Absolute ValueMeaning, How to Find Absolute Value, Examples
A lot of people think of absolute value as the length from zero to a number line. And that's not inaccurate, but it's nowhere chose to the whole 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 zero or number (0). Let's look at what absolute value is, how to calculate absolute value, some examples of absolute value, and the absolute value derivative.
Explanation 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. This refers that if you hold a negative number, the absolute value of that number is the number disregarding the negative sign.
Definition of Absolute Value
The previous definition states 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 length or distance a figure has from zero. You can see it if you look at a real number line:
As shown, the absolute value of a figure is the distance of the figure is from zero on the number line. The absolute value of -5 is five reason being it is five units apart from zero on the number line.
Examples
If we plot negative three on a line, we can observe that it is three units away from zero:
The absolute value of -3 is 3.
Well then, let's look at another absolute value example. Let's assume 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 states that absolute value is constantly positive, even if the number itself is negative.
How to Locate the Absolute Value of a Expression or Number
You should be aware of a couple of points prior going into how to do it. A few closely associated properties will help you understand how the expression within the absolute value symbol functions. Luckily, here we have an explanation of the ensuing four rudimental features of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of any real number is at all time zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Alternatively, the absolute value of a negative number is the non-negative value of that same expression.
Addition: The absolute value of a total is lower than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equal to the product of absolute values.
With these 4 fundamental properties in mind, let's check out two other helpful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always zero (0) or positive.
Triangle inequality: The absolute value of the difference among two real numbers is less 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 Find the Absolute Value of a Number
You are required to observe few steps to find the absolute value. These steps are:
Step 1: Write 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 change it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the figure is the expression you get after steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on both side of a figure or number, like this: |x|.
Example 1
To start out, let's consider an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To figure this out, we are required to find the absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we have to calculate the absolute value within the equation to get x.
Step 2: By utilizing the fundamental properties, we understand that the absolute value of the addition of these two figures is as same as 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 eliminate 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 distance from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's try one more absolute value example. We'll use the absolute value function to find a new equation, such as |x*3| = 6. To get there, we again need to obey the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We need to solve for x, so we'll start 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: Therefore, the first equation |x*3| = 6 also has two potential answers, x=2 and x=-2.
Absolute value can include a lot of complex figures 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 constant function, this refers it is distinguishable at any given point. The ensuing 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 increases 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 due to the 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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