Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people comprehend absolute value as the distance from zero to a number line. And that's not wrong, but it's nowhere chose to the entire story.
In mathematics, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is at all time a positive zero or number (0). Let's check at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the extent of a real number irrespective to its sign. That means if you have a negative number, the absolute value of that number is the number ignoring the negative sign.
Definition of Absolute Value
The last definition refers that the absolute value is the length of a number from zero on a number line. Hence, if you think about it, the absolute value is the distance or length a figure 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 the length of 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 graph negative three on a line, we can watch that it is three units away from zero:
The absolute value of negative three is three.
Well then, let's look at more absolute value example. Let's say we have an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. Therefore, what does this tell us? It states that absolute value is at all times positive, even if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should know a handful of things before going into how to do it. A few closely related characteristics will support you grasp how the expression within the absolute value symbol works. Luckily, what we have here is an definition of the following four essential features of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of ever 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 expression.
Addition: The absolute value of a sum is less than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these four basic properties 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 zero (0) or positive.
Triangle inequality: The absolute value of the difference within two real numbers is less than or equal to the absolute value of the total of their absolute values.
Considering that we learned these characteristics, we can in the end initiate learning how to do it!
Steps to Calculate the Absolute Value of a Number
You need to obey few steps to find the absolute value. These steps are:
Step 1: Note down the expression of whom’s absolute value you want to discover.
Step 2: If the number is negative, multiply it by -1. This will make the number positive.
Step3: If the expression is positive, do not convert it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the figure is the number you have after steps 2, 3 or 4.
Remember that the absolute value symbol 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, like |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we have to locate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We have the equation |x+5| = 20, and we have to find the absolute value inside the equation to solve x.
Step 2: By utilizing the basic properties, we know that the absolute value of the sum of these two numbers 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 get rid of the vertical bars: x+5 = 20
Step 4: Let's calculate for x: x = 20-5, x = 15
As we see, x equals 15, so its distance from zero will also be equivalent 15, and the equation above is right.
Example 2
Now let's check out another absolute value example. We'll utilize the absolute value function to find a new equation, such as |x*3| = 6. To get there, we again have to observe 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 gives 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 involve many complex figures or rational numbers in mathematical settings; however, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this refers it is varied everywhere. The following formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area 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 consistent 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 equal. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is given by:
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 zero (0).
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