Absolute ValueMeaning, 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 magnitude of a real number irrespective of its sign. So the absolute value is at all time a positive number or zero (0). Let's look at what absolute value is, how to calculate absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the magnitude of a real number irrespective to its sign. This signifies if you hold a negative number, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The prior 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 see it if you check out a real number line:
As shown, the absolute value of a figure is the distance of the figure is from zero on the number line. The absolute value of negative five is five because it is five units apart from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is 3 units away from zero:
The absolute value of negative three is three.
Now, let's check out another absolute value example. Let's assume we hold an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. So, what does this mean? It tells us that absolute value is at all times positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should know a couple of things before going into how to do it. A few closely related properties will help you understand how the figure inside the absolute value symbol functions. Fortunately, here we have an meaning of the following four rudimental properties of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of all real number is always zero (0) or positive.
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 expression.
Addition: The absolute value of a total is lower than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these four basic properties 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 difference within two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.
Considering that we learned these characteristics, we can finally start learning how to do it!
Steps to Discover the Absolute Value of a Expression
You need to obey few steps to discover the absolute value. These steps are:
Step 1: Jot down the number of whom’s 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 expression is positive, do not change it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the figure you obtain subsequently steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on either side of a expression or number, like this: |x|.
Example 1
To begin with, let's presume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To work 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 mentioned above:
Step 1: We have the equation |x+5| = 20, and we have to discover the absolute value inside the equation to solve x.
Step 2: By using the essential characteristics, we know that the absolute value of the sum of these two expressions is the same as the total of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unidentified, 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 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 check out another absolute value example. We'll use the absolute value function to solve a new equation, such as |x*3| = 6. To make it, we again have to observe the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required to calculate the value 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 possible results: x = 2 and x = -2.
Step 4: Therefore, the first equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can contain many complicated values 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, 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 0, and the length is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is constant at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 reason being the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is stated 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 distinctable at zero (0).
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