# Absolute ValueDefinition, How to Calculate Absolute Value, Examples

A lot of people comprehend absolute value as the distance from zero to a number line. And that's not wrong, but it's nowhere chose to the complete story.

In mathematics, an absolute value is the extent of a real number without considering its sign. So the absolute value is all the time a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, some examples of absolute value, and the absolute value derivative.

## Explanation 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. That means if you hold a negative figure, the absolute value of that figure is the number without the negative sign.

### Definition of Absolute Value

The prior explanation states that the absolute value is the distance of a number from zero on a number line. Hence, if you think about it, the absolute value is the distance or length a number has from zero. You can observe it if you take a look at a real number line:

As shown, the absolute value of a figure is the length of the figure is from zero on the number line. The absolute value of negative five is five reason being it is 5 units away from zero on the number line.

### Examples

If we graph negative three on a line, we can watch that it is 3 units apart from zero:

The absolute value of -3 is 3.

Now, let's check out another absolute value example. Let's say we have an absolute value of 6. We can plot this on a number line as well:

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

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

You should be aware of few things prior working on how to do it. A handful of closely linked properties will support you grasp how the expression within the absolute value symbol works. Thankfully, here we have an explanation of the ensuing 4 rudimental properties of absolute value.

### Essential Characteristics of Absolute Values

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

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

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

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

With these four essential properties in mind, let's look at two more helpful characteristics of the absolute value:

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

Triangle inequality: The absolute value of the difference between two real numbers is less than or equivalent to the absolute value of the total of their absolute values.

Now that we went through these properties, we can ultimately initiate learning how to do it!

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

You are required to observe few steps to calculate the absolute value. These steps are:

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

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

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

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

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

Keep in mind that the absolute value sign is two vertical bars on either side of a expression or number, like this: |x|.

### Example 1

To set out, let's presume an absolute value equation, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we need to find the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:

Step 1: We are provided with the equation |x+5| = 20, and we are required to discover the absolute value within the equation to get x.

Step 2: By utilizing the essential properties, we understand that the absolute value of the addition of these two numbers is as same as the total of each absolute value: |x|+|5| = 20

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

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

As we can observe, x equals 15, so its distance from zero will also be as same as 15, and the equation above is genuine.

### Example 2

Now let's try one more absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To do this, we again have to obey the steps:

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

Step 2: We need to solve for x, so we'll start by dividing 3 from both side of the equation. This step gives us |x| = 2.

Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.

Step 4: Therefore, the first equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.

Absolute value can involve many complicated expressions or rational numbers in mathematical settings; nevertheless, that is a story for another day.

## The Derivative of Absolute Value Functions

The absolute value is a continuous function, meaning it is distinguishable everywhere. The ensuing formula gives 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 distance 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 zero(0), so the derivative of the absolute value at 0 is 0.

The absolute value function is not distinctable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is provided as:

I'm →0−(|x|/x)

The right-hand limit is given by:

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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