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 not the whole story.
In math, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is always a positive zero or number (0). Let's look at what absolute value is, how to find absolute value, some 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 magnitude of a real number without considering its sign. That means if you hold a negative figure, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The prior explanation 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 figure has from zero. You can visualize 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 number is from zero on the number line. The absolute value of negative five is 5 reason being it is 5 units away from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is 3 units apart from zero:
The absolute value of -3 is 3.
Well then, let's check out one more absolute value example. Let's say we hold 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 shows us that absolute value is constantly positive, even though the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You should know a couple of points before working on how to do it. A couple of closely related features will support you understand how the expression inside the absolute value symbol works. Thankfully, here we have an meaning of the following 4 rudimental features of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of any real number is constantly positive or zero (0).
Identity: The absolute value of a positive number is the number itself. Otherwise, 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 equivalent to the product of absolute values.
With above-mentioned four basic properties in mind, let's take a look at two more helpful 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 variance between two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.
Considering that we learned these characteristics, we can in the end initiate learning how to do it!
Steps to Discover the Absolute Value of a Figure
You are required to follow a handful of steps to find 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 number is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not change it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the figure is the figure you get after steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a expression or number, like this: |x|.
Example 1
To start out, 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 work 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 are provided with the equation |x+5| = 20, and we have to discover the absolute value inside the equation to find x.
Step 2: By using the basic properties, we learn 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 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 distance from zero will also equal 15, and the equation above is true.
Example 2
Now let's work on another absolute value example. We'll utilize the absolute value function to solve a new equation, similar to |x*3| = 6. To make it, we again have to observe the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to find the value of 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 original equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can contain several complicated numbers or rational numbers in mathematical settings; nevertheless, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is varied 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 consistent at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not differentiable at 0 due to the 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 given by:
I'm →0+(|x|/x)
Considering the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinctable at 0.
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