November 02, 2022

Absolute ValueMeaning, How to Calculate Absolute Value, Examples

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

In math, 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 check at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.

What Is Absolute Value?

An absolute value of a number is constantly zero (0) or positive. It is the extent of a real number irrespective to its sign. This refers that if you have a negative figure, the absolute value of that number is the number ignoring the negative sign.

Definition of Absolute Value

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

As shown, the absolute value of a figure is the length 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 away from zero on the number line.


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.

Well then, let's look at another absolute value example. Let's assume we posses an absolute value of sin. We can graph this on a number line as well:

The absolute value of 6 is 6. So, what does this mean? It shows us that absolute value is constantly positive, regardless if the number itself is negative.

How to Find the Absolute Value of a Number or Expression

You should know few things before going into how to do it. A handful of closely associated characteristics will assist you grasp how the figure within the absolute value symbol works. Thankfully, here we have an explanation of the following 4 fundamental characteristics of absolute value.

Essential Properties of Absolute Values

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

Identity: The absolute value of a positive number is the figure 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 above-mentioned 4 fundamental 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 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.

Taking into account that we know these characteristics, we can ultimately begin learning how to do it!

Steps to Calculate the Absolute Value of a Expression

You are required to obey a couple of steps to calculate the absolute value. These steps are:

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

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 alter it.

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

Step 5: The absolute value of the figure is the number 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, similar to this: |x|.

Example 1

To start out, let's consider an absolute value equation, like |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we need to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:

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

Step 2: By utilizing the essential characteristics, we learn that the absolute value of the total 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 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 length from zero will also be as same as 15, and the equation above is true.

Example 2

Now let's work on another absolute value example. We'll use the absolute value function to solve a new equation, similar to |x*3| = 6. To do this, we again have to obey the steps:

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

Step 2: We are required to find the value of 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 answers: x = 2 and x = -2.

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

Absolute value can contain several complicated 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 constant function, this refers it is differentiable everywhere. The following formula gives the derivative of the absolute value function:


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 constant at zero(0), so the derivative of the absolute value at 0 is 0.

The absolute value function is not distinctable at 0 reason being 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 provided as:

I'm →0+(|x|/x)

Since the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinguishable at 0.

Grade Potential Can Guide You with Absolute Value

If the absolute value looks like a lot to take in, or if you're having a tough time with math, Grade Potential can assist you. We provide one-on-one tutoring from experienced and certified instructors. They can help you with absolute value, derivatives, and any other concepts that are confusing you.

Connect with us today to learn more about how we can guide you succeed.