Properties of absolute value

Here is a comprehensive list of some important properties of absolute value with some examples showing how to use them.

1. Non-negativity: The absolute value is always positive.

|x| ≥ 0

Example: |3| = 3 and |-2| = 2

2. Symmetry:

|x| = |-x|

Example: |5| = 5 and |-5| = 5

|5| = |-5|

3. Idempotence:

||x|| = |x|

Example: ||5|| = |5| = 5

4.

|x - y| = |y - x|

Example: |2 - 5| = |5 - 2| = 3

|2 - 5| = |-3| = 3 and |5 - 2| = |3| = 3

5. Multiplicativity

|xy| = |x||y|

Example: |-5 × 6| = |-5| × |6|

|-5 × 6| = | -30| = 30

|-5| × |6| = 5 × 6 = 30

6. Preservation of division

|x / y| = |x| / |y| if y ≠ 0

Example: |-24 / 3| = |-24| / |3|

|-24 / 3| = |-8| = 8

|-24| / |3| = 24 / 3 = 8

7.

|x| = √(x²)

Example: |-4| = 4 and √[(-4)²] = √(16) = 4

8. Subadditivity

|x + y| ≤ |x| + |y|

Example:

a. |-4 + -5| ≤ |-4| + |-5|

|-9| ≤ 4 + 5

9 ≤ 9

b. |-4 + 5| ≤ |4| + |-5|

|1| ≤ 4 + 5

1 ≤ 9

9. 

|x - y| ≤ |x| + |y|

Example:

a. |5 - 1| ≤ |5| + |1|

|4| ≤ |5| + |1|

4 ≤ 5 + 1

4 ≤ 6

b. |5 - -1| ≤ |5| + |-1|

|5 + 1| ≤ |5| + |-1|

|6| ≤ |5| + |-1|

6 ≤ 5 + 1

6 ≤ 6

10. Triangle of inequality

|x - y| ≤ |x - z| + |z - y|

Example:

a. |9 - 8| ≤ |9 - 2| + |2 - 8|

|9 - 8| ≤ |9 - 2| + |2 - 8|

|1| ≤ |7| + |-6|

1 ≤ 7 + 6

1 ≤ 13

b. |9 - -8| ≤ |9 - 2| + |2 - -8|

|9 + 8| ≤ |7| + |2 + 8|

|17| ≤ |7| + |10|

17 ≤ 7 + 10

17 ≤ 17

11. Positive-definiteness

|x| = 0 if and only if x  = 0

Example: |0| = 0

12. Identity of indiscernibles 

|x - y| = 0 if and only if x  =  y

Example: |12 - 12| = 0, then 12 = 12

13. 

|x|² = x²

Example: |-5|² = 5²

14. 

|x| - |y| ≤ |x - y|

Example:

a. |8| - |4| ≤ |8 - 4|

|8| - |4| ≤ |4|

8 - 4 ≤ 4

4 ≤ 4

b. |8| - |-4| ≤ |8 - -4|

|8| - |-4| ≤ |8 + 4|

8 - 4 ≤ |12|

4 ≤ 12

15. Absolute value inequalities

Let k be a positive real number

|x| ≥ k is equivalent to x ≤ -k or x ≥ k

Example: |x| ≥ 2 is equivalent to x ≤ -2 or x ≥ 2

16. Absolute value inequalities

Let k be a positive real number

|x| ≤ k is equivalent to -k ≤ x ≤ k 

Example: |x| ≤ 2 is equivalent to -2 ≤ x ≤ 2