Absolute ValueDefinition, How to Discover Absolute Value, Examples
A lot of people comprehend absolute value as the length from zero to a number line. And that's not wrong, but it's not the whole story.
In mathematics, an absolute value is the extent of a real number without regard to its sign. So the absolute value is always a positive zero or number (0). Let's look at what absolute value is, how to calculate absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is always positive or zero (0). It is the magnitude of a real number without considering its sign. This signifies if you possess a negative figure, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The prior explanation means that the absolute value is the length of a figure from zero on a number line. So, if you think about that, the absolute value is the length or distance a figure has from zero. You can observe it if you check out a real number line:
As you can see, the absolute value of a figure is how far away the figure is from zero on the number line. The absolute value of -5 is five due to the fact it is 5 units away from zero on the number line.
Examples
If we graph -3 on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is 3.
Well then, let's look at more 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. Hence, what does this tell us? It states that absolute value is at all times positive, even if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You need to know a couple of things prior going into how to do it. A couple of closely linked properties will help you understand how the number within the absolute value symbol works. Luckily, what we have here is an definition of the following 4 fundamental properties of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of all real number is always 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 figure.
Addition: The absolute value of a total is less than or equivalent to the sum of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned 4 basic characteristics 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 zero (0) or positive.
Triangle inequality: The absolute value of the difference between two real numbers is less than or equal to the absolute value of the total of their absolute values.
Now 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 observe a handful of steps to calculate the absolute value. These steps are:
Step 1: Write down the expression whose absolute value you desire to calculate.
Step 2: If the figure is negative, multiply it by -1. This will convert the number to positive.
Step3: If the number is positive, do not change it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the expression is the figure you have after steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on both side of a figure or expression, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we need 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 are given the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to find x.
Step 2: By using the essential properties, we learn that the absolute value of the sum of these two expressions 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 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 try one more absolute value example. We'll utilize the absolute value function to find a new equation, like |x*3| = 6. To get there, we again have to observe the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We have 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 results: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can involve many complicated figures or rational numbers in mathematical settings; still, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this refers it is differentiable at any given point. The ensuing 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 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 because 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 offered 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 differentiable at zero (0).
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