absolute value

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The absolute value of a real number is its distance from zero on the number line, regardless of direction. It is represented by two vertical bars on either side of the number (e.g., |x|). The absolute value of a number is always non-negative, as it reflects magnitude without regard to direction. For any real number xx, the absolute value is defined as:

  • ∣x∣=x∣x∣=x if x≥0x≥0
  • ∣x∣=−x∣x∣=−x if x<0x<0

Key Concepts:

  • Non-Negativity: The absolute value of a number is always positive or zero because it represents distance, which cannot be negative.
  • Zero’s Absolute Value: The absolute value of zero is zero (|0| = 0), as zero is neither positive nor negative and represents no distance from itself.
  • Symmetry: The absolute value function is symmetric about the y-axis on a graph, indicating that |x| = |-x| for any real number xx.

Examples:

  1. Positive Number: The absolute value of 5 is 5.∣5∣=5∣5∣=5Explanation: Since 5 is already positive, its absolute value is itself.
  2. Negative Number: The absolute value of -3 is 3.∣−3∣=3∣−3∣=3Explanation: The distance of -3 from 0 is 3 units, hence its absolute value is 3.
  3. Zero: The absolute value of 0 is 0.∣0∣=0∣0∣=0Explanation: Zero is at no distance from itself, so its absolute value is 0.
  4. Variable Expression: For a negative variable, such as x=−8x=−8, the absolute value of xx is 8.∣−8∣=8∣−8∣=8Explanation: The distance of -8 from 0 on the number line is 8 units.
  5. Use in Equations: Solving ∣x−4∣=2∣x−4∣=2 results in two possible equations: x−4=2x−4=2 and x−4=−2x−4=−2, yielding x=6x=6 or x=2x=2.
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