Key Properties:

• Non-negative: |x| ≥ 0 for all real numbers x
• Symmetric: |x| = |-x|
• Triangle Inequality: |x + y| ≤ |x| + |y|
• Multiplicative: |xy| = |x| × |y|
• Zero Property: |x| = 0 if and only if x = 0

Common Examples:

• |5| = 5 (positive number stays positive)
• |-5| = 5 (negative number becomes positive)
• |0| = 0 (zero stays zero)
• |-3.14| = 3.14 (negative decimal becomes positive)
• |2/3| = 2/3 (positive fraction stays positive)
• |-7/4| = 7/4 (negative fraction becomes positive)

Solving Absolute Value Equations:

For |x| = a (where a > 0):

• x = a or x = -a
• Example: |x| = 5 → x = 5 or x = -5

For |x + b| = a (where a > 0):

• x + b = a or x + b = -a
• x = a - b or x = -a - b
• Example: |x - 3| = 5 → x - 3 = 5 or x - 3 = -5 → x = 8 or x = -2

Absolute Value Inequalities:

For |x| < a (where a > 0):

• -a < x < a
• Example: |x| < 3 → -3 < x < 3

For |x| > a (where a > 0):

• x < -a or x > a
• Example: |x| > 3 → x < -3 or x > 3

Real-World Applications:

• Distance: |x₁ - x₂| gives distance between two points
• Error Analysis: |measured - actual| gives absolute error
• Temperature: |T - T₀| gives temperature deviation
• Finance: |profit - target| gives deviation from goal
• Physics: |velocity| gives speed (magnitude of velocity)