How to Calculate Absolute Value

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. The absolute value of a positive number is itself; the absolute value of a negative number is its positive counterpart.

The Formula

|x| = x    if x โ‰ฅ 0
|x| = โˆ’x   if x < 0

Examples

  • |5| = 5
  • |โˆ’5| = 5
  • |0| = 0
  • |โˆ’3.7| = 3.7
  • |โˆš2| = โˆš2 (since โˆš2 > 0)

Absolute Value in Equations

To solve |x| = 7:

  • Case 1: x = 7
  • Case 2: x = โˆ’7

Both solutions work: |7| = 7 โœ“ and |โˆ’7| = 7 โœ“

To solve |2x โˆ’ 3| = 11:

  • Case 1: 2x โˆ’ 3 = 11 โ†’ 2x = 14 โ†’ x = 7
  • Case 2: 2x โˆ’ 3 = โˆ’11 โ†’ 2x = โˆ’8 โ†’ x = โˆ’4

Absolute Value Inequalities

|x| < 5 means โˆ’5 < x < 5 (x is within 5 units of zero) |x| > 5 means x < โˆ’5 or x > 5 (x is more than 5 units from zero)

Applications

  • Error calculation: |measured โˆ’ actual| gives the absolute error
  • Statistics: Mean absolute deviation uses |xแตข โˆ’ xฬ„|
  • Physics: Magnitude of velocity (speed) is |velocity|
  • Distance: Distance between two points on a number line is |a โˆ’ b|

Use our absolute value calculator for any expression.