# Absolute ValueMeaning, How to Discover Absolute Value, Examples

Many think of absolute value as the length from zero to a number line. And that's not incorrect, but it's nowhere chose to the whole story.

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

## Explanation of Absolute Value?

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

### Meaning of Absolute Value

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

As shown, the absolute value of a number is how far away the number is from zero on the number line. The absolute value of negative five is five due to the fact it is 5 units apart from zero on the number line.

### Examples

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

The absolute value of negative three is 3.

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

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

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

You should be aware of few things before working on how to do it. A couple of closely related properties will assist you understand how the number within the absolute value symbol works. Fortunately, what we have here is an explanation of the ensuing 4 fundamental features of absolute value.

### Basic Properties of Absolute Values

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

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

Addition: The absolute value of a total is lower 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 above-mentioned 4 basic characteristics in mind, let's look at two other 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 among two real numbers is lower than or equal to the absolute value of the total of their absolute values.

Now that we know these characteristics, we can ultimately start learning how to do it!

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

You are required to observe a couple of steps to discover the absolute value. These steps are:

Step 1: Jot down the expression whose absolute value you desire to discover.

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 alter it.

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

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

Bear in mind that the absolute value sign is two vertical bars on both 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 locate the absolute value of the two numbers in the inequality. We can do this by following the steps above:

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

Step 2: By using the basic properties, we understand that the absolute value of the addition of these two expressions is the 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 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 genuine.

### Example 2

Now let's work on another absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To get there, 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 start by dividing 3 from each side of the equation. This step offers us |x| = 2.

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

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

Absolute value can involve several complicated expressions 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 states it is differentiable 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 increases 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 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)

Considering the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinguishable at zero (0).

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