Absolute ValueDefinition, How to Find Absolute Value, Examples
A lot of people think of absolute value as the distance from zero to a number line. And that's not inaccurate, but it's not the entire story.
In math, an absolute value is the extent of a real number irrespective of its sign. So the absolute value is all the time a positive number or zero (0). Let's look at what absolute value is, how to find absolute value, several examples of absolute value, and the absolute value derivative.
Definition 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 irrespective to its sign. This refers that if you hold a negative figure, the absolute value of that number is the number overlooking the negative sign.
Definition of Absolute Value
The previous explanation means that the absolute value is the length of a number from zero on a number line. Hence, if you think about it, the absolute value is the distance or length a figure has from zero. You can see it if you take a look at a real number line:
As demonstrated, the absolute value of a figure is the length of the figure 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 plot negative three on a line, we can see that it is three units apart from zero:
The absolute value of negative three is 3.
Presently, let's look at another absolute value example. Let's say we posses an absolute value of 6. We can plot this on a number line as well:
The absolute value of six is 6. Therefore, what does this refer to? It states that absolute value is constantly positive, regardless if the number itself is negative.
How to Locate the Absolute Value of a Number or Expression
You should know a handful of things before going into how to do it. A couple of closely related properties will help you understand how the number inside the absolute value symbol works. Thankfully, here we have an meaning of the following four fundamental properties of absolute value.
Essential Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is always zero (0) or positive.
Identity: The absolute value of a positive number is the figure itself. Alternatively, 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 equal to the product of absolute values.
With above-mentioned 4 basic properties in mind, let's check out two other helpful 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 lower than or equivalent to the absolute value of the sum of their absolute values.
Considering that we learned these properties, we can in the end begin learning how to do it!
Steps to Find the Absolute Value of a Number
You are required to obey few steps to calculate the absolute value. These steps are:
Step 1: Jot down the number whose absolute value you desire to find.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the number is positive, do not convert it.
Step 4: Apply all properties applicable to the absolute value equations.
Step 5: The absolute value of the figure is the number you obtain following steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on both side of a figure or expression, like this: |x|.
Example 1
To start out, 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 solve this, we have to locate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we have to discover the absolute value within the equation to get x.
Step 2: By utilizing the fundamental characteristics, we learn 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 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 right.
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 need to observe the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We have to solve for x, so we'll initiate by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential results: x = 2 and x = -2.
Step 4: So, the first equation |x*3| = 6 also has two possible results, x=2 and x=-2.
Absolute value can involve a lot of complicated values 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, meaning it is differentiable at any given point. 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 distance 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 zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinguishable at 0 because 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 provided 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 distinctable at 0.
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