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 incorrect, but it's by no means the entire 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 observe at what absolute value is, how to discover absolute value, few examples of absolute value, and the absolute value derivative.
What Is 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 hold a negative number, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The previous explanation means that the absolute value is the distance of a figure from zero on a number line. Hence, if you think about it, the absolute value is the length or distance a number has from zero. You can visualize it if you take a look at a real number line:
As you can see, the absolute value of a figure is how far away the number is from zero on the number line. The absolute value of -5 is five due to the fact it is 5 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.
Well then, let's check out 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 tell us? It tells us that absolute value is at all times positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Number or Expression
You should be aware of a handful of points prior working on how to do it. A handful of closely linked characteristics will help you grasp how the number inside the absolute value symbol functions. Thankfully, what we have here is an explanation of the ensuing four rudimental properties of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of any real number is at all time zero (0) or positive.
Identity: The absolute value of a positive number is the figure 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 lower than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned four fundamental properties in mind, let's take a look at two other helpful characteristics 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 less than or equivalent to the absolute value of the total of their absolute values.
Taking into account that we went through these characteristics, we can in the end start learning how to do it!
Steps to Discover the Absolute Value of a Expression
You need to obey a handful of steps to discover the absolute value. These steps are:
Step 1: Jot down the number of whom’s absolute value you desire to find.
Step 2: If the number is negative, multiply it by -1. This will convert the number to positive.
Step3: If the figure is positive, do not convert it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the expression is the figure you get after 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 assume 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 are required 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 are provided with the equation |x+5| = 20, and we are required to calculate the absolute value inside the equation to find x.
Step 2: By using the fundamental characteristics, we understand that the absolute value of the addition of these two expressions is as same as the sum 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 calculate for x: x = 20-5, x = 15
As we can observe, x equals 15, so its distance from zero will also equal 15, and the equation above is genuine.
Example 2
Now let's check out one more absolute value example. We'll use the absolute value function to find a new equation, like |x*3| = 6. To make it, we again have to follow the steps:
Step 1: We use 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 gives us |x| = 2.
Step 3: |x| = 2 has two possible answers: x = 2 and x = -2.
Step 4: Therefore, the original equation |x*3| = 6 also has two potential results, x=2 and x=-2.
Absolute value can contain many complicated expressions or rational numbers in mathematical settings; nevertheless, 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 differentiable 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 zero (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 zero(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 stated 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 0.
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