November 02, 2022

Absolute ValueMeaning, How to Calculate Absolute Value, Examples

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

In mathematics, an absolute value is the extent of a real number without considering 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 discover absolute value, several examples of absolute value, and the absolute value derivative.

Definition of Absolute Value?

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

Meaning of Absolute Value

The previous explanation refers that the absolute value is the length of a figure from zero on a number line. Hence, if you consider it, the absolute value is the distance or length a number has from zero. You can visualize it if you check out a real number line:

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

Examples

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 three.

Presently, let's look at 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 6 is 6. So, what does this tell us? It shows us that absolute value is constantly positive, even if the number itself is negative.

How to Locate the Absolute Value of a Number or Figure

You should know few things before going into how to do it. A handful of closely related characteristics will support you grasp how the number inside the absolute value symbol works. Thankfully, here we have an definition of the ensuing 4 fundamental features of absolute value.

Fundamental Characteristics of Absolute Values

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

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 figure.

Addition: The absolute value of a total is less than or equivalent to the sum of absolute values.

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

With these 4 fundamental characteristics in mind, let's check out two more useful characteristics 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 total of their absolute values.

Now that we learned these properties, we can finally begin learning how to do it!

Steps to Find the Absolute Value of a Expression

You are required to obey few steps to calculate 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 number is negative, multiply it by -1. This will convert the number to 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 figure is the figure you get after steps 2, 3 or 4.

Bear in mind that the absolute value sign is two vertical bars on either side of a figure or expression, like this: |x|.

Example 1

To begin with, let's presume 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 need to find the absolute value of the two numbers in the inequality. We can do this by following the steps above:

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

Step 2: By utilizing the fundamental characteristics, we know that the absolute value of the total of these two figures is the same as 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 eliminate 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 equivalent 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, like |x*3| = 6. To do this, we again have to obey the steps:

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

Step 2: We are required to calculate the value x, so we'll begin by dividing 3 from both side of the equation. This step offers us |x| = 2.

Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.

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

Absolute value can contain several intricate values 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 continuous function, this states it is varied everywhere. The ensuing formula offers the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the domain is all real numbers except 0, and the distance 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 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)

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 Help You with Absolute Value

If the absolute value seems like a lot to take in, or if you're having a tough time with mathematics, Grade Potential can help. We offer one-on-one tutoring by professional and authorized tutors. They can guide you with absolute value, derivatives, and any other concepts that are confusing you.

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