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.
|a| = √a^{2}
|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
If |a| < b, then there are two cases:
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.