## Absolute Value

*Absolute Values*represented using two |'s are common in Algebra. They are meant to signify the number's distance from 0 on a number line. So, if the number is negative, it becomes positive. And if the number was positive, it remains positive:

|8| = 8 and |−8| = 8

A formal definition of |x| is:

If x ≥ 0, then |x| = x

If x ≤ 0, then |x| = −x

If x ≤ 0, then |x| = −x

Graphical Representation of |x| using formal definition provided above:

Properties:

Example #1:

Find x for |x| =19

As the absolute value of either +19 or −19 will result in 19, it follows that x = −19 or 19

Solution #1:

As the absolute value of either +19 or −19 will result in 19, it follows that x = −19 or 19

Example #2:

Find x for
|3x + 9| = 6

Knowing that the contents within the absolute value symbol should equal 4, it follows that:

3x + 9 = 6 and 3x +9 = −6

∴ x = −1, −5

Solution #2:

Knowing that the contents within the absolute value symbol should equal 4, it follows that:

3x + 9 = 6 and 3x +9 = −6

∴ x = −1, −5

Example #3:

Find x for −2 |3x + 9| = 6

Dividing both sides by -2, results in the following:

|3x + 9| = −2

As an absolute value can never be negative, correct answer is 'No Solution'

Solution #3:

Dividing both sides by -2, results in the following:

|3x + 9| = −2

As an absolute value can never be negative, correct answer is 'No Solution'