# Absolute ValueMeaning, How to Discover 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 not the complete story.

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

## Explanation of Absolute Value?

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

### Meaning of Absolute Value

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

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

### Examples

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

The absolute value of -3 is three.

Well then, let's check out one more absolute value example. Let's say we posses an absolute value of sin. We can graph this on a number line as well:

The absolute value of six is 6. Hence, what does this mean? It states that absolute value is always positive, even if the number itself is negative.

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

You should know a handful of things prior going into how to do it. A few closely related characteristics will support you understand how the number within the absolute value symbol functions. Fortunately, here we have an meaning of the ensuing four rudimental features 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 equal to the product of absolute values.

With above-mentioned four basic characteristics in mind, let's check out two more beneficial properties 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 difference within two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.

Now that we learned these characteristics, we can ultimately begin learning how to do it!

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

You have to follow few steps to discover the absolute value. These steps are:

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

Step 2: If the expression 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 properties applicable to the absolute value equations.

Step 5: The absolute value of the number is the expression you obtain subsequently 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 start out, let's assume 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 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 are required to calculate the absolute value within the equation to get x.

Step 2: By utilizing the fundamental characteristics, we understand 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 unknown, so let's eliminate 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 equal 15, and the equation above is true.

### Example 2

Now let's work on one more absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To make it, we again need to observe the steps:

Step 1: We hold 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 possible results: x = 2 and x = -2.

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

Absolute value can contain several complicated values 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 continuous function, this refers it is distinguishable at any given point. The ensuing formula offers 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 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 because 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 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 distinctable at 0.

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