# Absolute ValueDefinition, How to Find Absolute Value, Examples

A lot of people think of absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the whole story.

In math, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is all the time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.

## Definition of Absolute Value?

An absolute value of a figure is constantly zero (0) or positive. It is the extent of a real number irrespective to its sign. This signifies if you have a negative number, the absolute value of that figure is the number without the negative sign.

### Meaning of Absolute Value

The last explanation means that the absolute value is the distance of a number from zero on a number line. So, if you consider it, the absolute value is the distance or length a number has from zero. You can see it if you take a look at a real number line:

As you can see, the absolute value of a number is how far away the number is from zero on the number line. The absolute value of -5 is 5 because it is 5 units away from zero on the number line.

### Examples

If we plot -3 on a line, we can see that it is 3 units apart from zero:

The absolute value of negative three is 3.

Presently, let's look at more absolute value example. Let's suppose we have an absolute value of 6. We can graph this on a number line as well:

The absolute value of six is 6. So, what does this refer to? It states that absolute value is always positive, regardless if the number itself is negative.

## How to Find the Absolute Value of a Number or Expression

You should be aware of few points before working on how to do it. A couple of closely related properties will support you comprehend how the expression within the absolute value symbol functions. Luckily, what we have here is an explanation of the ensuing 4 fundamental characteristics of absolute value.

### Fundamental Properties of Absolute Values

Non-negativity: The absolute value of ever real number is constantly zero (0) or positive.

Identity: The absolute value of a positive number is the expression itself. Otherwise, the absolute value of a negative number is the non-negative value of that same expression.

Addition: The absolute value of a sum is less than or equal to the total of absolute values.

Multiplication: The absolute value of a product is equal to the product of absolute values.

With above-mentioned four fundamental properties in mind, let's look at two other beneficial characteristics of the absolute value:

Positive definiteness: The absolute value of any real number is constantly positive or zero (0).

Triangle inequality: The absolute value of the difference among two real numbers is less than or equal to the absolute value of the sum of their absolute values.

Taking into account that we know these properties, we can in the end initiate learning how to do it!

### Steps to Discover the Absolute Value of a Expression

You are required to obey few steps to find the absolute value. These steps are:

Step 1: Note down the figure of whom’s absolute value you want to calculate.

Step 2: If the expression is negative, multiply it by -1. This will convert the number to positive.

Step3: If the expression is positive, do not change it.

Step 4: Apply all characteristics significant to the absolute value equations.

Step 5: The absolute value of the expression is the number you have after steps 2, 3 or 4.

Remember that the absolute value sign is two vertical bars on either side of a expression or number, similar to this: |x|.

### Example 1

To set out, let's assume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we have to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:

Step 1: We have the equation |x+5| = 20, and we are required to calculate the absolute value inside the equation to find x.

Step 2: By utilizing the fundamental properties, we understand that the absolute value of the addition of these two figures is equivalent to the sum of each absolute value: |x|+|5| = 20

Step 3: The absolute value of 5 is 5, and the x is unknown, so let's remove the vertical bars: x+5 = 20

Step 4: Let's solve for x: x = 20-5, x = 15

As we can observe, x equals 15, so its distance from zero will also equal 15, and the equation above is right.

### Example 2

Now let's work on another absolute value example. We'll use the absolute value function to find a new equation, like |x*3| = 6. To make it, we again have to follow the steps:

Step 1: We hold the equation |x*3| = 6.

Step 2: We have to calculate the value x, so we'll start by dividing 3 from both side of the equation. This step offers us |x| = 2.

Step 3: |x| = 2 has two potential results: x = 2 and x = -2.

Step 4: So, the original equation |x*3| = 6 also has two potential results, x=2 and x=-2.

Absolute value can include many intricate numbers or rational numbers in mathematical settings; however, that is something we will work on separately to this.

## The Derivative of Absolute Value Functions

The absolute value is a constant function, meaning it is differentiable at any given point. The ensuing formula provides the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the domain is all real numbers except zero (0), and the length is all positive real numbers. The absolute value function increases for all x<0 and all x>0. The absolute value function is consistent at 0, so the derivative of the absolute value at 0 is 0.

The absolute value function is not differentiable at 0 reason being the left-hand limit and the right-hand limit are not uniform. The left-hand limit is provided as:

I'm →0−(|x|/x)

The right-hand limit is provided as:

I'm →0+(|x|/x)

Considering the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not differentiable at 0.

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