Absolute Value (or modulus) is the value of a number without regard to its sign. The absolute value of any number is always positive or zero.

|a| = a, if a ≥ 0 

|a| = -a, if a < 0

For example, |x| = 5 indicates that x = 5 or –5.

Properties of Absolute Value

  • |a| = √a2

  • |a| ≥ 0

  • |a| = 0 if and only if a = 0

  • |-a| = |a|

  • |a+b| ≤ |a| + |b|

  • |a| ≤ b iff -b ≤ a ≤ b

  • |a| ≥ b iff a ≤ -b or b ≤ a

Absolute Value and Inequalities

If |a| < b, then there are two cases:

  1. a < b
  2. a < -b

In the first case, remove the absolute value brackets and solve. In the second case, remove the absolute value brackets, put a negative sign on the other side of the inequality, and change the sign.