Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people perceive absolute value as the length from zero to a number line. And that's not inaccurate, but it's nowhere chose to the whole story.
In mathematics, an absolute value is the extent of a real number without considering its sign. So the absolute value is at all time a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a number is at all times zero (0) or positive. It is the magnitude of a real number irrespective to its sign. This refers that if you have a negative figure, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The prior explanation refers that the absolute value is the length of a number from zero on a number line. Therefore, if you consider it, the absolute value is the distance or length a number has from zero. You can visualize it if you look at a real number line:
As demonstrated, the absolute value of a number is the distance of the number is from zero on the number line. The absolute value of -5 is five due to the fact 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.
Presently, let's look at 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 mean? It tells us that absolute value is at all times positive, even though the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You need to know a couple of things before going into how to do it. A few closely associated features will assist you comprehend how the figure within the absolute value symbol works. Fortunately, what we have here is an meaning of the following four rudimental features of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is always positive or zero (0).
Identity: The absolute value of a positive number is the number 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 sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned 4 essential characteristics in mind, let's look at two more helpful 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 between two real numbers is lower than or equal to the absolute value of the sum of their absolute values.
Considering that we learned these characteristics, we can finally start learning how to do it!
Steps to Find the Absolute Value of a Number
You have to follow few steps to calculate the absolute value. These steps are:
Step 1: Note down the number whose absolute value you desire to find.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the figure is positive, do not convert it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the expression is the figure you obtain subsequently steps 2, 3 or 4.
Keep in mind that the absolute value symbol is two vertical bars on either side of a figure or expression, like this: |x|.
Example 1
To begin with, let's presume an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To figure this out, we are required to locate 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 must find the absolute value inside the equation to find x.
Step 2: By utilizing the essential characteristics, we know that the absolute value of the addition of these two numbers 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 solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's work on one more absolute value example. We'll utilize the absolute value function to solve a new equation, similar to |x*3| = 6. To get there, we again have to follow the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to solve for x, so we'll begin 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: Hence, the initial equation |x*3| = 6 also has two potential solutions, x=2 and x=-2.
Absolute value can include several complicated numbers or rational numbers in mathematical settings; still, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this refers it is distinguishable at any given point. The following formula provides 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 range is all positive real numbers. The absolute value function increases for all x<0 and all x>0. The absolute value function is consistent at 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable 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 provided 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 distinctable at 0.
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