Absolute Value Linear Inequalities by Example 

-- Part 1.6 --

Case 1 

 | x – a | < b

 means  - b <  | x – a | < b 
  Solve for x:  | x – 2 | < 5   

Solve

Step

Graph

| x – 2 | < 5  

-5 < x – 2 < 5

-5 + 2 < x – 2 + 2 < 5 + 2

-3 < x < 7

[-3, 7]

Use this technique when 
|x| < a
-a < x < a 


Add 2 to each part

Inequality Notation

Interval Notation

 

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This means x has to be between -3 and 7 (x could be -3 or 7 because of the < symbol)

     Solve for x:  | x – 2 | < 3   

[Solution]

  Solve for x:  | 3x + 2 | < 7   

Solve

Step

Graph

| 3x + 2 | < 7  

-7 < 3x + 2 < 7

-7 – 2 < 3x + 2 –  2 < 7 – 2

-9 < 3x < 5

-3 < x < 5/3

(-3, 5/3)

Use this technique when 
   |x| < a   
-a < x < a 

Subtract 2 from each part

Divide each part by 3

Inequality Notation

Interval Notation

 

[Click Here]

Click Link for Inequality Grapher.

This means x has to be between -3 and 5/3 (x could not be -3 or 5/3 because of the < symbol)

     Solve for x:  | 4x + 1 | < 9   

[Solution]

 Case 2 

 | x – a | > b

 means   x – a  < -b  or   x – a  > b
  Solve for x:  | x + 2 | > 4   

Solve

Step

Graph

| x + 2 | > 4  

x + 2 < -4 or x + 2 > 4

x + 2 –  2< -4 –  2 

x < -6

x + 2 –  2 > 4 –  2

x > 2

Use this technique when 
|x| > a
x < -a  or x > a

Solve each inequality separately.

First inequality

 

Second Inequality

Inequality Notation

Interval Notation

 

[Click Here]

Click Link for Inequality Grapher.

x < -6  or x > 2
This means x has to be less than or equal to -6 or greater than or equal to 2

     Solve for x:  | x – 1 | > 5   

[Solution]

  Solve for x:  | 2x + 3 | > 3   

Solve

Step

Graph

| 2x + 3 | > 3  

2x + 3 < -3 or 2x + 3 > 3

2x + 3 –  3 < -3  – 3 

2x < -6
x < -3

2x + 3 – 3 > 3 – 3

2x > 0
x > 0

Use this technique when 
|x| > a
x < -a  or x > a

Solve each inequality separately.

First inequality


Second Inequality

Inequality Notation

Interval Notation

 

[Click Here]

Click Link for Inequality Grapher.

x < -3  or x > 0
This means x has to be less than -3 or greater than 0

     Solve for x:   | 4x – 1 | > 7   

[Solution]

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joe mcdonald

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Last Update 03.15.2006