# Absolute ValueDefinition, How to Discover Absolute Value, Examples

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

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

## Definition of Absolute Value?

An absolute value of a figure is always zero (0) or positive. It is the magnitude of a real number irrespective to its sign. That means if you have a negative figure, the absolute value of that number is the number disregarding the negative sign.

### Definition of Absolute Value

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

As you can see, the absolute value of a number is the length of the number is from zero on the number line. The absolute value of negative five is five reason being it is 5 units apart from zero on the number line.

### Examples

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

The absolute value of -3 is 3.

Presently, let's look at one more absolute value example. Let's assume we hold an absolute value of sin. We can plot this on a number line as well:

The absolute value of 6 is 6. So, what does this mean? It tells us that absolute value is always positive, even though the number itself is negative.

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

You should know few points before going into how to do it. A couple of closely associated properties will help you understand how the number within the absolute value symbol works. Luckily, what we have here is an explanation of the ensuing four fundamental characteristics of absolute value.

### Fundamental Characteristics of Absolute Values

Non-negativity: The absolute value of ever real number is at all time 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 expression.

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

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

With above-mentioned four fundamental characteristics in mind, let's check out two more useful 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 between two real numbers is lower than or equal to the absolute value of the sum of their absolute values.

Considering that we learned these characteristics, we can in the end begin learning how to do it!

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

You are required to follow a handful of steps to calculate the absolute value. These steps are:

Step 1: Jot down the expression whose absolute value you want to find.

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

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

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

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

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

### Example 1

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

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

Step 2: By using the fundamental characteristics, we understand that the absolute value of the sum of these two expressions is equivalent to 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 eliminate 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 equal 15, and the equation above is true.

### Example 2

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

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

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

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

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

Absolute value can contain many intricate numbers or rational numbers in mathematical settings; nevertheless, that is something we will work on another day.

## The Derivative of Absolute Value Functions

The absolute value is a continuous function, this refers it is distinguishable at any given point. The ensuing formula gives the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the area is all real numbers except zero (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 consistent at 0, so the derivative of the absolute value at 0 is 0.

The absolute value function is not differentiable at 0 due to the the left-hand limit and the right-hand limit are not equal. The left-hand limit is stated as:

I'm →0−(|x|/x)

The right-hand limit is offered 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 distinctable at 0.

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