Understanding Absolute Values and Their Properties

This page provides a comprehensive overview of absolute values and their applications in mathematics. The content explores fundamental concepts and solving equations involving absolute values.

Definition: The absolute value of a negative number is its positive counterpart, represented as |x| = x if x ≥ 0, and |x| = -x if x < 0.

Example: |-5| = 5, |-11.3| = 11.3, |-1-√3| = 1+√3

Highlight: When solving equations with absolute values, consider both positive and negative possibilities for the solution.

Vocabulary: Key properties of absolute values include:

• |a| = a for positive numbers
• |a·b| = |a|·|b| for multiplication
• |a-b| = |b-a| for subtraction
• √(a²) = |a| for square roots

Example: Solving |x-3| = 6:

1. Consider both cases: x-3 = 6 or x-3 = -6
2. Solve each equation: x = 9 or x = -3
3. Both solutions are valid

The page concludes with multiple worked examples demonstrating the application of these concepts in solving various absolute value equations.