Absolute ValueDefinition, How to Calculate 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 whole story.
In math, an absolute value is the extent of a real number without considering its sign. So the absolute value is always a positive number or zero (0). Let's look at what absolute value is, how to calculate absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is always zero (0) or positive. It is the extent of a real number irrespective to its sign. This refers that if you have a negative number, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The last definition means that the absolute value is the length of a number from zero on a number line. Therefore, if you think about that, the absolute value is the length or distance a number 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 the length of the figure is from zero on the number line. The absolute value of -5 is five because it is five units away from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is three units apart from zero:
The absolute value of -3 is three.
Presently, let's look at one more absolute value example. Let's suppose we have an absolute value of 6. We can graph this on a number line as well:
The absolute value of 6 is 6. Therefore, what does this mean? It shows us that absolute value is constantly positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You should be aware of a handful of points before going into how to do it. A handful of closely linked characteristics will assist you grasp how the number inside the absolute value symbol works. Thankfully, here we have an definition of the following 4 rudimental characteristics of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is at all time 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 figure.
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 4 basic properties in mind, let's look at two more beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is at all times zero (0) or positive.
Triangle inequality: The absolute value of the difference within two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Considering that we learned these properties, we can in the end start learning how to do it!
Steps to Find the Absolute Value of a Figure
You need to follow a handful of steps to calculate the absolute value. These steps are:
Step 1: Jot down the number of whom’s absolute value you want to find.
Step 2: If the number is negative, multiply it by -1. This will convert the number to positive.
Step3: If the expression is positive, do not change it.
Step 4: Apply all properties significant to the absolute value equations.
Step 5: The absolute value of the number is the number you get after steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on both side of a figure or number, similar to this: |x|.
Example 1
To set out, 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 need to locate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:
Step 1: We are provided with the equation |x+5| = 20, and we must discover the absolute value within the equation to solve x.
Step 2: By utilizing the essential properties, we understand that the absolute value of the total 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 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 right.
Example 2
Now let's try another absolute value example. We'll use the absolute value function to get a new equation, like |x*3| = 6. To get there, we again have to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to calculate the value 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 potential solutions: x = 2 and x = -2.
Step 4: Therefore, the first equation |x*3| = 6 also has two potential results, x=2 and x=-2.
Absolute value can include many complicated numbers 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 ensuing formula offers 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 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 distinctable at 0 due to the 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 given by:
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 differentiable at zero (0).
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