November 02, 2022

Absolute ValueDefinition, How to Discover Absolute Value, Examples

Many think of absolute value as the length from zero to a number line. And that's not inaccurate, 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 all the time a positive zero or number (0). Let's look at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.

What Is Absolute Value?

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

Definition of Absolute Value

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

As you can see, the absolute value of a number is how far away the figure is from zero on the number line. The absolute value of negative five is five reason being it is five units away from zero on the number line.

Examples

If we graph -3 on a line, we can observe that it is 3 units apart from zero:

The absolute value of negative three is 3.

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

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

How to Locate the Absolute Value of a Expression or Figure

You should know a couple of points prior working on how to do it. A handful of closely linked features will assist you understand how the figure within the absolute value symbol works. Thankfully, here we have an definition of the following four essential characteristics of absolute value.

Basic Characteristics of Absolute Values

Non-negativity: The absolute value of any 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 equal to the sum of absolute values.

Multiplication: The absolute value of a product is equal to the product of absolute values.

With above-mentioned four basic properties in mind, let's check out two other useful characteristics of the absolute value:

Positive definiteness: The absolute value of any real number is constantly positive or zero (0).

Triangle inequality: The absolute value of the variance within 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 Discover the Absolute Value of a Figure

You have to follow a handful of steps to find the absolute value. These steps are:

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

Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.

Step3: If the figure is positive, do not change it.

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

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

Remember that the absolute value symbol 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 work this out, we have to find the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:

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

Step 2: By utilizing the essential properties, we understand 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 unknown, so let's remove the vertical bars: x+5 = 20

Step 4: Let's solve for x: x = 20-5, x = 15

As we see, x equals 15, so its length from zero will also be as same as 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 do this, we again have to observe the steps:

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

Step 2: We are required to calculate the value x, so we'll initiate by dividing 3 from each side of the equation. This step offers 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 possible solutions, x=2 and x=-2.

Absolute value can include a lot of complex figures or rational numbers in mathematical settings; nevertheless, that is a story for another day.

The Derivative of Absolute Value Functions

The absolute value is a constant function, meaning it is distinguishable everywhere. The following formula offers the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the area is all real numbers except 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 distinctable 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)

Because 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 Help You with Absolute Value

If the absolute value seems like complicated task, or if you're having problem with mathematics, Grade Potential can help. We offer one-on-one tutoring from experienced and authorized tutors. They can help you with absolute value, derivatives, and any other concepts that are confusing you.

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