Absolute ValueDefinition, How to Calculate Absolute Value, Examples
A lot of people comprehend absolute value as the length from zero to a number line. And that's not incorrect, but it's by no means 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 discover absolute value, several examples of absolute value, and the absolute value derivative.
Explanation 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. That means if you have a negative figure, the absolute value of that figure is the number without the negative sign.
Meaning of Absolute Value
The previous definition states that the absolute value is the length of a number from zero on a number line. Therefore, if you think about it, the absolute value is the distance or length a number has from zero. You can see it if you take a look at a real number line:
As demonstrated, the absolute value of a figure is how far away the figure is from zero on the number line. The absolute value of negative five is five due to the fact it is five units away from zero on the number line.
Examples
If we graph negative three on a line, we can see that it is three units away from zero:
The absolute value of -3 is three.
Presently, let's look at more absolute value example. Let's say 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 refer to? 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 Figure or Expression
You should be aware of a handful of things before going into how to do it. A few closely linked features will support you understand how the number inside the absolute value symbol functions. Fortunately, here we have an meaning of the following four fundamental properties of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is constantly zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Otherwise, 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 total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned 4 basic properties in mind, let's look at two more useful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is at all times positive or zero (0).
Triangle inequality: The absolute value of the difference within two real numbers is less 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 Calculate the Absolute Value of a Number
You need to follow a handful of steps to find the absolute value. These steps are:
Step 1: Jot down the figure of whom’s absolute value you want to calculate.
Step 2: If the expression 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 applicable to the absolute value equations.
Step 5: The absolute value of the figure is the figure you get after steps 2, 3 or 4.
Bear in mind that the absolute value sign is two vertical bars on either side of a figure or expression, similar to this: |x|.
Example 1
To begin with, let's consider an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we have to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:
Step 1: We are provided with the equation |x+5| = 20, and we are required to find the absolute value within the equation to solve x.
Step 2: By using the essential properties, we know that the absolute value of the total 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 get rid of 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 be as same as 15, and the equation above is right.
Example 2
Now let's work on one more absolute value example. We'll use the absolute value function to solve a new equation, similar to |x*3| = 6. To make it, we again need to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We have to solve for x, so we'll initiate by dividing 3 from both side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two potential solutions: x = 2 and x = -2.
Step 4: Hence, the original equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can contain a lot of complex expressions or rational numbers in mathematical settings; still, that is something we will work on separately to this.
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 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 length 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 differentiable 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 distinguishable at 0.
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