Absolute ValueMeaning, How to Find 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 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 discover absolute value, few examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the extent of a real number without considering its sign. This refers that if you have a negative number, the absolute value of that figure is the number without the negative sign.
Definition 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 figure is the distance of the figure is from zero on the number line. The absolute value of negative five is five reason being 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 -3 is 3.
Now, let's check out another absolute value example. Let's suppose we have an absolute value of 6. We can plot this on a number line as well:
The absolute value of six is 6. Hence, what does this tell us? It shows us 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 Number
You need to know few points before going into how to do it. A couple of closely linked properties will help you comprehend how the expression inside the absolute value symbol functions. Fortunately, what we have here is an meaning of the following four essential features of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is always positive or zero (0).
Identity: The absolute value of a positive number is the number itself. Instead, 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 four basic characteristics in mind, let's take a look at two other beneficial characteristics 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 equal to the absolute value of the sum of their absolute values.
Taking into account that we went through these characteristics, we can finally begin learning how to do it!
Steps to Find the Absolute Value of a Expression
You are required to observe a couple of steps to calculate the absolute value. These steps are:
Step 1: Jot down the number whose 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 alter it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the number is the expression you obtain subsequently steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on either side of a figure or number, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To figure this out, we have to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:
Step 1: We have the equation |x+5| = 20, and we are required to calculate the absolute value within the equation to get x.
Step 2: By utilizing the basic properties, we know that the absolute value of the total of these two expressions is the same as the sum of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unidentified, so let's remove 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 be equivalent 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 get a new equation, like |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 are required to find the value of x, so we'll initiate by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible answers: x = 2 and x = -2.
Step 4: Hence, the first equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can include a lot of intricate expressions 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 varied at any given point. The ensuing formula provides 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 range is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is consistent 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 provided 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 distinguishable at zero (0).
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