Absolute Value Linear Inequalities by Example 

-- Section 1.8 --

Case 1 

| x a | < b     means  - b <  x < b 

  Solve for x:  | x 2 | < 5   

Solve

Step

Graph

| x 2 | < 5  

Use this technique when 

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-5 < x 2 < 5

|x| < a  Û  -a < x <

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-5 + 2 < x 2 + 2 < 5 + 2

Add 2 to each part

-3 < x < 7

Inequality Notation

[-3, 7]

Interval Notation
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    Û    -a < x < a 

Subtract 2 from each part

Divide each part by 3

Inequality Notation

Interval Notation

 

[Click Here]

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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 < -b  or   x > b

  Solve for x:  | x + 2 | > 4   

Solve

Step

Graph

| x + 2 | > 4  

Use this technique when 

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x + 2 < -4  or  x + 2 > 4

|x| > a   Û    x < -a  or x > a

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Solve each inequality separately.

x + 2   2 < -4   2 

First inequality

x < -6

 

x + 2   2 > 4   2

Second Inequality  

x > 2

 

x < -6  or x > 2

Inequality Notation

Interval Notation
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  

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

2x + 3 < -3  or  2x + 3 > 3

Solve each inequality separately.

[Click Here]

 

Click Link for Inequality Grapher.

2x + 3   3 < -3   3 

First inequality

2x < -6

x < -3

2x + 3 3 > 3  3

Second Inequality

2x > 0

x > 0

x < -3  or x > 0

Inequality Notation

Interval Notation
This means x has to be less than -3 or greater than 0

     Solve for x:   | 4x  1 | > 7   

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada