Absolute Value Linear Inequalities by Example -- Section 1.8 --
 Case 1 | x – a | < b     means  - b <  x – a  < b
 Solve for x:  | x – 2 | < 5
 Solve Step Graph | x – 2 | < 5 Use this technique when -5 < x – 2 < 5 |x| < a  Û  -a < x < a Click Link for Inequality Grapher -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 3Inequality Notation Interval Notation 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 Use this technique when x + 2 < -4  or  x + 2 > 4 |x| > a   Û    x < -a  or x > a Click Link for Inequality Grapher. 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 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