Absolute ValueDefinition, How to Find Absolute Value, Examples
Many think of absolute value as the length from zero to a number line. And that's not incorrect, but it's by no means the complete story.
In mathematics, an absolute value is the magnitude of a real number irrespective of its sign. So the absolute value is at all time a positive zero or number (0). Let's observe at what absolute value is, how to discover absolute value, few examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the magnitude of a real number without considering its sign. This refers that if you possess a negative number, the absolute value of that number is the number ignoring the negative sign.
Definition of Absolute Value
The last definition refers that the absolute value is the distance of a figure from zero on a number line. So, if you think about it, the absolute value is the distance or length a number has from zero. You can observe it if you take a look at a real number line:
As shown, 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 reason being it is 5 units apart from zero on the number line.
Examples
If we plot negative three on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is 3.
Presently, let's look at more absolute value example. Let's suppose we hold an absolute value of 6. We can graph this on a number line as well:
The absolute value of 6 is 6. Hence, what does this mean? It tells us that absolute value is constantly positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Expression or Number
You should be aware of a couple of things prior working on how to do it. A couple of closely linked characteristics will help you grasp how the expression inside the absolute value symbol works. Thankfully, what we have here is an explanation of the following four essential characteristics of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is constantly zero (0) or positive.
Identity: The absolute value of a positive number is the expression itself. Instead, the absolute value of a negative number is the non-negative value of that same figure.
Addition: The absolute value of a sum is less than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned 4 fundamental characteristics in mind, let's check out two other useful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is at all times zero (0) or positive.
Triangle inequality: The absolute value of the variance among two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Considering that we know these properties, we can finally start learning how to do it!
Steps to Discover the Absolute Value of a Expression
You need to observe a couple of steps to calculate the absolute value. These steps are:
Step 1: Jot down the expression whose absolute value you desire to calculate.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all properties significant to the absolute value equations.
Step 5: The absolute value of the figure is the figure you obtain after steps 2, 3 or 4.
Keep in mind that the absolute value symbol is two vertical bars on either side of a number or expression, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, like |x + 5| = 20. As we can see, 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 mentioned priorly:
Step 1: We are given the equation |x+5| = 20, and we must find the absolute value inside the equation to find x.
Step 2: By utilizing the essential characteristics, we learn 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 unidentified, so let's remove the vertical bars: x+5 = 20
Step 4: Let's solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also equal 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 get 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 are required to calculate the value x, so we'll start by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential solutions: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can involve a lot of complicated figures or rational numbers in mathematical settings; nevertheless, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous 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 domain is all real numbers except zero (0), and the distance is all positive real numbers. The absolute value function rises 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 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)
Considering the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinguishable at zero (0).
Grade Potential Can Help You with Absolute Value
If the absolute value looks like a difficult topic, or if you're struggling with math, Grade Potential can guide you. We provide face-to-face tutoring from experienced and qualified tutors. They can help you with absolute value, derivatives, and any other theories that are confusing you.
Contact us today to learn more about how we can guide you succeed.
