# 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| **