Absolute ValueMeaning, How to Calculate Absolute Value, Examples
Many perceive absolute value as the distance from zero to a number line. And that's not incorrect, but it's nowhere chose to the entire story.
In mathematics, 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 observe at what absolute value is, how to discover absolute value, few examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is at all times positive or zero (0). It is the extent of a real number without regard to its sign. This signifies if you have a negative number, the absolute value of that figure is the number ignoring the negative sign.
Meaning of Absolute Value
The prior definition refers that the absolute value is the distance of a figure from zero on a number line. Hence, if you consider it, the absolute value is the distance or length a number has from zero. You can visualize it if you check out a real number line:
As demonstrated, 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 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 observe that it is three units apart from zero:
The absolute value of negative three is three.
Presently, let's check out more absolute value example. Let's assume we have an absolute value of 6. We can graph this on a number line as well:
The absolute value of 6 is 6. So, what does this mean? It states that absolute value is at all times positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Number or Expression
You need to know a couple of points before going into how to do it. A handful of closely linked features will help you comprehend how the figure inside the absolute value symbol functions. Luckily, here we have an definition of the following four essential features of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of all real number is constantly zero (0) or positive.
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 figure.
Addition: The absolute value of a sum is less than or equivalent to the sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With above-mentioned 4 basic properties in mind, let's take a look at two other beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is constantly zero (0) or positive.
Triangle inequality: The absolute value of the difference between two real numbers is lower than or equivalent to the absolute value of the sum of their absolute values.
Taking into account that we know these properties, we can finally start learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You need to follow a handful of steps to discover the absolute value. These steps are:
Step 1: Write down the number whose absolute value you want to discover.
Step 2: If the number is negative, multiply it by -1. This will make the number positive.
Step3: If the figure 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 number you have following steps 2, 3 or 4.
Keep in mind that the absolute value symbol is two vertical bars on either side of a expression or number, similar to 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 solve this, we are required to calculate the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned priorly:
Step 1: We are provided with the equation |x+5| = 20, and we must discover the absolute value inside the equation to find x.
Step 2: By using the essential properties, we know that the absolute value of the total of these two figures 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 calculate for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also equal 15, and the equation above is right.
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 make it, we again need to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need to find the value of x, so we'll start by dividing 3 from both side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two possible solutions: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two likely solutions, x=2 and x=-2.
Absolute value can involve several intricate values or rational numbers in mathematical settings; still, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this refers it is distinguishable everywhere. The ensuing 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 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 offered 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 differentiable at 0.
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