# Absolute Value

The absolute value of a real number, a, is written as |a|. It is the equivalent number if the number is already positive, but the opposite if the number is negative. |5| = 5    |-5 | = 5    |0| = 0

|x| = 2    strong>x = −2    x = 2

|x| < 2    − 2 < x < 2    x (−2, 2 )

|x |> 2            x < −2 or x > 2     (−∞, −2 ) (2, +∞)

|x −2 | < 5     − 5 < x − 2 < 5

− 5 + 2 < x <  5 + 2     − 3 < x < 7

− 5 + 2 < x < 5 + 2     − 3 < x < 7

### Properties of the Absolute Value

1 Opposite numbers have equal absolute value.

|a| = |−a|

|5| = |−5| = 5

2 The absolute value of a product is equal to the product of the absolute values of the factors.

|a · b| = |a| ·|b|

|5 · (−2)| = |5| · |(−2)|      |− 10| = |5| · |2|     10 = 10

3 The absolute value of a sum is less than or equal to the sum of the absolute values of the addends.

|a + b| ≤ |a| + |b|

|5 + (−2)| ≤ |5| + |(−2)|      |3| = |5| + |2|     3 ≤ 7

## Distance

The distance between two real numbers a and b, which writes d(a, b), is defined as the absolute value of the difference in both numbers:

d(a, b) = |b − a|

The distance between −5 and 4 is:

d(−5, 4) = |4 − (−5)| = |4 + 5| = |9|