# Absolute ValueMeaning, How to Find Absolute Value, Examples

Many think of absolute value as the distance from zero to a number line. And that's not incorrect, but it's by no means the entire story.

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

## Definition of Absolute Value?

An absolute value of a figure is at all times zero (0) or positive. It is the magnitude of a real number irrespective to its sign. This signifies if you possess a negative figure, the absolute value of that number is the number without the negative sign.

### Definition of Absolute Value

The previous definition states that the absolute value is the distance of a number from zero on a number line. Therefore, if you think about it, the absolute value is the length or distance a figure has from zero. You can see it if you take a look at a real number line:

As demonstrated, the absolute value of a figure is the distance of the figure is from zero on the number line. The absolute value of -5 is 5 due to the fact it is five units away from zero on the number line.

### Examples

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

The absolute value of negative three is three.

Now, let's look at one more absolute value example. Let's suppose we hold an absolute value of 6. We can graph 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 at all times positive, even if the number itself is negative.

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

You should be aware of a handful of things prior going into how to do it. A couple of closely linked features will support you grasp how the expression inside the absolute value symbol works. Fortunately, here we have an explanation of the ensuing 4 essential features of absolute value.

### Essential Properties of Absolute Values

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

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

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 equivalent to the product of absolute values.

With these 4 essential characteristics in mind, let's check out two more helpful characteristics of the absolute value:

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

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

Now that we know these characteristics, we can finally begin learning how to do it!

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

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

Step 1: Note down the expression of whom’s absolute value you desire to find.

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

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

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

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

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

### Example 1

To begin with, let's consider an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To figure this out, we are required to calculate 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 must find the absolute value within the equation to solve x.

Step 2: By utilizing the fundamental properties, we understand that the absolute value of the sum of these two figures is as same as 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 see, x equals 15, so its length from zero will also be as same as 15, and the equation above is genuine.

### Example 2

Now let's check out another absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To do this, we again need to observe the steps:

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

Step 2: We are required to find the value of x, so we'll initiate by dividing 3 from each side of the equation. This step gives us |x| = 2.

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

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

Absolute value can contain many intricate figures or rational numbers in mathematical settings; still, that is a story for another day.

## The Derivative of Absolute Value Functions

The absolute value is a constant function, this refers it is varied everywhere. The following formula provides the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the area is all real numbers except 0, and the range is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is constant at zero(0), so the derivative of the absolute value at 0 is 0.

The absolute value function is not distinguishable at 0 reason being the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is stated as:

I'm →0−(|x|/x)

The right-hand limit is provided as:

I'm →0+(|x|/x)

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

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