November 02, 2022

Absolute ValueDefinition, How to Discover Absolute Value, Examples

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

In mathematics, an absolute value is the magnitude of a real number without regard to 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, several examples of absolute value, and the absolute value derivative.

Explanation of Absolute Value?

An absolute value of a number is at all times positive or zero (0). It is the magnitude of a real number irrespective to its sign. That means if you possess a negative number, the absolute value of that number is the number overlooking the negative sign.

Definition of Absolute Value

The last explanation states 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 distance or length a number has from zero. You can observe it if you check out 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 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 watch that it is 3 units away from zero:

The absolute value of -3 is three.

Now, let's look at one more absolute value example. Let's suppose we posses an absolute value of 6. We can graph this on a number line as well:

The absolute value of six is 6. So, what does this refer to? It shows us that absolute value is at all times positive, even if the number itself is negative.

How to Find the Absolute Value of a Figure or Expression

You need to know a couple of points before going into how to do it. A handful of closely linked characteristics will help you grasp how the number inside the absolute value symbol functions. Thankfully, here we have an definition of the ensuing 4 fundamental characteristics of absolute value.

Essential Properties of Absolute Values

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

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 figure.

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 look at two other helpful properties of the absolute value:

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

Triangle inequality: The absolute value of the difference among two real numbers is less than or equal to the absolute value of the sum of their absolute values.

Taking into account that we learned these characteristics, we can finally start learning how to do it!

Steps to Calculate the Absolute Value of a Figure

You need to observe a couple of steps to discover the absolute value. These steps are:

Step 1: Write down the expression whose absolute value you desire to discover.

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

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

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

Step 5: The absolute value of the expression is the number you obtain subsequently 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 observe, there are two real numbers and a variable inside. To solve this, we are required to locate the absolute value of the two numbers in the inequality. We can do this by following the steps above:

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

Step 2: By utilizing the fundamental properties, we understand that the absolute value of the sum 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 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 distance from zero will also be equivalent 15, and the equation above is genuine.

Example 2

Now let's try another absolute value example. We'll use the absolute value function to get a new equation, such as |x*3| = 6. To get there, we again need to observe the steps:

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

Step 2: We need to find the value of x, so we'll start 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 original equation |x*3| = 6 also has two potential answers, x=2 and x=-2.

Absolute value can include many complicated figures 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 constant function, meaning it is differentiable at any given point. The following formula gives the derivative of the absolute value function:

f'(x)=|x|/x

For absolute value functions, the area 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 consistent at zero(0), so the derivative of the absolute value at 0 is 0.

The absolute value function is not differentiable at 0 because 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 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 distinctable at zero (0).

Grade Potential Can Guide You with Absolute Value

If the absolute value seems like a difficult topic, or if you're having problem with math, Grade Potential can guide you. We provide one-on-one tutoring from experienced and authorized instructors. They can help you with absolute value, derivatives, and any other theories that are confusing you.

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