Absolute ValueDefinition, How to Find Absolute Value, Examples
Many think of absolute value as the distance from zero to a number line. And that's not wrong, but it's nowhere chose to the whole story.
In math, an absolute value is the extent of a real number without regard to its sign. So the absolute value is always a positive number or zero (0). Let's check at what absolute value is, how to discover absolute value, few examples of absolute value, and the absolute value derivative.
Definition 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 without considering its sign. That means if you have a negative figure, the absolute value of that figure is the number ignoring the negative sign.
Definition 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 consider it, the absolute value is the length or distance a number has from zero. You can see it if you look at a real number line:
As you can see, the absolute value of a number is the distance of the figure is from zero on the number line. The absolute value of negative five is 5 because it is five units apart from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is 3.
Presently, let's look at one more absolute value example. Let's say we hold an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. So, what does this tell us? It states that absolute value is constantly positive, regardless if the number itself is negative.
How to Locate the Absolute Value of a Expression or Figure
You should be aware of a couple of things prior going into how to do it. A couple of closely linked features will help you grasp how the number inside the absolute value symbol works. Fortunately, what we have here is an meaning of the following 4 rudimental properties of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of all real number is at all time zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Instead, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a total is lower 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 four essential properties in mind, let's take a look at two other helpful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the difference among two real numbers is less than or equivalent to the absolute value of the total of their absolute values.
Now that we went through these characteristics, we can finally begin learning how to do it!
Steps to Find the Absolute Value of a Figure
You have to observe few steps to find the absolute value. These steps are:
Step 1: Note down the figure whose absolute value you want to calculate.
Step 2: If the number is negative, multiply it by -1. This will convert the number to positive.
Step3: If the number is positive, do not change it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the expression is the number you obtain after steps 2, 3 or 4.
Keep in mind that the absolute value symbol is two vertical bars on both side of a expression or number, like this: |x|.
Example 1
To begin with, let's presume an absolute value equation, like |x + 5| = 20. As we can see, 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 following the steps mentioned priorly:
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 characteristics, we learn that the absolute value of the addition of these two expressions is equivalent to 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 length from zero will also equal 15, and the equation above is genuine.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to get 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 have 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: Therefore, the initial equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can contain many complicated expressions or rational numbers in mathematical settings; however, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is differentiable everywhere. The following formula gives 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 range is all positive real numbers. The absolute value function increases 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 differentiable at 0 reason being 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 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 0.
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