Lecture on Absolute Value Inequalities

Key Concepts

• Absolute Value Definition: The absolute value of a real number is the number's distance from zero on the number line.

Solving Absolute Value Inequalities

Example 1: (|x| < 3)

• Objective: Find numbers whose distance from zero is less than 3 units.
• Graphical Representation:
  • Open points at (-3) and (3).
  • Interval: ((-3, 3)).
• Interval Notation:
  • Use parentheses since endpoints are not included: ((-3, 3)).

Example 2: (|x| \leq 6)

• Objective: Find numbers whose distance from zero is less than or equal to 6 units.
• Graphical Representation:
  • Closed points at (-6) and (6).
  • Interval includes endpoints.
• Interval Notation:
  • Use square brackets since endpoints are included: ([-6, 6]).

Example 3: (-|x| \geq -6)

• Steps to Solve:
  • Multiply both sides by (-1) (Note: reverse inequality symbol).
  • Result: (|x| \leq 6).
• Solution Set:
  • Closed interval ([-6, 6]).
• Interval Notation:
  • Same as Example 2: ([-6, 6]).

Example 4: (2|x| < 10)

• Steps to Solve:
  • Divide both sides by 2.
  • Result: (|x| < 5).
• Solution Set:
  • Open interval between (-5) and (5).
• Interval Notation:
  • Use parentheses: ((-5, 5)).

Additional Notes

• Types of Intervals:
  • Open Interval: Endpoints are not included, use parentheses in notation.
  • Closed Interval: Endpoints are included, use square brackets in notation.