Absolute ValueDefinition, How to Discover Absolute Value, Examples
Many perceive absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the whole story.
In mathematics, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is all the time 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.
Definition of Absolute Value?
An absolute value of a figure is always zero (0) or positive. It is the extent of a real number irrespective to its sign. That means if you have a negative number, the absolute value of that figure is the number without the negative sign.
Definition of Absolute Value
The previous definition refers that the absolute value is the length of a number from zero on a number line. Therefore, if you think about it, the absolute value is the length or distance a number has from zero. You can observe it if you look at a real number line:
As you can see, the absolute value of a number is the length of the number is from zero on the number line. The absolute value of -5 is 5 due to the fact it is five units away from zero on the number line.
Examples
If we plot negative three on a line, we can observe that it is 3 units apart from zero:
The absolute value of negative three is 3.
Well then, let's look at another 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 tell us? It tells us that absolute value is constantly positive, even if the number itself is negative.
How to Locate the Absolute Value of a Number or Figure
You should know a handful of points prior going into how to do it. A handful of closely associated properties will support you comprehend how the expression within the absolute value symbol functions. Thankfully, here we have an meaning of the ensuing 4 fundamental features of absolute value.
Basic 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 expression itself. Otherwise, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a total is lower than or equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 basic properties in mind, let's look at two more useful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always positive or zero (0).
Triangle inequality: The absolute value of the variance within two real numbers is less than or equivalent to the absolute value of the sum of their absolute values.
Considering that we went through these characteristics, we can finally start learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You are required to follow few steps to find the absolute value. These steps are:
Step 1: Note down the expression of whom’s absolute value you desire to calculate.
Step 2: If the figure is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not alter it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the expression is the number you obtain subsequently steps 2, 3 or 4.
Remember 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, like |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To work this out, we have to find 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 are required to find the absolute value within the equation to get x.
Step 2: By utilizing the fundamental characteristics, we understand that the absolute value of the addition of these two expressions is equivalent to 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 be equivalent 15, and the equation above is true.
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 follow the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We are required to find the value of x, so we'll initiate by dividing 3 from both side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two potential answers: x = 2 and x = -2.
Step 4: So, the first equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can contain several complex expressions or rational numbers in mathematical settings; still, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a constant function, this states it is differentiable everywhere. The following formula offers the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except 0, and the length 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 equivalent. The left-hand limit is provided as:
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 differentiable at 0.
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