Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people perceive absolute value as the distance from zero to a number line. And that's not wrong, but it's not the whole story.
In math, an absolute value is the extent of a real number without considering its sign. So the absolute value is at all time a positive number or zero (0). Let's observe 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 number is constantly zero (0) or positive. It is the magnitude of a real number without regard to its sign. This signifies if you hold a negative figure, the absolute value of that number is the number ignoring the negative sign.
Meaning of Absolute Value
The previous explanation means that the absolute value is the length of a number from zero on a number line. Hence, if you think about that, the absolute value is the length or distance a figure has from zero. You can see it if you check out a real number line:
As shown, the absolute value of a number 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 apart from zero on the number line.
Examples
If we plot negative three on a line, we can observe that it is three units away from zero:
The absolute value of negative three is 3.
Well then, let's check out one more absolute value example. Let's assume we hold an absolute value of 6. We can graph this on a number line as well:
The absolute value of 6 is 6. So, what does this tell us? It shows us that absolute value is at all times positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You should be aware of a couple of points before working on how to do it. A handful of closely linked characteristics will assist you comprehend how the number within the absolute value symbol works. Fortunately, what we have here is an explanation of the following four rudimental characteristics of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of ever real number is constantly zero (0) or positive.
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 number.
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 equal to the product of absolute values.
With these four fundamental properties in mind, let's check out two more beneficial 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 difference among two real numbers is less than or equivalent to the absolute value of the total of their absolute values.
Now that we know these characteristics, we can in the end begin learning how to do it!
Steps to Calculate the Absolute Value of a Number
You have to obey a handful of steps to calculate the absolute value. These steps are:
Step 1: Jot down the figure whose absolute value you desire to discover.
Step 2: If the expression is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not change it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the number is the number you obtain subsequently steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on both side of a expression or number, like this: |x|.
Example 1
To begin with, let's assume 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 are required to find the absolute value of the two numbers in the inequality. We can do this by following the steps above:
Step 1: We have the equation |x+5| = 20, and we must discover the absolute value within the equation to find x.
Step 2: By using the essential characteristics, we understand that the absolute value of the addition 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 unidentified, 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 distance from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's work on another absolute value example. We'll utilize the absolute value function to solve a new equation, like |x*3| = 6. To get there, we again need to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We are required to solve for x, so we'll begin by dividing 3 from both 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: So, the original equation |x*3| = 6 also has two potential answers, x=2 and x=-2.
Absolute value can include many complex numbers 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, meaning it is distinguishable at any given point. The ensuing formula offers 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 constant at 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is given by:
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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