Absolute ValueMeaning, How to Calculate Absolute Value, Examples
Many perceive absolute value as the distance from zero to a number line. And that's not inaccurate, but it's by no means the complete story.
In mathematics, an absolute value is the extent of a real number without considering its sign. So the absolute value is always a positive number or zero (0). Let's observe 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 extent of a real number without considering its sign. That means if you possess a negative figure, the absolute value of that figure is the number disregarding the negative sign.
Meaning of Absolute Value
The last 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 length or distance a number has from zero. You can see it if you check out 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 five because 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.
Now, let's look at more absolute value example. Let's say 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 Locate the Absolute Value of a Expression or Number
You should know few things prior going into how to do it. A couple of closely related features will help you comprehend how the figure within the absolute value symbol functions. Luckily, here we have an explanation of the ensuing four rudimental properties of absolute value.
Essential 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 figure 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 less than or equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these 4 essential properties in mind, let's check out two other helpful properties of the absolute value:
Positive definiteness: The absolute value of any real number is at all times positive or zero (0).
Triangle inequality: The absolute value of the difference among two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Now that we went through these characteristics, we can in the end start learning how to do it!
Steps to Find the Absolute Value of a Figure
You are required to follow a couple of steps to discover the absolute value. These steps are:
Step 1: Jot down the number of whom’s absolute value you desire to calculate.
Step 2: If the number 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 characteristics applicable to the absolute value equations.
Step 5: The absolute value of the expression is the expression you get following steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a number or expression, like this: |x|.
Example 1
To begin with, 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 figure this out, we have to locate the absolute value of the two numbers in the inequality. We can do this by observing the steps above:
Step 1: We are provided with the equation |x+5| = 20, and we have to calculate the absolute value inside the equation to solve x.
Step 2: By utilizing the fundamental properties, we learn that the absolute value of the total of these two numbers 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 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 need to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required to calculate the value x, so we'll initiate by dividing 3 from each 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 first equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can include several complex 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 constant function, this states it is distinguishable at any given point. The ensuing formula gives 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 increases 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 differentiable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is given by:
I'm →0−(|x|/x)
The right-hand limit is given by:
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 0.
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