Absolute ValueMeaning, How to Discover Absolute Value, Examples
Many perceive absolute value as the distance from zero to a number line. And that's not inaccurate, but it's not the whole story.
In math, an absolute value is the magnitude of a real number without regard to its sign. So the absolute value is always a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, few examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the magnitude of a real number without considering its sign. This refers that if you possess a negative figure, the absolute value of that number is the number ignoring the negative sign.
Definition of Absolute Value
The prior definition refers that the absolute value is the distance of a number from zero on a number line. Therefore, if you think about that, 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 shown, 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 five due to the fact it is five units apart from zero on the number line.
Examples
If we plot -3 on a line, we can watch that it is 3 units apart from zero:
The absolute value of negative three is 3.
Now, let's look at another absolute value example. Let's assume we hold an absolute value of sin. 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 constantly positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Expression or Number
You need to know a couple of points prior going into how to do it. A couple of closely associated features will assist you grasp how the figure within the absolute value symbol functions. Fortunately, what we have here is an definition of the ensuing 4 essential properties of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of all real number is always positive or zero (0).
Identity: The absolute value of a positive number is the expression itself. Instead, the absolute value of a negative number is the non-negative value of that same figure.
Addition: The absolute value of a total is less 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 these 4 fundamental properties in mind, let's check out two more beneficial 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 total of their absolute values.
Now that we know these properties, we can finally begin learning how to do it!
Steps to Find the Absolute Value of a Figure
You are required to follow a couple of steps to calculate the absolute value. These steps are:
Step 1: Write down the number of whom’s absolute value you desire to discover.
Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not alter it.
Step 4: Apply all characteristics significant to the absolute value equations.
Step 5: The absolute value of the figure is the number you obtain subsequently steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on either side of a figure or expression, like this: |x|.
Example 1
To begin with, 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 locate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:
Step 1: We are given the equation |x+5| = 20, and we have to calculate the absolute value within the equation to solve x.
Step 2: By using the essential properties, we learn that the absolute value of the sum of these two numbers is the 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 get rid of the vertical bars: x+5 = 20
Step 4: Let's calculate 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 true.
Example 2
Now let's try another absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To get there, we again have to follow the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We are required to solve for x, so we'll initiate by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible results: x = 2 and x = -2.
Step 4: Hence, the first equation |x*3| = 6 also has two likely results, x=2 and x=-2.
Absolute value can involve several 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, this refers it is varied 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 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 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 equal. The left-hand limit is stated as:
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 distinguishable at zero (0).
Grade Potential Can Help You with Absolute Value
If the absolute value seems like a lot to take in, or if you're struggling with mathematics, Grade Potential can help. We offer one-on-one tutoring by professional and qualified teachers. They can guide you with absolute value, derivatives, and any other theories that are confusing you.
Call us today to know more with regard to how we can assist you succeed.
