November 02, 2022

Absolute ValueDefinition, How to Calculate Absolute Value, Examples

Many perceive absolute value as the distance from zero to a number line. And that's not wrong, but it's not the complete 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 observe 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 figure is at all times zero (0) or positive. It is the magnitude of a real number without regard to its sign. This refers that if you hold a negative figure, the absolute value of that figure is the number ignoring the negative sign.

Meaning of Absolute Value

The previous explanation means that the absolute value is the distance of a number 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 observe it if you take a look at a real number line:

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


If we plot negative three on a line, we can see that it is 3 units apart from zero:

The absolute value of -3 is 3.

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

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

How to Find the Absolute Value of a Number or Figure

You need to know a couple of points before working on how to do it. A couple of closely linked properties will assist you grasp how the number within the absolute value symbol functions. Thankfully, here we have an definition of the ensuing four fundamental features of absolute value.

Basic Properties of Absolute Values

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

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

Addition: The absolute value of a total 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 4 basic properties in mind, let's check out two more helpful 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 difference between two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.

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

Steps to Calculate the Absolute Value of a Number

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

Step 1: Write down the expression of whom’s absolute value you desire 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 change it.

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

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

Bear in mind that the absolute value sign is two vertical bars on both side of a figure or expression, like 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 solve this, we are required to find the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:

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

Step 2: By using the basic characteristics, we learn that the absolute value of the sum of these two expressions is equivalent to the sum 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 can observe, x equals 15, so its distance from zero will also be equivalent 15, and the equation above is genuine.

Example 2

Now let's try one more absolute value example. We'll use the absolute value function to get a new equation, such as |x*3| = 6. To get there, we again have to obey the steps:

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

Step 2: We have to calculate the value x, so we'll start by dividing 3 from each 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: Therefore, the first equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.

Absolute value can contain a lot of complex values or rational numbers in mathematical settings; still, that is something we will work on another day.

The Derivative of Absolute Value Functions

The absolute value is a constant function, this states it is differentiable everywhere. The ensuing formula offers the derivative of the absolute value function:


For absolute value functions, the area is all real numbers except 0, and the length 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 due to the the left-hand limit and the right-hand limit are not equal. The left-hand limit is provided 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 distinctable at zero (0).

Grade Potential Can Help You with Absolute Value

If the absolute value seems like a lot to take in, or if you're struggling with mathematics, Grade Potential can assist you. We provide one-on-one tutoring by professional and qualified teachers. They can help you with absolute value, derivatives, and any other theories that are confusing you.

Contact us today to know more about how we can assist you succeed.