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What are Double Absolute Value Inequalities?


In mathematics, the algebric expressions that contains two times absolute value and inequality symbols, are double absolute values inequalities. This guide will help you to understand more about double absolute value inequalitiesvand how to represent them.

Double Absolute Value Inequalities

The double absolute value inequalities refer to the algebraic expressions with double absolute values on both sides i.e. in other words the expression of the form |ax + b| < |cx + d| is said to be double absolute value inequality. There are four types of inequality symbols involved in algebraic expression such as ( >,<,≥,≤).

The double absolute value inequality is similar to the absolute value inequality, so the same concept is used to understand it. In double absolute value, the expression has absolute value 2 times either it is on both sides and just on one side of the inequality equation.

Representation of Double Absolute Value Inequality

Similar to the absolute value inequality, the double absolute value inequality can be expressed using the inequality signs ( >,<,≥,≤). The ways of representation are

  • |ax + b| > |cx + d| (greater than sign)
  • |ax + b| < |cx + d| (less than sign)
  • |ax + b| ≤ |cx + d| (less than or equal)
  • |ax + b| ≥ |cx + d| (greater than or equal)

These are general representations of double absolute value inequalities. The general form can also include any constant such as

|ax + b| + c ≤ |dx + e| + f

How to Solve a Double Absolute Value Inequality?

The method of solving a double absolute value inequality is different from absolute value inequality which has norm (| |) only one time. There are two methods to solve these inequalities.

In the first method, the four cases are discussed for both sides. The second method is an alternative method in which we have to vanish the norm by taking square on both sides and solving for the variable involved in the inequality equation.

See the following example to understand the method of solving double absolute value inequality equations.

Example of Double Absolute Value Inequality

Solve for x.

|x - 2| > |x - 4|

  • Case I: Both Positive

  • |x - 2| > |x - 4|

    x - 2 > x - 4

    0 > - 2

    So this equation is no use to us. There is no solution in this case.

  • Case II One is negative

  • |x - 2| > |x - 4|

    x - 2 > - (x - 4)

    x - 2 > 4 - x

    2x > 6

    x > 3

  • Case III Both negative

  • |x - 2| > |x - 4|

    - (x - 2) > - (x - 4)

    x - 2 > x - 4

    0 > - 2

    So this equation is no use to us. There is no solution in this case. It means that the solution is only possible when one is taken negative. So, lets try to solve absolute value of inequality with some alternative method. The detailed discussion of this methos is given below

  • Alternative method

  • Since both sides have an absolute value, we can just square both sides and remove the absolute value symbol. So we get

    〖(x - 2)〗^ 2 >〖(x - 4)〗^ 2

    x ^ 2 - 4x + 4 > x ^ 2 - 8x + 16

    -4x + 4 > - 8x + 16

    4x > 12

    x > 3

    Rules of Double Absolute Value Inequality

    We should remember some important rules to solve double absolute value inequalities. These are

  • When one is positive in |x|>|y|

  • In this case

    |x| > |y||x| > y


    |x| > |y|x > |y|

  • When one is negative in |x| > |y|

  • In this case,

    |x| > |y||x| > - y


    |x| > |y|- x > |y|

  • When both are negative in |x| > |y|

  • In this case,

    |x| > |y|- x > - y

    - x > - yx > y

  • When both are positive in |x| > |y|

In this case,

|x| > |y|x > y


What does 2 Absolute Value Signs mean?

It means that the inequality equation has two times norms either it is on one side or both sides of the equation.

What are absolute value equations and inequalities?

The absolute value equations are those algebraic expression that contain absolute values in it. Whereas the expressions contain inequality symbols are inequalities.

How do you write an absolute value inequality?

To write absolute value inequality, we use inequality symbols along wiht absolute value symbol, such that,

|ax + b| > c

How do you Solve Double Absolute Value Inequality?

You can solve it with two methods. In the first method, all cases of inequality signs are discussed to find a solution. But in the second method, the inequality equation is solved by taking squares on both sides of the equation.You can also calculate it with online tool.