# Absolute ValueDefinition, How to Find Absolute Value, Examples

A lot of people think of absolute value as the distance from zero to a number line. And that's not inaccurate, but it's by no means the whole story.

In math, an absolute value is the extent of a real number without regard to 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 constantly positive or zero (0). It is the magnitude of a real number without considering its sign. That means if you possess a negative figure, the absolute value of that figure is the number ignoring the negative sign.

### Definition of Absolute Value

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

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

### Examples

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

The absolute value of -3 is three.

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

The absolute value of 6 is 6. Therefore, what does this tell us? It states that absolute value is at all times positive, even though the number itself is negative.

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

You should know a handful of things prior going into how to do it. A couple of closely associated characteristics will assist you grasp how the expression within the absolute value symbol works. Thankfully, here we have an definition of the following four essential characteristics of absolute value.

### Basic Characteristics of Absolute Values

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

Identity: The absolute value of a positive number is the number 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 total of absolute values.

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

With these 4 fundamental properties in mind, let's look at two more useful properties of the absolute value:

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

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

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

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

You need to follow a couple of steps to discover the absolute value. These steps are:

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

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 significant to the absolute value equations.

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

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

### Example 1

To set out, let's presume 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 need to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:

Step 1: We have the equation |x+5| = 20, and we must find the absolute value within the equation to solve x.

Step 2: By utilizing the essential characteristics, we learn that the absolute value of the addition of these two figures is the 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 see, 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 check out one more absolute value example. We'll utilize the absolute value function to solve a new equation, similar to |x*3| = 6. To get there, we again need to observe the steps:

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

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

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

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

Absolute value can include many complex 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 distinguishable at any given point. The following 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 rises 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 differentiable at 0 because the left-hand limit and the right-hand limit are not equal. The left-hand limit is given by:

I'm →0−(|x|/x)

The right-hand limit is provided as:

I'm →0+(|x|/x)

Because 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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