November 02, 2022

Absolute ValueDefinition, How to Find Absolute Value, Examples

Many perceive absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the entire story.

In mathematics, an absolute value is the magnitude of a real number irrespective of 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, few examples of absolute value, and the absolute value derivative.

What Is Absolute Value?

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

Meaning of Absolute Value

The last definition states 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 figure has from zero. You can visualize it if you check out a real number line:

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


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

The absolute value of negative three 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. Hence, what does this refer to? It states that absolute value is constantly positive, even if the number itself is negative.

How to Calculate the Absolute Value of a Number or Figure

You need to know few points prior working on how to do it. A couple of closely linked properties will assist you grasp how the number inside the absolute value symbol works. Fortunately, here we have an definition of the following four essential characteristics of absolute value.

Fundamental 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 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 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 four essential properties in mind, let's look at 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 variance among two real numbers is less than or equal to the absolute value of the total of their absolute values.

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

Steps to Calculate the Absolute Value of a Number

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

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

Step 2: If the figure is negative, multiply it by -1. This will make the number positive.

Step3: If the number is positive, do not alter it.

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

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

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

Example 1

To start out, 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 figure this out, we are required to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:

Step 1: We are provided with the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to find x.

Step 2: By using the basic characteristics, we learn that the absolute value of the total of these two numbers 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 unknown, so let's remove 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 equal 15, and the equation above is genuine.

Example 2

Now let's check out another absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To make it, we again need to obey the steps:

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

Step 2: We need to solve for x, so we'll begin by dividing 3 from both 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: Hence, the initial equation |x*3| = 6 also has two possible answers, x=2 and x=-2.

Absolute value can involve a lot of intricate expressions 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 continuous function, this refers it is varied everywhere. The ensuing formula provides the derivative of the absolute value function:


For absolute value functions, the area is all real numbers except 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 constant at zero(0), so the derivative of the absolute value at 0 is 0.

The absolute value function is not distinctable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is provided as:

I'm →0−(|x|/x)

The right-hand limit is given by:

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 distinguishable at zero (0).

Grade Potential Can Guide 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 guide you. We offer face-to-face tutoring by professional and certified teachers. They can guide you with absolute value, derivatives, and any other theories that are confusing you.

Call us today to learn more about how we can assist you succeed.