The tutor defines absolute value in two ways.

The absolute value of x is normally written as |x|; a calculator might call it abs(x) but show it either way. It can be defined as follows:

Then we have |6| = 6, since 6≥0; |-3| = -(-3) = 3, since -3<0

Note: |x| ≥ 0 for all x.

Another definition of absolute value is

From this definition, we have |-4| = ((-4)²)^0.5 = (16)^0.5 = 4

By definition, square root is positive only, unless ± is written in front. Therefore, (9)^0.5 = 3, not -3. Hence, by the second definition as well, |x|≥0.

Each definition of absolute value has its advantage; for example, the first is probably easier to understand. The second definition has the advantage that it always does the same operations, regardless of input. Therefore, it’s more convenient to use in combination with other formulas.

I’ll be showing an application of the second definition in a coming post.

Source:

Roland, Larson E. and Robert P. Hostetler. Calculus, 3rd ed. Toronto: D C Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.