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 

Recent Articles

  1. Pressure in a Liquid - How to Derive Formula

    Jul 02, 22 07:53 AM

    This lesson will show clearly how to derive the formula to find the pressure in a liquid.

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.