Absolute ValueDefinition, How to Calculate Absolute Value, Examples
A lot of people comprehend absolute value as the distance from zero to a number line. And that's not inaccurate, but it's by no means the entire story.
In math, an absolute value is the extent of a real number without regard to its sign. So the absolute value is at all time a positive number or zero (0). Let's observe at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the magnitude of a real number without regard to its sign. That means if you have a negative number, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The last explanation states that the absolute value is the distance of a figure from zero on a number line. So, if you think about it, the absolute value is the length or distance a figure has from zero. You can observe 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 5 reason being it is five units away from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is three units away from zero:
The absolute value of -3 is 3.
Presently, let's check out more absolute value example. Let's say 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 tell us? It tells us that absolute value is always positive, even if the number itself is negative.
How to Locate the Absolute Value of a Figure or Expression
You should know few things before going into how to do it. A couple of closely associated characteristics will help you understand how the expression within the absolute value symbol works. Fortunately, here we have an explanation of the following four essential features of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is always 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 equivalent to the product of absolute values.
With above-mentioned four fundamental characteristics in mind, let's check out two more useful 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 among two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Now that we know these properties, we can ultimately start learning how to do it!
Steps to Discover the Absolute Value of a Number
You are required to follow a couple of steps to calculate the absolute value. These steps are:
Step 1: Jot down the expression of whom’s absolute value you desire to calculate.
Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.
Step3: If the expression is positive, do not alter it.
Step 4: Apply all characteristics significant to the absolute value equations.
Step 5: The absolute value of the expression is the figure you obtain after steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on either side of a number or expression, similar to this: |x|.
Example 1
To start out, let's consider an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To solve this, we need to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:
Step 1: We have the equation |x+5| = 20, and we are required to calculate the absolute value inside the equation to find x.
Step 2: By using the fundamental properties, we learn that the absolute value of the sum of these two figures 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 unidentified, so let's eliminate 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 length from zero will also equal 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 need to follow the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required 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 potential answers: x = 2 and x = -2.
Step 4: So, the original equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can involve many complex numbers or rational numbers in mathematical settings; however, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is differentiable at any given point. 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 0, and the range 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 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable at 0 due to the the left-hand limit and the right-hand limit are not equal. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is offered as:
I'm →0+(|x|/x)
Because 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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