Absolute ValueMeaning, How to Find Absolute Value, Examples
Many think of absolute value as the distance 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 extent of a real number without regard to its sign. So the absolute value is always a positive number or zero (0). Let's observe at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a figure 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 hold a negative figure, the absolute value of that figure is the number overlooking the negative sign.
Definition of Absolute Value
The last definition means that the absolute value is the distance of a figure from zero on a number line. So, if you consider it, the absolute value is the distance or length a figure has from zero. You can visualize it if you look at a real number line:
As shown, the absolute value of a number is the length of the figure is from zero on the number line. The absolute value of negative five is 5 due to the fact it is five units away from zero on the number line.
Examples
If we plot -3 on a line, we can see that it is 3 units away from zero:
The absolute value of -3 is three.
Now, let's check out one more absolute value example. Let's say we have an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. Therefore, what does this tell us? It tells us that absolute value is constantly positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Number or Expression
You should be aware of a handful of points before going into how to do it. A couple of closely related features will support you grasp how the expression within the absolute value symbol functions. Thankfully, what we have here is an meaning of the following 4 essential features of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of all real number is always positive or zero (0).
Identity: The absolute value of a positive number is the expression 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 equal to the product of absolute values.
With above-mentioned 4 essential characteristics in mind, let's check out two other useful 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 within two real numbers is less than or equal to the absolute value of the sum of their absolute values.
Considering that we went through these properties, we can ultimately start learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You need to follow a couple of steps to calculate the absolute value. These steps are:
Step 1: Note down the number of whom’s absolute value you want to find.
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 alter it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the figure is the number you get following steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on either side of a expression or number, like 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 work this out, we are required to find 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 are required to calculate the absolute value within the equation to solve x.
Step 2: By utilizing the fundamental properties, we learn that the absolute value of the sum of these two numbers 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 eliminate 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 distance from zero will also be equivalent 15, and the equation above is genuine.
Example 2
Now let's check out one more absolute value example. We'll use the absolute value function to solve a new equation, similar to |x*3| = 6. To do this, we again have to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We are required 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 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 many complicated values or rational numbers in mathematical settings; however, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this states it is differentiable at any given point. The ensuing formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except zero (0), and the range 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 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 uniform. The left-hand limit is given by:
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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