Absolute ValueMeaning, How to Find Absolute Value, Examples
A lot of people think of absolute value as the length from zero to a number line. And that's not wrong, 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 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.
What Is Absolute Value?
An absolute value of a number is at all times positive or zero (0). It is the magnitude of a real number irrespective to its sign. This refers that if you hold a negative number, the absolute value of that number is the number without the negative sign.
Definition of Absolute Value
The last definition refers that the absolute value is the distance of a figure from zero on a number line. Therefore, if you consider it, 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 number is how far away the figure is from zero on the number line. The absolute value of -5 is 5 because it is five units away from zero on the number line.
Examples
If we graph -3 on a line, we can watch that it is three units away from zero:
The absolute value of -3 is 3.
Now, let's check out more absolute value example. Let's suppose we posses an absolute value of sin. We can plot this on a number line as well:
The absolute value of six is 6. Hence, what does this tell us? It shows us that absolute value is at all times positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should be aware of a handful of points prior going into how to do it. A handful of closely associated characteristics will assist you comprehend how the figure within the absolute value symbol functions. Thankfully, here we have an definition of the ensuing four essential properties of absolute value.
Fundamental Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is constantly 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 figure.
Addition: The absolute value of a total is lower than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these 4 fundamental characteristics in mind, let's look at two more beneficial characteristics 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 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 know these characteristics, we can ultimately start learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You have to observe few steps to find the absolute value. These steps are:
Step 1: Note down the expression whose absolute value you desire to find.
Step 2: If the expression is negative, multiply it by -1. This will make the number positive.
Step3: If the expression is positive, do not change it.
Step 4: Apply all properties significant to the absolute value equations.
Step 5: The absolute value of the figure is the expression you have following steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a figure or number, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we need to find 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 are required to find the absolute value inside the equation to get x.
Step 2: By utilizing the fundamental properties, we learn that the absolute value of the addition of these two numbers is equivalent to 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 length from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to get a new equation, like |x*3| = 6. To do this, we again have to observe the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We need to solve for x, so we'll initiate by dividing 3 from each side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two potential solutions: x = 2 and x = -2.
Step 4: So, the original equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can involve a lot of complex expressions or rational numbers in mathematical settings; nevertheless, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is differentiable everywhere. The following formula offers 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 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 differentiable at 0 because 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 given by:
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 distinctable at zero (0).
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