## Introduction to Absolute Value Inequalities in Algebra

The absolute value inequality is a special expression in mathesmatics that include inequality symbols. In this article, you will learn more about absolute values inequalities and their representation.

### Definition

The algebraic expressions involving inequality symbols are called absolute values inequality. There are four types of inequality symbols involved in algebraic expression such as >,<,≥,≤. In other words, the algebraic expressions that have inequality sign are absolute value inequality.

In mathematics, the absolute value inequality and double absolute value inequality follows the same rule as the absolute number. There is only one difference between them which is that absolute value inequality involves a variable but absolute number involves only constants.

## How do you write an absolute value inequality?

There are four types of inequality symbols involved in absolute value inequalities that are used to write these inequalities, such that,

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

For example, the absolute value inequality can be generally expressed as

ax + b < c

Or,

ax + b > c

Also,

ax + b ≥ c

And

ax + b ≤ c

## How to solve absolute value inequalities?

You can solve absolute value inequalities by converting the equation in isolated form and then solving for the variable involved in the equation.

See the following examples to understand how to solve absolute value inequalities equations.

### Absolute Value Inequality Example#1

Solve |x-1|<2 and express on the number line.

The given equation is already in isolated form. So,

x - 1 > 2 and x - 1 < - 2

Solving them for x, we get

x > 3 and x < - 1

Expressing the values of x on number line,

### Absolute Value Inequality Example#2

Solve 3|x| - 2 ≤ 1 and express on the number line.

Convert the given equation in isolated form such as,

3 |x| - 2 ≤ 1

3 |x| ≤ 1 + 2

3 |x| ≤ 3

Dividing by 3 on both sides of the equation,

|x| ≤ 1

Now we will write it in 2 parts using absolute value so,

x ≤ 1 and x ≥ - 1

On number line the solution for x is;

## Absolute Value Inequality Rules

When you are solving the absolute value inequality equations, you have to remember some rules to express the solution of a given equation. These rules are

• When the inequality is in the form of |x| < a or |x| < a

In this case,

If

|x| < a-a < x < a,

And if

|x| < a-a < x < a.

• When the inequality is in the form of |x| > a or |x| > a

If

|x|> ax < -a or x > a

And if

|x| > ax < -a or x > a.

• When the inequality is in the form of |x| < - a or |x| < - a

We know that the absolute value always results in a positive value. But this case shows that the value of x is less than or equal to the negative number. It means that there will be no solution for x.

• When the inequality is in the form of |x| > - a or |x| > - a

We know that the absolute value always results in a positive value. So the absolute value of x will be positive, which is always true. Here, in this case, the value of x is greater than or equal to –a, so the solution of x will be all real numbers. The online Absolute Value Calculator also follow the same rules.

## FAQ’s

### Is Absolute Value Always Positive?

Yes, the absolute value of a number is always a positive number. It is because the absolute value always gives a positive number even if the number is negative.

### What is the Absolute Value of -10?

The absolute value of -10 is always 10.

### What does this ≤ mean?

The inequality symbol ≤ means less than or equal. It means that a number cannot exceed to another number.

### What is a Number Inequality?

In mathematics, an inequality between two numbers or expressions makes a relation of non-equal comparison between them.