Absolute ValueDefinition, How to Find Absolute Value, Examples
Many perceive absolute value as the length from zero to a number line. And that's not wrong, but it's by no means the entire story.
In math, an absolute value is the extent of a real number without considering its sign. So the absolute value is all the time a positive number or zero (0). Let's check at what absolute value is, how to discover absolute value, several examples of absolute value, and the absolute value derivative.
Explanation of Absolute Value?
An absolute value of a number is constantly zero (0) or positive. It is the extent of a real number without regard to its sign. That means if you hold a negative figure, the absolute value of that figure is the number overlooking the negative sign.
Meaning of Absolute Value
The prior definition refers that the absolute value is the distance of a number 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 you can see, the absolute value of a figure is the length of the figure is from zero on the number line. The absolute value of -5 is five 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 watch that it is 3 units apart from zero:
The absolute value of negative three is 3.
Well then, let's check out one more absolute value example. Let's suppose we hold an absolute value of 6. We can graph this on a number line as well:
The absolute value of six is 6. Hence, what does this mean? It tells us that absolute value is at all times positive, even though the number itself is negative.
How to Locate the Absolute Value of a Expression or Figure
You should know a handful of points before working on how to do it. A handful of closely associated properties will assist you understand how the figure within the absolute value symbol functions. Fortunately, what we have here is an explanation of the following 4 rudimental features of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is at all time positive or zero (0).
Identity: The absolute value of a positive number is the expression itself. Otherwise, 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 sum of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned four fundamental characteristics in mind, let's check out two more useful properties 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 variance between two real numbers is less than or equivalent to the absolute value of the total 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 Number
You need 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 want to discover.
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 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.
Bear in mind that the absolute value sign is two vertical bars on both side of a number or expression, like 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 figure this out, we need to locate 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 inside the equation to find x.
Step 2: By using the fundamental properties, we learn that the absolute value of the total of these two figures is as same as 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 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 equal 15, and the equation above is genuine.
Example 2
Now let's check out another absolute value example. We'll use the absolute value function to find a new equation, similar to |x*3| = 6. To get there, we again need to follow the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We need to calculate the value x, so we'll start by dividing 3 from each side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two potential results: x = 2 and x = -2.
Step 4: Therefore, the original equation |x*3| = 6 also has two likely answers, x=2 and x=-2.
Absolute value can include a lot of 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 continuous function, meaning it is differentiable at any given point. The following formula provides the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except zero (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 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 uniform. The left-hand limit is stated as:
I'm →0−(|x|/x)
The right-hand limit is offered as:
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 distinguishable at 0.
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