Absolute ValueMeaning, How to Find Absolute Value, Examples
Many think of absolute value as the distance from zero to a number line. And that's not incorrect, but it's not the whole story.
In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is all the time a positive zero or number (0). Let's check at what absolute value is, how to find absolute value, some examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a number is always positive or zero (0). It is the extent of a real number without regard to its sign. This signifies if you hold a negative number, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The previous definition states 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 number has from zero. You can visualize it if you look at a real number line:
As shown, the absolute value of a figure is the length of the figure is from zero on the number line. The absolute value of negative five is five reason being it is five units apart from zero on the number line.
Examples
If we plot -3 on a line, we can observe that it is 3 units away from zero:
The absolute value of -3 is three.
Well then, let's check out one more absolute value example. Let's assume we have an absolute value of 6. We can plot this on a number line as well:
The absolute value of six is 6. Hence, what does this mean? It shows us that absolute value is always positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You should be aware of a couple of points prior working on how to do it. A few closely associated features will support you comprehend how the expression within the absolute value symbol works. Fortunately, what we have here is an definition of the following 4 fundamental properties of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of any real number is at all time zero (0) or positive.
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 figure.
Addition: The absolute value of a sum is less than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 basic characteristics in mind, let's take a look at two other helpful 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 difference within two real numbers is less than or equal to the absolute value of the sum of their absolute values.
Taking into account that we went through these characteristics, we can finally start learning how to do it!
Steps to Discover the Absolute Value of a Number
You are required to observe few steps to calculate the absolute value. These steps are:
Step 1: Write down the number whose absolute value you desire to calculate.
Step 2: If the expression is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not change it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the expression you obtain after steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on either side of a figure or number, like this: |x|.
Example 1
To start out, let's consider an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we need 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 have the equation |x+5| = 20, and we have to discover the absolute value inside the equation to find x.
Step 2: By utilizing the basic properties, we learn that the absolute value of the sum 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 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 see, x equals 15, so its length from zero will also equal 15, and the equation above is true.
Example 2
Now let's check out one more absolute value example. We'll utilize the absolute value function to get a new equation, such as |x*3| = 6. To make it, we again have to observe the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We need to calculate the value x, so we'll initiate 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 original equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can contain many complicated values 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 states it is distinguishable at any given point. The following formula provides 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 distance 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 due to the the left-hand limit and the right-hand limit are not uniform. The left-hand limit is stated as:
I'm →0−(|x|/x)
The right-hand limit is provided 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 zero (0).
Grade Potential Can Help You with Absolute Value
If the absolute value looks like complicated task, or if you're having problem with mathematics, Grade Potential can guide you. We offer one-on-one tutoring from experienced and qualified tutors. They can assist you with absolute value, derivatives, and any other theories that are confusing you.
Call us today to know more about how we can assist you succeed.
