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# Absolute Value Inequalities – Explanation & Examples

The **absolute value of inequalities **follows the same rules as the** absolute value of numbers. **The difference is that we have a variable in the prior and a constant in the latter.

This article will show a brief overview of the absolute value inequalities, followed by the **step-by-step method to solve the absolute value inequalities**.

Finally, there are examples of different scenarios for better understanding.

## What is Absolute Value Inequality?

Before we can learn how to solve absolute value inequalities, let’s remind ourselves about a number’s absolute value.

**By definition, the absolute value of a number is the distance of a value from the origin, regardless of the direction. Absolute value is denoted by two vertical lines enclosing the number or expression. **

**For example**, the absolute value of x is expressed as | x | = a, which implies that, x = +a and -a. Now let’s see what the absolute value inequalities entail.

An absolute value inequality is an expression with absolute functions as well as inequality signs. For example, the expression |x + 3| > 1 is an absolute value inequality containing a greater than symbol.

There are four different inequality symbols to choose from. These are less than (**<**), greater than (**>**), less than or equal (**≤**), and greater than or equal (**≥**). So, the absolute value inequalities can possess any one of these four symbols.

## How to Solve Absolute Value Inequalities?

**The steps for solving absolute value inequalities** are much similar to solving absolute value equations. However, there is some extra information you need to keep in mind when solving absolute value inequalities.

*The following are the general rules to consider when solving absolute value inequalities:*

- Isolate on the left the absolute value expression.
- Solve the positive and negative versions of the absolute value inequality.
- When the number on the other side of the inequality sign is negative, we either conclude all real numbers as the solutions, or the inequality has no solution.
- When the number on the other side is positive, we proceed by setting up a compound inequality by removing the absolute value bars.
- The type of inequality sign determines the format of the compound inequality to be formed. For instance, if a problem contains greater than or greater than/equals to sign, set up a compound inequality that has the following formation:

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

- Similarly, if a problem contains a less than or less than/equals to sign, set up a 3- part compound inequality of the following form:

– (The number on the other side of inequality sign) < (quantity within the absolute value bars) < (The number on the other side of the inequality sign)

* **Example 1*

Solve the inequality for x: | 5 + 5x| − 3 > 2.

__Solution__

Isolate the absolute value expression by adding 3 to both sides of the inequality;

=> | 5 + 5x| − 3 (+ 3) > 2 (+ 3)

=> | 5 + 5x | > 5.

Now solve both the positive and negative “versions” of the inequality as follows;

We’ll assume absolute value symbols by solving the equation the normal way.

=> | 5 + 5x| > 5 → 5 + 5x > 5.

=> 5 + 5_x_> 5

Subtract 5 from both sides

5 + 5x (− 5) > 5 (− 5) 5x > 0

Now, divide both sides by 5

5x/5 > 0/5

*x* > 0.

Thus, *x* > 0 is one of the possible solutions.

To solve for negative version of the absolute value inequality, multiply the number on the other side of the inequality sign by -1, and reverse the inequality sign:

| 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.

*x* > 0 or *x* < −2 are the two possible solutions to the inequality. Alternatively, we can solve | 5 + 5x | > 5 using the formula:

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

Illustration:

(5 + 5x) < – 5 OR (5 + 5x) > 5

Solve the expression above to get;

*x* < −2 or *x* > 0

*Example 2*

Solve |x + 4| – 6 < 9

__Solution__

Isolate the absolute value.

|x + 4| – 6 < 9 → |x + 4| < 15

Since our absolute value expression has a less than inequality sign, we set up the a 3-part compound inequality solution as:

-15 < x + 4 < 15

-19 < x < 11

* *

*Example 3*

Solve |2x – 1| – 7 ≥ -3

__Solution__

First, isolate the variable

|2x – 1| – 7≥-3 → |2x – 1|≥4

We will set up an “or” compound inequality because of the greater than or equal to sign in our equation.

2 – 1≤ – 4 or 2x – 1 ≥ 4

Now, solve the inequalities;

2x – 1 ≤ -4 or 2x – 1 ≥ 4

2x ≤ -3 or 2x ≥ 5

x ≤ -3/2 or x ≥ 5/2

* *

*Example 4*

Solve |5x + 6| + 4 < 1

__Solution__

Isolate the absolute value.

|5x + 6| + 4 < 1 → |5x + 6| < -3

Since the number on the other side is negative, check also the opposite to determine the solution.

|5x + 6| < -3

Positive < negative (false). Therefore, this absolute value inequality has no solution.

* *

*Example 5*

Solve |3x – 4| + 9 > 5

__Solution__

Isolate the absolute value.

|3x – 4| + 9 > 5 → |3x – 4| > -4

|5x + 6| < -3

Since, positive < negative (true). Therefore, the solutions to this absolute value inequality are all real numbers.

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