algebra 2 solve absolute value inequality and graph the solution

*Let's start with definition of absolute value*

*1. l a l = a , if a ≥ 0 (if "a" is positive or 0) , for example*

*l 5 l = 5*

*2. l a l = -a , if a < 0 (if "a" is negative) , for example*

*l -5 l = - (-5) = 5*

*So, absolute value is always positive or zero.*

**~~~~~~~~~~~**So, we need to consider two cases, because expression on the left side of inequality contain variable and can be positive, or negative.

*Let's assume that (x - 3) ≥ 0 , then*

**1.**x - 3 < 5 ---> x < 5 + 3 --->

**x < 8***If (x - 3) < 0 , then we have to put the "–" before (x - 3)*

**2.**– (x - 3) < 5 or

x - 3 > - 5 ---> x > - 5 + 3 --->

**x > - 2****x ∈ (-2, 8)**

<-------|----

**o**

**===|=======***-----|------->*

**o***- ∞ - 3 - 2*

*0*

*8 9 ∞*

*Now let's check our answer:*

*For example:*

if x = -3, then

| - 3 - 3 | < 5

| - 6| < 5

6 < 5 (false)

if x = 0, then

| 0 - 3 | < 5

| - 3 | < 5

3 < 5 (true)

if x = 9, then

| 9 - 3 | < 5

| 6 | < 5

6 < 5 (false)