Absolute ValueDefinition, How to Calculate Absolute Value, Examples
A lot of people perceive absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the complete story.
In math, an absolute value is the magnitude of a real number without regard to its sign. So the absolute value is all the time a positive number or zero (0). Let's observe 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 number is always positive or zero (0). It is the magnitude of a real number without regard to its sign. This signifies if you hold a negative number, the absolute value of that figure is the number overlooking the negative sign.
Meaning of Absolute Value
The prior definition means that the absolute value is the length of a number from zero on a number line. Therefore, if you think about it, the absolute value is the distance or length a figure has from zero. You can visualize it if you look at a real number line:
As you can see, the absolute value of a number is how far away the figure is from zero on the number line. The absolute value of -5 is five reason being it is 5 units away from zero on the number line.
Examples
If we plot -3 on a line, we can watch that it is 3 units away from zero:
The absolute value of negative three is three.
Presently, let's check out another absolute value example. Let's say we hold 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 refer to? It tells us 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 need to know a handful of points before working on how to do it. A couple of closely associated features will assist you comprehend how the figure inside the absolute value symbol functions. Luckily, what we have here is an explanation of the ensuing four rudimental characteristics of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of all 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 figure.
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 equal to the product of absolute values.
With above-mentioned 4 essential properties in mind, let's take a look at two more beneficial properties 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 among two real numbers is lower than or equal to the absolute value of the sum of their absolute values.
Considering that we know these properties, we can ultimately initiate learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You need to follow a handful of steps to find the absolute value. These steps are:
Step 1: Write down the figure whose absolute value you desire to discover.
Step 2: If the expression is negative, multiply it by -1. This will convert the number to positive.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the number is the number you have following steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on both side of a number or expression, similar to 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 figure this out, we are required 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 are given the equation |x+5| = 20, and we are required to find the absolute value inside the equation to solve x.
Step 2: By utilizing the essential properties, we learn that the absolute value of the sum of these two numbers is the 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 get rid of 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 distance from zero will also be as same as 15, and the equation above is true.
Example 2
Now let's try another absolute value example. We'll utilize the absolute value function to solve a new equation, similar to |x*3| = 6. To make it, we again need to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to find the value of 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 first equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can involve many intricate figures or rational numbers in mathematical settings; nevertheless, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is distinguishable 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 length is all positive real numbers. The absolute value function rises 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 distinctable at 0 due to the the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is stated 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 differentiable at zero (0).
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