Absolute ValueDefinition, How to Find Absolute Value, Examples
Many comprehend absolute value as the length from zero to a number line. And that's not wrong, 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 always a positive zero or number (0). Let's check 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 always zero (0) or positive. It is the extent of a real number irrespective to its sign. This refers that if you possess a negative number, the absolute value of that figure is the number without the negative sign.
Meaning of Absolute Value
The previous explanation refers that the absolute value is the length of a number from zero on a number line. Hence, if you think about it, the absolute value is the distance or length 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 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 plot negative three on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is three.
Well then, let's look at another absolute value example. Let's assume we hold an absolute value of 6. We can graph 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 at all times positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Expression or Number
You should know a handful of things before working on how to do it. A few closely associated characteristics will assist you grasp how the figure within the absolute value symbol works. Luckily, what we have here is an explanation of the following 4 rudimental properties of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is constantly zero (0) or positive.
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 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 equivalent to the product of absolute values.
With these four fundamental characteristics in mind, let's check out two more beneficial properties 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 within two real numbers is lower than or equal to the absolute value of the total of their absolute values.
Considering that we learned these characteristics, we can ultimately initiate learning how to do it!
Steps to Find the Absolute Value of a Number
You are required to obey few steps to discover the absolute value. These steps are:
Step 1: Note down the number whose absolute value you desire to discover.
Step 2: If the figure is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not convert it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the expression is the number you have subsequently steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on both side of a number or expression, similar to this: |x|.
Example 1
To set out, 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 work this out, we need to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to solve x.
Step 2: By utilizing the fundamental characteristics, we know that the absolute value of the sum of these two numbers is as same as 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 see, x equals 15, so its distance from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's check out one more 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 follow the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to calculate the value x, so we'll begin by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.
Step 4: Therefore, the first equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can include several intricate figures or rational numbers in mathematical settings; however, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this states it is distinguishable everywhere. The following formula provides 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 distance 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 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 distinctable at 0.
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