Absolute ValueDefinition, How to Discover 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 not the whole story.
In mathematics, an absolute value is the magnitude of a real number irrespective of its sign. So the absolute value is at all time a positive number or zero (0). Let's look at what absolute value is, how to find absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a number is at all times zero (0) or positive. It is the extent of a real number without regard to its sign. That means if you have a negative number, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The previous explanation refers that the absolute value is the length of a number from zero on a number line. Hence, if you consider it, the absolute value is the length or distance a figure 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 length of the number is from zero on the number line. The absolute value of -5 is five due to the fact it is five units away from zero on the number line.
Examples
If we graph -3 on a line, we can observe that it is 3 units away from zero:
The absolute value of -3 is three.
Now, let's check out more absolute value example. Let's say we have 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 refer to? It tells us that absolute value is constantly positive, even if the number itself is negative.
How to Find the Absolute Value of a Number or Figure
You should know few points before working on how to do it. A couple of closely related features will assist you grasp how the expression within the absolute value symbol functions. Luckily, what we have here is an definition of the ensuing 4 fundamental features of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of any real number is at all time positive or zero (0).
Identity: The absolute value of a positive number is the number itself. Alternatively, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a sum is lower 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 above-mentioned four basic 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 constantly positive or zero (0).
Triangle inequality: The absolute value of the variance 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 characteristics, we can finally initiate learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You have to obey a couple of steps to discover the absolute value. These steps are:
Step 1: Note down the figure 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 expression is positive, do not convert it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the figure is the expression you obtain following steps 2, 3 or 4.
Bear in mind that the absolute value sign is two vertical bars on both side of a expression or number, like this: |x|.
Example 1
To begin with, let's consider 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 need to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps above:
Step 1: We are provided with the equation |x+5| = 20, and we must calculate the absolute value within the equation to solve x.
Step 2: By utilizing the basic characteristics, we know 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 unidentified, so let's remove 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 distance from zero will also be equivalent 15, and the equation above is true.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to find a new equation, like |x*3| = 6. To make it, we again need to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We have to solve for 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 solutions: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can involve 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 constant function, meaning it is distinguishable everywhere. The following 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 increases 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 distinguishable at 0 reason being the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is given by:
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 distinguishable at 0.
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