Definition of absolute value
Definition of absolute value: the absolute value of a number is the distance the number is from zero
Look at the following two graphs:
The first one shows 6 located at a distance of 6 units from zero.
We write |6| = 6
The second one shows -8 located at a distance of 8 units from zero.
We write |-8| = 8
You can see from this that the absolute value of a number is always positive with the exception of taking the absolute value of 0 (|0| = 0)
Therefore, do not write |5| = -5 or |4| = -4 please!
|6| means distance from 0 to 6
|-8| means distance from 0 to -8
For |-8| = 8, you could also argue that to get 8, you have to take the negative of -8 since - - 8 = 8
So, | - 8| = - - 8 = 8
This observation helps us to come up with a formal definition of absolute value
| x | = x if x is positive or zero, but -x if x is negative
This definition is important to understand before solving absolute value equations or absolute value inequalities
Calculate the absolute value of the following numerical expressions
1) 4
^{2} − 4 × 2
2) -5 + 5 × 2 − 15
3) -8 + 2 × 5
1)
| -8 + 2 × 5 | = | -8 + 10 |
| -8 + 2 × 5 | = | 2 |
| -8 + 2 × 5 | = 2
2)
| 4
^{2} − 4 × 2| = | 16 − 4 × 2|
| 4
^{2} − 4 × 2| = | 16 − 8|
| 4
^{2} − 4 × 2| = | 8 |
| 4
^{2} − 4 × 2| = 8
3)
|-5 + 5 × 2 − 15| = | -5 + 10 − 15 |
|(-5 + 5 × 2 − 15)| = | 5 − 15 |
|(-5 + 5 × 2 − 15)| = | -15 |
|(-5 + 5 × 2 − 15)| = 15