Absolute ValueMeaning, How to Calculate Absolute Value, Examples
Many think of absolute value as the length from zero to a number line. And that's not incorrect, but it's not the whole story.
In math, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is all the time a positive number or zero (0). Let's look at what absolute value is, how to discover absolute value, some examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the magnitude of a real number irrespective to its sign. This signifies if you possess a negative figure, the absolute value of that figure is the number ignoring the negative sign.
Definition of Absolute Value
The previous definition refers 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 check out a real number line:
As demonstrated, the absolute value of a figure is the length of the number is from zero on the number line. The absolute value of negative five is five reason being it is 5 units apart from zero on the number line.
Examples
If we graph negative three on a line, we can watch that it is three units apart from zero:
The absolute value of negative three is 3.
Now, let's look at more absolute value example. Let's assume 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 refer to? It shows us that absolute value is at all times positive, even though the number itself is negative.
How to Find the Absolute Value of a Number or Figure
You should be aware of a couple of points before working on how to do it. A couple of closely linked properties will help you grasp how the number inside the absolute value symbol works. Luckily, what we have here is an explanation of the following 4 essential characteristics of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is always positive or zero (0).
Identity: The absolute value of a positive number is the expression itself. Otherwise, the absolute value of a negative number is the non-negative value of that same expression.
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 fundamental characteristics in mind, let's check out two other helpful properties of the absolute value:
Positive definiteness: The absolute value of any real number is constantly zero (0) or positive.
Triangle inequality: The absolute value of the difference between two real numbers is less than or equivalent to the absolute value of the total of their absolute values.
Now that we know these properties, we can finally begin learning how to do it!
Steps to Discover the Absolute Value of a Expression
You are required to follow a couple of steps to discover the absolute value. These steps are:
Step 1: Jot down the number whose absolute value you want to discover.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the figure is positive, do not convert it.
Step 4: Apply all characteristics significant to the absolute value equations.
Step 5: The absolute value of the figure is the number you have following steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on either side of a figure or number, similar to this: |x|.
Example 1
To start out, 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 need to find the absolute value of the two numbers in the inequality. We can do this by observing the steps above:
Step 1: We are provided with the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to solve x.
Step 2: By using the fundamental characteristics, we know that the absolute value of the addition of these two figures is as 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 see, x equals 15, so its distance from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's work on one more absolute value example. We'll utilize the absolute value function to find a new equation, like |x*3| = 6. To do this, we again have to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required to calculate the value x, so we'll begin by dividing 3 from each side of the equation. This step gives 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 potential solutions, x=2 and x=-2.
Absolute value can involve a lot of complex expressions or rational numbers in mathematical settings; still, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, meaning it is distinguishable 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 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 consistent at 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 provided as:
I'm →0−(|x|/x)
The right-hand limit is provided as:
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 distinctable at 0.
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