Solve Inequalities With Absolute Value - Tutorials



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

Example 1: Solve the inequality.

|x + 2| < 3

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.

|- 2x - 4| > 9

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.

x + 2 < |x2 - 4|

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).

graph of x + 2 and |x^2 - 4| to check answer to inequality above.

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


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Updated: 3 April 2011

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