Absolute ValueDefinition, How to Find Absolute Value, Examples
Many comprehend absolute value as the distance from zero to a number line. And that's not incorrect, but it's nowhere chose to the entire story.
In math, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is always a positive number or zero (0). Let's check 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 number is always zero (0) or positive. It is the magnitude 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 overlooking the negative sign.
Meaning of Absolute Value
The last definition states that the absolute value is the distance of a number from zero on a number line. So, if you think about 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 shown, the absolute value of a number is the distance of the figure is from zero on the number line. The absolute value of -5 is five because it is five units apart from zero on the number line.
Examples
If we graph -3 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 check out another 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 six is 6. Therefore, what does this tell us? It tells us that absolute value is constantly positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You should be aware of a handful of things before working on how to do it. A handful of closely related features will help you comprehend how the figure within the absolute value symbol works. Luckily, here we have an explanation of the ensuing four essential features of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of ever real number is at all time positive or zero (0).
Identity: The absolute value of a positive number is the figure itself. Instead, 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 equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these 4 basic properties in mind, let's look at two more useful 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 between two real numbers is less than or equal to the absolute value of the total of their absolute values.
Now that we know these properties, we can in the end initiate learning how to do it!
Steps to Calculate the Absolute Value of a Number
You are required to observe a handful of steps to calculate the absolute value. These steps are:
Step 1: Note down the number of whom’s absolute value you want to discover.
Step 2: If the figure is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not change it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the number is the figure you obtain subsequently steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on either side of a expression or number, like this: |x|.
Example 1
To begin with, let's presume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To figure this out, we need to locate the absolute value of the two numbers in the inequality. We can do this by following the steps above:
Step 1: We have the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to find x.
Step 2: By utilizing the basic properties, we know that the absolute value of the addition of these two figures is equivalent to 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 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 true.
Example 2
Now let's try another absolute value example. We'll use the absolute value function to find a new equation, similar to |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 start by dividing 3 from both 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: So, the original equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can involve several complex numbers 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 varied at any given point. The ensuing formula provides the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except 0, and the length 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 distinguishable at 0 because the left-hand limit and the right-hand limit are not uniform. The left-hand limit is given by:
I'm →0−(|x|/x)
The right-hand limit is offered 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 distinctable at 0.
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