Absolute ValueMeaning, How to Find 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 nowhere chose to the complete story.
In mathematics, 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 find absolute value, some examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is constantly zero (0) or positive. It is the magnitude of a real number irrespective to its sign. This refers that if you possess a negative number, the absolute value of that number is the number ignoring the negative sign.
Meaning of Absolute Value
The previous explanation states that the absolute value is the length of a number from zero on a number line. Hence, if you think about that, the absolute value is the length or distance a number has from zero. You can visualize it if you take a look at a real number line:
As you can see, the absolute value of a figure is the distance of 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 plot -3 on a line, we can see that it is three units away from zero:
The absolute value of -3 is three.
Now, let's look at more absolute value example. Let's assume we hold an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. Therefore, what does this mean? It states that absolute value is constantly positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should know a handful of points before working on how to do it. A few closely related features will assist you comprehend how the number inside the absolute value symbol works. Thankfully, here we have an meaning of the ensuing four rudimental properties of absolute value.
Basic Properties 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 number itself. Otherwise, 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 sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned 4 basic characteristics in mind, let's look at two more useful properties 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 variance among two real numbers is lower than or equal to the absolute value of the sum of their absolute values.
Taking into account that we went through these properties, we can finally begin learning how to do it!
Steps to Find the Absolute Value of a Expression
You have to follow a handful of steps to discover the absolute value. These steps are:
Step 1: Write down the number of whom’s absolute value you desire to calculate.
Step 2: If the expression is negative, multiply it by -1. This will make the number positive.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the figure is the figure you have subsequently steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on either side of a number or expression, similar to this: |x|.
Example 1
To begin with, let's presume 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 have to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we are required to find the absolute value inside the equation to find x.
Step 2: By using the basic properties, we understand that the absolute value of the total of these two numbers is as same as 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 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 distance from zero will also be as same as 15, and the equation above is right.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to find a new equation, such as |x*3| = 6. To get there, we again need to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to solve for x, so we'll initiate by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible answers: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two potential answers, x=2 and x=-2.
Absolute value can contain several intricate expressions 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 continuous function, meaning 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 zero (0), and the length 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 zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is stated 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 distinguishable at 0.
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