Absolute ValueDefinition, How to Find Absolute Value, Examples
Many comprehend absolute value as the length from zero to a number line. And that's not inaccurate, 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 all the time a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is at all times zero (0) or positive. It is the magnitude of a real number irrespective to its sign. That means if you possess a negative figure, the absolute value of that figure is the number disregarding the negative sign.
Definition of Absolute Value
The previous 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 length or distance a number has from zero. You can visualize it if you check out a real number line:
As demonstrated, the absolute value of a number is how far away the number is from zero on the number line. The absolute value of -5 is 5 because it is five units away from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is three units away from zero:
The absolute value of negative three is 3.
Now, let's look at another absolute value example. Let's assume we have an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. Therefore, what does this mean? It states that absolute value is always positive, regardless if the number itself is negative.
How to Locate the Absolute Value of a Figure or Expression
You need to know few things before going into how to do it. A few closely associated characteristics will support you understand how the expression inside the absolute value symbol works. Luckily, here we have an explanation of the following 4 rudimental properties of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is constantly 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 number.
Addition: The absolute value of a sum is lower than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned four basic properties in mind, let's take a 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 between two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.
Considering that we went through these characteristics, we can ultimately start learning how to do it!
Steps to Discover the Absolute Value of a Number
You need to observe a handful of steps to discover the absolute value. These steps are:
Step 1: Write down the figure whose absolute value you desire to calculate.
Step 2: If the number 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 applicable to the absolute value equations.
Step 5: The absolute value of the number is the number you get subsequently 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, like this: |x|.
Example 1
To begin with, 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 locate 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 have to calculate the absolute value within the equation to get x.
Step 2: By utilizing the fundamental characteristics, we understand that the absolute value of the total 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 remove 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 be as same as 15, and the equation above is true.
Example 2
Now let's try one more absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To make it, we again have to follow the steps:
Step 1: We have 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 results: x = 2 and x = -2.
Step 4: Hence, the first equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can include several complicated figures or rational numbers in mathematical settings; still, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this states it is distinguishable at any given point. The ensuing formula provides 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 consistent at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is given by:
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 distinctable at zero (0).
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