Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people perceive absolute value as the distance from zero to a number line. And that's not wrong, but it's not the complete story.
In math, an absolute value is the extent of a real number without considering its sign. So the absolute value is at all time a positive zero or number (0). Let's look at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the magnitude of a real number without considering its sign. This refers that if you have a negative number, the absolute value of that figure is the number ignoring the negative sign.
Definition of Absolute Value
The previous explanation states that the absolute value is the length of a number from zero on a number line. Therefore, if you consider it, the absolute value is the length or distance a number has from zero. You can visualize it if you look at a real number line:
As demonstrated, the absolute value of a figure is the distance of the number is from zero on the number line. The absolute value of negative five is 5 due to the fact it is 5 units apart from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is 3 units apart from zero:
The absolute value of negative three is three.
Now, let's check out more absolute value example. Let's assume we have an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. Hence, what does this refer to? It tells us that absolute value is always positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Number or Expression
You should know few points before going into how to do it. A couple of closely linked features will help you grasp how the expression within the absolute value symbol functions. Thankfully, what we have here is an meaning of the ensuing 4 essential features of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of all real number is at all time 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 total is lower than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these four basic properties in mind, let's look at two more beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the variance among 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 properties, we can ultimately begin learning how to do it!
Steps to Discover the Absolute Value of a Expression
You are required to follow a couple of steps to discover the absolute value. These steps are:
Step 1: Write down the number of whom’s absolute value you desire to find.
Step 2: If the figure is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not change it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the number 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 figure or number, similar to this: |x|.
Example 1
To begin with, let's presume 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 have to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps above:
Step 1: We have the equation |x+5| = 20, and we have to discover the absolute value within the equation to find x.
Step 2: By utilizing the fundamental characteristics, we understand that the absolute value of the addition of these two expressions 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 unidentified, so let's remove the vertical bars: x+5 = 20
Step 4: Let's solve for x: x = 20-5, x = 15
As we see, x equals 15, so its length from zero will also be equivalent 15, and the equation above is true.
Example 2
Now let's check out one more absolute value example. We'll utilize the absolute value function to find a new equation, similar to |x*3| = 6. To make it, we again have to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We are required to calculate the value 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 possible solutions: x = 2 and x = -2.
Step 4: So, the original equation |x*3| = 6 also has two potential results, x=2 and x=-2.
Absolute value can include a lot of complicated numbers 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 states 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 constant 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 uniform. The left-hand limit is stated as:
I'm →0−(|x|/x)
The right-hand limit is given by:
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).
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