Absolute ValueDefinition, How to Discover 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 nowhere chose to the entire story.
In mathematics, 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 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 regard to its sign. That means if you possess a negative figure, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The previous explanation means that the absolute value is the length of a number from zero on a number line. So, if you think about it, the absolute value is the distance or length a figure has from zero. You can observe it if you take a look at a real number line:
As demonstrated, the absolute value of a figure is how far away the number is from zero on the number line. The absolute value of -5 is 5 because it is five units apart from zero on the number line.
Examples
If we graph -3 on a line, we can watch that it is three units away from zero:
The absolute value of -3 is 3.
Presently, let's look at another 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 tells us that absolute value is constantly positive, even if the number itself is negative.
How to Find the Absolute Value of a Expression or Number
You should know few points prior going into how to do it. A handful of closely associated features will help you grasp how the expression inside the absolute value symbol functions. Fortunately, here we have an definition of the following 4 rudimental properties of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of any real number is constantly positive or zero (0).
Identity: The absolute value of a positive number is the number itself. Alternatively, 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 sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned four basic characteristics in mind, let's take a look at two other useful 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 variance within two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Taking into account that we learned these properties, we can finally begin learning how to do it!
Steps to Discover the Absolute Value of a Number
You need to follow few steps to discover 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 expression is negative, multiply it by -1. This will convert the number to positive.
Step3: If the number is positive, do not convert it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the expression is the number you obtain subsequently steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a figure or number, like this: |x|.
Example 1
To set out, let's presume 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 find the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:
Step 1: We are given the equation |x+5| = 20, and we must find the absolute value within the equation to find x.
Step 2: By using the fundamental characteristics, we understand that the absolute value of the addition of these two numbers 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 solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also be as same as 15, and the equation above is true.
Example 2
Now let's work on one more absolute value example. We'll utilize the absolute value function to solve a new equation, like |x*3| = 6. To make it, we again have to follow 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 offers us |x| = 2.
Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can include a lot of complex figures or rational numbers in mathematical settings; nevertheless, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this states it is distinguishable everywhere. The ensuing formula provides 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 distance 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 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 provided as:
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 0.
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