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 inaccurate, 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 zero or number (0). Let's check at what absolute value is, how to discover absolute value, several examples of absolute value, and the absolute value derivative.
Definition of Absolute Value?
An absolute value of a figure is always positive or zero (0). It is the magnitude of a real number irrespective to its sign. This refers that if you possess a negative figure, the absolute value of that number is the number without the negative sign.
Meaning of Absolute Value
The last definition states that the absolute value is the length of a figure from zero on a number line. Therefore, if you think about it, the absolute value is the distance or length a figure has from zero. You can observe it if you look at a real number line:
As demonstrated, the absolute value of a number is the distance of the number is from zero on the number line. The absolute value of negative five is five due to the fact it is five units apart 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 negative three is three.
Now, let's look at one more absolute value example. Let's say we posses an absolute value of sin. We can graph this on a number line as well:
The absolute value of 6 is 6. So, what does this mean? It states that absolute value is constantly positive, even if the number itself is negative.
How to Find the Absolute Value of a Expression or Number
You should know few points prior working on how to do it. A couple of closely linked features will support you comprehend how the number within the absolute value symbol functions. Fortunately, what we have here is an explanation of the ensuing four essential properties of absolute value.
Essential Properties of Absolute Values
Non-negativity: The absolute value of all real number is constantly positive or zero (0).
Identity: The absolute value of a positive number is the figure 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 less 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 essential properties in mind, let's look at two other helpful properties of the absolute value:
Positive definiteness: The absolute value of any real number is at all times zero (0) or positive.
Triangle inequality: The absolute value of the variance among two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Now that we went through these characteristics, we can in the end begin learning how to do it!
Steps to Calculate the Absolute Value of a Expression
You are required to obey a couple of steps to discover the absolute value. These steps are:
Step 1: Jot down the number whose absolute value you want to find.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the expression is positive, do not convert it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the expression is the figure you obtain subsequently steps 2, 3 or 4.
Remember that the absolute value symbol is two vertical bars on both side of a figure or number, similar to this: |x|.
Example 1
To begin with, let's consider an absolute value equation, like |x + 5| = 20. As we can see, 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 observing the steps mentioned above:
Step 1: We have the equation |x+5| = 20, and we must discover the absolute value within the equation to solve x.
Step 2: By utilizing the basic properties, we know that the absolute value of the addition of these two figures 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 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 right.
Example 2
Now let's try another absolute value example. We'll use the absolute value function to get a new equation, like |x*3| = 6. To make it, we again have to follow the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We have to calculate the value x, so we'll begin by dividing 3 from both 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: Therefore, the original equation |x*3| = 6 also has two potential results, x=2 and x=-2.
Absolute value can contain many complicated expressions or rational numbers in mathematical settings; nevertheless, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is differentiable everywhere. The following formula offers 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 length 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 zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 reason being 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 offered as:
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 differentiable at 0.
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