Absolute ValueDefinition, How to Discover Absolute Value, Examples
A lot of people comprehend absolute value as the length from zero to a number line. And that's not wrong, but it's not the complete story.
In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is at all time a positive zero or number (0). Let's observe at what absolute value is, how to calculate absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the extent of a real number without regard to its sign. This signifies if you have a negative figure, the absolute value of that figure is the number disregarding the negative sign.
Definition of Absolute Value
The last definition states that the absolute value is the distance 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 see it if you check out a real number line:
As demonstrated, the absolute value of a figure is how far away the number is from zero on the number line. The absolute value of negative five is 5 due to the fact 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 negative three is 3.
Well then, 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 6 is 6. So, what does this mean? It tells us that absolute value is constantly positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Expression or Number
You should know a couple of things before going into how to do it. A handful of closely related properties will assist you understand how the expression within the absolute value symbol functions. Luckily, here we have an definition of the following four fundamental properties of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is always zero (0) or positive.
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 equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these four fundamental characteristics in mind, let's check out two more helpful 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 variance among two real numbers is less than or equal to the absolute value of the total of their absolute values.
Considering that we learned these properties, we can finally begin learning how to do it!
Steps to Calculate the Absolute Value of a Figure
You have to obey a handful of steps to find 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 convert the number to positive.
Step3: If the expression is positive, do not convert it.
Step 4: Apply all characteristics significant to the absolute value equations.
Step 5: The absolute value of the expression is the expression you get following steps 2, 3 or 4.
Remember that the absolute value sign 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 are required 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 have the equation |x+5| = 20, and we are required to discover the absolute value within the equation to find x.
Step 2: By utilizing the fundamental characteristics, we know that the absolute value of the addition 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 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 equivalent 15, and the equation above is right.
Example 2
Now let's try one more absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To get there, we again have to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to calculate the value 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 solutions: x = 2 and x = -2.
Step 4: So, the first equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can contain several intricate expressions 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 varied everywhere. The ensuing formula gives 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 constant 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 equal. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is provided as:
I'm →0+(|x|/x)
Since 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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