rule: For absolute value greater than or greater than or equal to, it's a "split" scenario.
   • Interval Notation:
     • ( (-\infty, -5) \cup (5, \infty) )

Key Concepts

• Absolute Value: Reflects the distance from zero.
• Between Scenario: When the absolute value inequality is less than, indicating a range between two values.
• Split Scenario: When the absolute value inequality is greater than, indicating two separate ranges outside a specific range.
• Interval Notation: Used to express the solution set of inequalities.
  • "Union" is the same as "OR" for the solution sets.

Conclusion

• Understand the two main cases:
  • Absolute value less than ("between" scenario)
  • Absolute value greater than ("split" scenario)
• Practice these concepts to solve problems involving absolute value inequalities effectively.

In the next video, examples will be provided to illustrate these concepts further.