Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people comprehend absolute value as the distance from zero to a number line. And that's not inaccurate, but it's nowhere chose to the complete story.
In mathematics, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is at all time a positive zero or number (0). Let's check 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 hold a negative number, the absolute value of that number is the number ignoring the negative sign.
Meaning of Absolute Value
The prior definition states that the absolute value is the length of a number from zero on a number line. So, if you think about that, the absolute value is the length or distance a figure has from zero. You can see it if you look at a real number line:
As shown, the absolute value of a figure is how far away the figure is from zero on the number line. The absolute value of -5 is five because it is 5 units away from zero on the number line.
Examples
If we graph -3 on a line, we can see that it is three units away from zero:
The absolute value of -3 is 3.
Well then, 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 6 is 6. So, what does this refer to? 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 Number
You should know a couple of things before working on how to do it. A few closely linked features will support you grasp how the figure within the absolute value symbol works. Luckily, here we have an meaning of the ensuing 4 fundamental characteristics of absolute value.
Basic Characteristics 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 number itself. Instead, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a total is lower 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 4 basic characteristics in mind, let's take a look at two other beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the variance between two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Considering that we know these properties, we can in the end initiate learning how to do it!
Steps to Discover the Absolute Value of a Number
You are required to observe few steps to calculate the absolute value. These steps are:
Step 1: Note down the number whose absolute value you want to calculate.
Step 2: If the expression is negative, multiply it by -1. This will convert the number to positive.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all properties significant to the absolute value equations.
Step 5: The absolute value of the number is the figure you have following 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 start out, let's consider 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 calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We have the equation |x+5| = 20, and we are required to calculate the absolute value inside the equation to solve x.
Step 2: By using the basic characteristics, we learn that the absolute value of the sum 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 unknown, so let's eliminate the vertical bars: x+5 = 20
Step 4: Let's solve 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 check out one more absolute value example. We'll use the absolute value function to solve a new equation, such as |x*3| = 6. To make it, we again need to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We need to calculate the value x, so we'll start by dividing 3 from each side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.
Step 4: So, the initial equation |x*3| = 6 also has two possible answers, x=2 and x=-2.
Absolute value can involve many complicated figures 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, meaning it is distinguishable everywhere. The following formula offers 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 rises 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 distinctable at 0 because the left-hand limit and the right-hand limit are not equal. The left-hand limit is provided as:
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 zero (0).
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