Absolute ValueMeaning, How to Calculate Absolute Value, Examples
A lot of people think of 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 regard to its sign. So the absolute value is all the time a positive number or zero (0). Let's look 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 figure is at all times positive or zero (0). It is the magnitude 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 definition refers that the absolute value is the length of a number from zero on a number line. So, if you think about it, the absolute value is the distance or length 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 number is how far away the number is from zero on the number line. The absolute value of negative five is 5 because it is 5 units apart from zero on the number line.
Examples
If we plot -3 on a line, we can watch that it is 3 units away from zero:
The absolute value of negative three is 3.
Now, let's look at one more absolute value example. Let's assume we have an absolute value of 6. We can graph this on a number line as well:
The absolute value of six is 6. So, what does this tell us? It states that absolute value is always positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Number or Figure
You should be aware of a handful of things before going into how to do it. A few closely linked properties will support you grasp how the figure inside the absolute value symbol works. Thankfully, what we have here is an definition of the ensuing four rudimental characteristics of absolute value.
Fundamental 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 figure itself. Otherwise, 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 equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 basic properties in mind, let's check out two more helpful properties of the absolute value:
Positive definiteness: The absolute value of any real number is constantly zero (0) or positive.
Triangle inequality: The absolute value of the variance among two real numbers is lower than or equal to the absolute value of the sum of their absolute values.
Taking into account that we know these properties, we can ultimately start learning how to do it!
Steps to Discover the Absolute Value of a Figure
You need to obey a couple of steps to find the absolute value. These steps are:
Step 1: Write down the figure of whom’s absolute value you desire to calculate.
Step 2: If the number is negative, multiply it by -1. This will make the number positive.
Step3: If the number is positive, do not change it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the figure is the number you get subsequently steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on either side of a number or expression, like this: |x|.
Example 1
To start 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 solve this, 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 have the equation |x+5| = 20, and we must discover the absolute value within the equation to solve x.
Step 2: By using the fundamental characteristics, we learn that the absolute value of the sum of these two figures 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 eliminate the vertical bars: x+5 = 20
Step 4: Let's calculate 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 right.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to solve a new equation, such as |x*3| = 6. To get there, we again need to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to find the value of 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 solutions: 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 include several intricate expressions or rational numbers in mathematical settings; nevertheless, 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 differentiable at any given point. The following formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except 0, and the distance 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 due to the the left-hand limit and the right-hand limit are not uniform. The left-hand limit is given by:
I'm →0−(|x|/x)
The right-hand limit is offered 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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