Absolute ValueDefinition, How to Find Absolute Value, Examples
A lot of people perceive absolute value as the length from zero to a number line. And that's not wrong, but it's by no means the entire story.
In mathematics, an absolute value is the extent of a real number without regard to its sign. So the absolute value is all the time a positive number or zero (0). Let's observe at what absolute value is, how to find absolute value, few 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 without considering its sign. This signifies if you have a negative number, the absolute value of that figure is the number without the negative sign.
Meaning of Absolute Value
The previous explanation means that the absolute value is the length of a figure from zero on a number line. So, if you think about it, the absolute value is the length or distance a number 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 number is the length of the figure is from zero on the number line. The absolute value of negative five is 5 due to the fact it is five units apart from zero on the number line.
Examples
If we graph negative three on a line, we can watch that it is three units apart from zero:
The absolute value of negative three is three.
Presently, let's look at more absolute value example. Let's assume we hold an absolute value of sin. We can plot this on a number line as well:
The absolute value of 6 is 6. Hence, what does this tell us? It shows us that absolute value is always positive, even if the number itself is negative.
How to Find the Absolute Value of a Figure or Expression
You need to know few things prior going into how to do it. A few closely linked characteristics will assist you comprehend how the figure within the absolute value symbol works. Fortunately, what we have here is an definition of the following four rudimental features of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is at all time positive or zero (0).
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 expression.
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 these 4 basic characteristics in mind, let's take a look at two more beneficial properties 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 variance between two real numbers is less than or equal to the absolute value of the sum of their absolute values.
Considering that we know these properties, we can ultimately begin learning how to do it!
Steps to Find the Absolute Value of a Figure
You are required to follow few 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 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 figure is the number you obtain after steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on either side of a figure or expression, like this: |x|.
Example 1
To begin with, 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 figure this out, we are required to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:
Step 1: We are given the equation |x+5| = 20, and we are required to calculate the absolute value within the equation to solve x.
Step 2: By utilizing the fundamental properties, we understand that the absolute value of the total of these two expressions is as same as the total 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 calculate 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 one more absolute value example. We'll utilize the absolute value function to solve a new equation, like |x*3| = 6. To make it, we again have to observe the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to calculate the value x, so we'll initiate 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 initial equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can contain a lot of complicated numbers 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 constant function, this states it is differentiable everywhere. 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 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 zero(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 equal. The left-hand limit is stated 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 distinguishable at zero (0).
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