Absolute Value Inequalities – Explanation & Examples

absolute value inequality

Absolute value for inequality follows similar rules to an absolute number. The distinction is that there is a variable in the first case and a constant in the second.

This article will provide an introduction to absolute value inequalities and then the step-by-step procedure to solve the absolute value equations.

In addition, there are examples of various scenarios for greater understanding.

What is the Absolute Value of Inequality?

Before we master the art of solving absolute value inequality, let’s be reminded of the absolute value of a number.

According to the definition, the “absolute value” of a value is the distance a number is from its origin regardless of its direction. Absolute value is indicated in two horizontal lines surrounding the expression or number.

For instance, an absolute value for x can be = a. This implies that there is a +a and a. Let’s look at the absolute value inequality entails.

An absolute value inequality can be described as an expression that has absolute functions and inequality indicators. For instance, the expression | + 1 can be described as an absolute inequality with the symbol greater than.

There are four distinct inequalities symbols that you can choose from. They are: lesser than ( <), greater than ( >) and equal or less ( <=), and greater than or equal to ( >=). Thus, the value absolute inequalities may have any of the four symbols.

How can I Solve the Absolute Value Problem?

The steps to solve absolute value equations are very alike to solving equations of absolute value that you may solve with this absolute value equation calculator with steps. But, there are some additional details you must be aware of while solving absolute value equations. 

Here are some general guidelines to take into consideration when solving absolute value inequality:

  • On the left, isolate the expression for absolute value.
  • Find the positive and negative versions of the inequality in absolute value.
  • When the number on another side of the mark is not positive we can either take all the true numbers to be the solutions or the inequality is no solution.
  • If the value on the other hand can be positive, you move on to creating an inequality compound by taking out the bars of absolute value.
  • The kind of inequality sign is the basis for the compound inequality that will be created. For example, if a problem has more than or equal to signify, you can create an inequality compound that follows the following form:

(The values inside the bars of absolute values) < – (The number on the other side) OR (The values within absolute value bars) > (The numbers on the opposite side).

The same formula is used by absolute value inequalities calculator with steps for solving the inequalities online.

  • Also, if the issue has a sign that is less than, less than, or equal to sign, create a 3-part compound inequality in the following format:

The number on the other side of the inequality sign (The number that is on the opposite side of the inequality sign) * (quantity within the bars for absolute value) (quantity within the absolute value bars) (The numbers on the opposite end of an inequality sign)


Example1: Find the equation for x: |5 + 5x| – 3 > 2.

Solution: Determine the expression for absolute value using the addition of 3 on both ends.

=> | 5 + 5x| – 3 (+ 3) > 2 (+ 3)

=> | 5 + 5x | > 5.

Then, you can solve both the negative and positive “versions” of this inequality in the following manner:

We’ll assume that absolute value symbols are used through solving the equation the usual method.

=> | 5 + 5x| > 5 – 5 + 5x > 5.

=> 5 + 5_x_> 5

Add 5 to both sides.

5 + 5x (- 5) > 5 (- 5) 5x > 0

Then take both sides and divide them by 5

5x/5 > 0/5

x greater than 0.

So, x > 0 . is one of the solutions that could be found.

Related: Also read best approaches to solve maths.

To determine the negative variant of absolute value inequalities multiply the number on the opposite edge of the sign with -1 and reverse the inequality symbol:

|5 + 5x |> 5

5 + 5x < -5 => 5 + 5x < -5

Subtract 5 from both sides = 5 + 5x(5) < -5 (-5) => 5x < -10 => 5x/5 < -10/5 => x < -2.

the x > 0, or x < -2 are two possibilities for solving the inequality. Alternately, we could solve 5 + 5x = 5 by using the formula:

(The values contained within the bars of absolute values) < – (The number on another side) OR (The values within absolute value bars) > (The number on the other side).


(5 + 5x) < – 5 OR (5 + 5x) > 5

Find the expression above and solve it to obtain:

with x < -2 or x > 0

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