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 wrong, but it's not the complete story.

In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is always a positive zero or number (0). Let's observe at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.

Explanation of Absolute Value?

An absolute value of a figure is always positive or zero (0). It is the magnitude of a real number irrespective to its sign. This signifies if you hold a negative figure, the absolute value of that figure is the number ignoring the negative sign.

Definition of Absolute Value

The previous explanation states 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 length or distance 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 figure is the length of the number is from zero on the number line. The absolute value of -5 is five because it is five units away from zero on the number line.

Examples

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

The absolute value of -3 is 3.

Now, 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. Hence, what does this refer to? It states that absolute value is always positive, regardless if the number itself is negative.

How to Find the Absolute Value of a Expression or Figure

You should be aware of a handful of things before working on how to do it. A few closely associated characteristics will support you grasp how the figure inside the absolute value symbol functions. Fortunately, here we have an definition of the ensuing four fundamental characteristics of absolute value.

Basic Properties of Absolute Values

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

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

Addition: The absolute value of a total is lower 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 look at 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 difference between two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.

Now that we learned these characteristics, we can finally start learning how to do it!

Steps to Discover the Absolute Value of a Expression

You are required to follow a couple of steps to discover the absolute value. These steps are:

Step 1: Write down the expression whose absolute value you want to calculate.

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

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

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

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

Remember that the absolute value sign is two vertical bars on either side of a expression or number, like 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 work this out, we have to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:

Step 1: We have the equation |x+5| = 20, and we must discover the absolute value within the equation to find x.

Step 2: By utilizing the essential characteristics, we know that the absolute value of the addition 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 get rid of 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 true.

Example 2

Now let's check out another absolute value example. We'll use the absolute value function to find a new equation, similar to |x*3| = 6. To get there, we again need to follow the steps:

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

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

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

Step 4: So, the first equation |x*3| = 6 also has two potential answers, x=2 and x=-2.

Absolute value can include many complex expressions 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 continuous function, this refers it is varied everywhere. The ensuing formula gives the derivative of the absolute value function:

f'(x)=|x|/x

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 consistent at 0, so the derivative of the absolute value at 0 is 0.

The absolute value function is not differentiable at 0 reason being 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)

Because the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinctable at zero (0).

Grade Potential Can Assist 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 guide you. We provide one-on-one tutoring from experienced and qualified teachers. They can guide you with absolute value, derivatives, and any other theories that are confusing you.

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