Absolute Values Lecture Notes

Definition of Absolute Value

• Absolute Value: Distance from zero to a number on a number line.
  • Example:
    • |3| = 3 (3 units away from zero)
    • |-3| = 3 (also 3 units away from zero)
• Practical Definition: Magnitude of a number disregarding its sign.
  • Positive numbers remain the same.
  • Negative numbers become positive by dropping the negative sign.

Properties of Absolute Values

• Absolute values obey properties similar to other operations:
  • |AB| = |A| * |B| (Product property)
  • |A/B| = |A| / |B| (Division property)
• Important to note:
  • |A + B| ≠ |A| + |B|
    • Example: Let A = 1, B = -1
    • |1 + (-1)| = |0| = 0
    • |1| + |-1| = 1 + 1 = 2*

Solving Equations with Absolute Values

• Example 1: |X| = 2
  • Solutions: X = 2 or X = -2
• Example 2: |2X - 1| = 5
  • Split into two equations:
    1. 2X - 1 = 5
    2. 2X - 1 = -5
  • Solve each:
    • For 2X - 1 = 5:
      • 2X = 6
      • X = 3
    • For 2X - 1 = -5:
      • 2X = -4
      • X = -2
• Conclusion: Two solutions are X = 3 and X = -2.

Graphing Absolute Values

• Graph of Y = X:

  • Line with slope of 1, Y-intercept of 0.

• Graph of Y = |X|:

  • Positive values remain unchanged.
  • Negative values reflect across the X-axis.
  • Results in a V-shaped graph.
    • Example points:
      • X = -1 → Y = 1
      • X = -2 → Y = 2

Summary

• Basics of absolute values covered:
  • Definitions, properties, solving equations, and graphing.

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