Absolute ValueMeaning, How to Find Absolute Value, Examples
A lot of people think of absolute value as the distance from zero to a number line. And that's not wrong, but it's by no means the whole story.
In mathematics, an absolute value is the magnitude of a real number irrespective of its sign. So the absolute value is always a positive zero or number (0). Let's observe at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a number is constantly zero (0) or positive. It is the extent of a real number irrespective to its sign. This signifies if you possess a negative number, the absolute value of that number is the number overlooking the negative sign.
Definition of Absolute Value
The last explanation states that the absolute value is the length of a figure from zero on a number line. Therefore, if you think about that, the absolute value is the distance or length a number has from zero. You can observe it if you look at a real number line:
As you can see, the absolute value of a number is how far away the number is from zero on the number line. The absolute value of negative five is 5 reason being it is five units away from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is three units apart from zero:
The absolute value of negative three is three.
Now, let's look at another absolute value example. Let's suppose we posses an absolute value of sin. We can graph this on a number line as well:
The absolute value of six is 6. Therefore, what does this tell us? It states that absolute value is always positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Expression or Number
You need to know a handful of points prior working on how to do it. A handful of closely linked properties will assist you understand how the expression inside the absolute value symbol functions. Luckily, what we have here is an meaning of the following 4 essential characteristics of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is always zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Instead, the absolute value of a negative number is the non-negative value of that same expression.
Addition: The absolute value of a sum is less than or equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these four basic characteristics in mind, let's check out two other beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is always positive or zero (0).
Triangle inequality: The absolute value of the difference among two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Taking into account that we learned these properties, we can ultimately initiate learning how to do it!
Steps to Calculate the Absolute Value of a Number
You are required to obey few steps to discover the absolute value. These steps are:
Step 1: Note down the figure of whom’s absolute value you desire to find.
Step 2: If the figure 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 characteristics applicable to the absolute value equations.
Step 5: The absolute value of the number is the figure you obtain subsequently steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on both side of a number or expression, similar to this: |x|.
Example 1
To start out, let's assume an absolute value equation, like |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we have to locate 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 calculate the absolute value inside the equation to solve x.
Step 2: By using the basic characteristics, we know that the absolute value of the addition of these two figures 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 solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length 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 use the absolute value function to solve a new equation, like |x*3| = 6. To get there, we again need to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We need to calculate the value x, so we'll begin by dividing 3 from both 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 original equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can involve many complex figures or rational numbers in mathematical settings; still, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this states it is varied everywhere. The following formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except 0, and the length is all positive real numbers. The absolute value function rises 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 differentiable at 0 reason being the left-hand limit and the right-hand limit are not equal. The left-hand limit is given by:
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 0.
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