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Common Core Standard: A-CED.A.1
​Absolute value is the distance or number of units a number is from zero. Absolute value is represented with the following symbol, |  | An inequality is when two numbers are not equal to one another, but they have a relationship. For example a number can be less than another number.The four inequality symbols are less than <, greater than >, less than and equal to ≤  , and greater than and equal to ≥. If a problem has an inequality symbol then it is an inequality. Therefore, an absolute inequality will have an absolute value symbol, and an inequality symbol. Because of the properties of absolute values you will have two answers to these types of problems.
white box case 1 absolute value
white box case 1 absolute valuewhite box case 1 absolute valuewhite box case 1 absolute value
white box case 1 absolute value
white box case 2 absolute value
white box case 2 absolute valuewhite box case 2 absolute valuewhite box case 2 absolute value
white box case 2 absolute value
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white box 3x-5 greater than 4
white box 3x-5 greater than 4white box 3x-5 greater than 4white box 3x-5 greater than 4
white box 3x-5 greater than 4
Step 2. Create  " two cases "


we will absolute value of x<a
we will absolute value of x<a we will absolute value of x<a we will absolute value of x<a
we will absolute value of x<a

Solving Absolute Value Inequalities

Follow these four easy steps to solve absolute value inequalities

Step 1. Determine if you are going to use "or" or "and"

Use "or"  for the > , ≥ sign

Use "and" for the < , ≤ sign
Case 1: Write the problem without the absolute value sign, and solve the inequality

x<a
Case 2:    Flip and Change                                                                                                   Write the problem without the absolute value sign, flip the inequality and change the sign

x> -a


 We will use |x| < a   for our example
Step 3. Place the solutions on a number line

Step 4. Write it as an inequality

Example problem solving absolute value inequalities

Solve |3x – 5| ≥ 4
Step 1. Determine if you will use "or" or "and"  You have  ≥ therefore use "or"


Step 2. Set up two cases and rewrite without the absolute value signs
white box 3x-5 less than -4
white box 3x-5 less than -4white box 3x-5 less than -4white box 3x-5 less than -4
white box 3x-5 less than -4

3x – 5 ≥ 4

add 5 to each side

3x – 5 ≥ 4
     +5    +5

3x ≥9

Divide by 3

x≥ 3
3x – 5 ≤ -4

add 5 to each side

3x – 5 ≤ -4
       +5    +5

3x ≤ 1

Divide by 3

x   ≤ 1/3
Step 4. ​Rewrite as an inequality   x ≤ 1/3 or x≥ 3
number line
1/3
right arrow
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Step 3. Place the solutions on a number line

Absolute Value Inequalities Video Tutorials

This video gives step by step directions on how to solve:

|12-3x| ≥ 6
Video solution to the sample problem worked above:

|3x – 5| ≥ 4
This video gives step by step directions on how to solve:

|8x-15|< 9