A while ago, I did a video introducing absolute values and what they were but I did not go through some very useful properties and rules of absolute values. In this video I go over these properties and show, for example, that sqrt(a2) = absolute value of a. I go over some other useful properties so make sure you watch the video!
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Absolute Values: Properties
(1). √(a2) = |a|
(2). |ab| = |a||b|
(3). |a/b| = |a|/|b| where b ≠ 0
(4). |an| = |a|n
For the following rules, suppose a > 0, then:
(5). |x| = a if and only if x = ± a
(6). |x| < a if and only if – a < x < a
(7). |x| > a if and only if x > a or x < – a
Same as Rule 2 because division can be changed into multiplication:
Let c = 1/b where b ≠ 0.
|a/b| = |ac| and this then follows the same procedure as Rule 2.
(-a)n and (+a)n can only differ in sign but have the same absolute value: ±|a|n.
Thus, |a|n just means that the result is positive, n-power doesn't affect the sign, thus it is equal to |an|.
Rules 5, 6, 7