Absolute ValueDefinition, How to Find Absolute Value, Examples
A lot of people perceive absolute value as the distance 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 without regard to its sign. So the absolute value is all the time a positive zero or number (0). Let's observe at what absolute value is, how to find absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is constantly positive or zero (0). It is the extent of a real number without considering its sign. This signifies if you hold a negative number, the absolute value of that figure is the number disregarding the negative sign.
Meaning of Absolute Value
The previous definition states that the absolute value is the distance of a number from zero on a number line. Hence, if you consider it, the absolute value is the length or distance a figure 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 the length of the figure is from zero on the number line. The absolute value of -5 is 5 because it is five units apart 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 3.
Well then, let's check out another absolute value example. Let's say we have an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. Hence, what does this mean? It states that absolute value is constantly positive, even if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You need to know few points before going into how to do it. A few closely associated properties will support you grasp how the figure within the absolute value symbol functions. Thankfully, here we have an meaning of the following 4 essential characteristics of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of any real number is always positive or zero (0).
Identity: The absolute value of a positive number is the figure itself. Otherwise, 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 total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned 4 fundamental 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 constantly positive or zero (0).
Triangle inequality: The absolute value of the difference within two real numbers is lower than or equal to the absolute value of the sum of their absolute values.
Taking into account that we know these characteristics, we can in the end begin learning how to do it!
Steps to Discover the Absolute Value of a Expression
You need to observe a handful of steps to discover 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 change it to a positive number.
Step3: If the expression is positive, do not convert it.
Step 4: Apply all properties relevant 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 both side of a expression or number, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To work 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 priorly:
Step 1: We are given the equation |x+5| = 20, and we must find the absolute value within the equation to solve x.
Step 2: By using the essential characteristics, we know 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 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 equal 15, and the equation above is genuine.
Example 2
Now let's work on another 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 find the value of x, so we'll begin by dividing 3 from both side of the equation. This step offers 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 answers, x=2 and x=-2.
Absolute value can contain a lot of intricate figures or rational numbers in mathematical settings; nevertheless, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, meaning it is distinguishable everywhere. 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 zero (0), and the distance 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 zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 due to the the left-hand limit and the right-hand limit are not uniform. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is provided 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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