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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
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