Equations - Absolute

Absolute Value

Definition and Properties

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value of a number \( x \) is denoted as \( |x| \).

Graphical representation of absolute value

Key Properties

  • \( |x| = x \) if \( x \geq 0 \)
  • \( |x| = -x \) if \( x < 0 \)
  • \( |a - b| \) represents the distance between numbers \( a \) and \( b \) on the number line
  • \( |x \cdot y| = |x| \cdot |y| \), and \( \left|\frac{x}{y}\right| = \frac{|x|}{|y|} \) (if \( y \neq 0 \))

Applications

  • Measuring distance and error in mathematics and physics.
  • Used in inequalities, absolute equations, and real-world problem solving.
  • Widely applied in computer science, economics, and statistics.
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