November 02, 2022

Absolute ValueMeaning, How to Find Absolute Value, Examples

A lot of people perceive absolute value as the length from zero to a number line. And that's not incorrect, but it's not the entire story.

In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is at all time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, few examples of absolute value, and the absolute value derivative.

Definition of Absolute Value?

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

Definition of Absolute Value

The previous explanation means that the absolute value is the length of a number from zero on a number line. So, if you think about that, the absolute value is the distance or length a number 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 five because it is five units away from zero on the number line.


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

The absolute value of -3 is three.

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

The absolute value of six is 6. Therefore, what does this refer to? 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 few points prior going into how to do it. A couple of closely related characteristics will support you comprehend how the expression within the absolute value symbol functions. Fortunately, what we have here is an explanation of the ensuing four rudimental properties of absolute value.

Basic Properties of Absolute Values

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

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

Addition: The absolute value of a sum 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 these 4 basic characteristics in mind, let's take a look at two more beneficial properties of the absolute value:

Positive definiteness: The absolute value of any real number is constantly 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 sum of their absolute values.

Considering that we learned these properties, we can ultimately begin learning how to do it!

Steps to Discover the Absolute Value of a Expression

You need to obey a handful of steps to discover the absolute value. These steps are:

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

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

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

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

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

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

Example 1

To begin with, let's consider an absolute value equation, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To solve this, we have to locate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:

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

Step 2: By utilizing the basic properties, we learn that the absolute value of the sum of these two numbers is as 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 get rid of 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 be equivalent 15, and the equation above is genuine.

Example 2

Now let's try one more absolute value example. We'll utilize the absolute value function to solve a new equation, like |x*3| = 6. To do this, we again have to obey the steps:

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

Step 2: We have to solve for x, so we'll begin 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: Therefore, the first equation |x*3| = 6 also has two possible results, x=2 and x=-2.

Absolute value can include many complex numbers or rational numbers in mathematical settings; however, that is a story for another day.

The Derivative of Absolute Value Functions

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


For absolute value functions, the domain is all real numbers except zero (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 constant 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 uniform. The left-hand limit is provided as:

I'm →0−(|x|/x)

The right-hand limit is offered 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 differentiable at zero (0).

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