Solve Inequalities With Absolute Value - Tutorials

This is a tutorial on solving inequalities with absolute value. Detailed solutions and explanations are provided.

Example 1: Solve the inequality.

Solution to example 1

• The above inequality is solved by writing a double inequality equivalent to the given inequality but without absolute value
  
  - 3 < x + 2 < 3
  
  
• Solve the double inequality to obtain
  
  - 5 < x < 1
  
  
• The above solution set is written in interval form as follows
  
  (-5 , 1)

Example 2: Solve the inequality.

Solution to example 2

• Solving the above inequality is equivalent to solving
  
  -2x - 4 > 9 or -2x - 4 < - 9
  
  
• Which gives
  
  x < -13 / 2 or x > 5 / 2
  
  
• The above solution set is written in interval form as follows
  
  (-inf , -13 / 2) U (5 / 2 , + inf)

Example 3: Solve the inequality.

Solution to example 3

• Condition 1 - For x² - 4 > = 0, or x in the interval (-inf , -2] U [2 , +inf), we can write
  
  |x² - 4| = x² - 4
  
  
• Substitute the expression with the absolute value in the given inequality and solve
  
  x + 2 < x² - 4
  
  x² - x - 6 > 0
  
  
• The solution set to above inequality is given by the interval
  (-inf , -2) U (3 , +inf)
  
  
• The intersection of intervals(-inf , -2] U [2 , +inf) and (-inf , -2) U (3 , +inf) gives the solution set
  
  (-inf , -2) U (3 , +inf)
  
  Condition 2 - For x² - 4 < 0, or x in the interval (-2 , 2), we can write
  
  |x² - 4| = -(x² - 4)
  
  
• Substitute the expression with the absolute value in the given inequality and solve
  
  x + 2 < -(x² - 4)
  
  x² + x - 2 < 0
  
  
• The solution set to above inequatlity is given by the intersection of the intervals (-2 , 1) and (-2 , 2)
  
  (-2 , 1)
  
  
• Conclusion: The solution set to the given inequality is (-inf, -2) U (-2 , 1) U (3 , +inf)

Check the above answer to the inequality graphically.(see graph below).

More references and links on how to Solve Equations, Systems of Equations and Inequalities.