Absolute ValueMeaning, How to Discover Absolute Value, Examples
A lot of people perceive absolute value as the distance 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 magnitude of a real number without considering its sign. So the absolute value is all the time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is constantly positive or zero (0). It is the magnitude of a real number irrespective to its sign. This refers that if you have a negative figure, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The prior definition means that the absolute value is the length of a figure from zero on a number line. Hence, if you think about it, the absolute value is the distance or length a number has from zero. You can visualize it if you look at a real number line:
As you can see, the absolute value of a number is how far away the figure is from zero on the number line. The absolute value of negative five is five because it is 5 units apart from zero on the number line.
Examples
If we plot negative three on a line, we can watch that it is three units apart from zero:
The absolute value of -3 is 3.
Now, let's look at more absolute value example. Let's assume 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 at all times positive, even though the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should know a handful of things prior going into how to do it. A handful of closely linked features will support you comprehend how the number within the absolute value symbol works. Luckily, here we have an meaning of the ensuing 4 rudimental properties of absolute value.
Basic Properties of Absolute Values
Non-negativity: The absolute value of any real number is at all time positive or zero (0).
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 figure.
Addition: The absolute value of a total is less 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 above-mentioned four essential characteristics in mind, let's look at 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 within two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Considering that we learned these properties, we can in the end begin learning how to do it!
Steps to Discover the Absolute Value of a Figure
You are required to follow a couple of steps to discover the absolute value. These steps are:
Step 1: Write down the number whose 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 change it.
Step 4: Apply all properties relevant to the absolute value equations.
Step 5: The absolute value of the number is the figure you have subsequently steps 2, 3 or 4.
Keep in mind that the absolute value sign is two vertical bars on either side of a figure or number, similar to this: |x|.
Example 1
To set out, 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 solve this, we need to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned above:
Step 1: We are given the equation |x+5| = 20, and we are required to calculate the absolute value inside the equation to solve x.
Step 2: By using the basic characteristics, we know that the absolute value of the sum of these two numbers 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 remove 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 length from zero will also be equivalent 15, and the equation above is genuine.
Example 2
Now let's try one more absolute value example. We'll utilize the absolute value function to get a new equation, similar to |x*3| = 6. To get there, we again need to observe 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 begin by dividing 3 from both side of the equation. This step offers us |x| = 2.
Step 3: |x| = 2 has two possible answers: 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 involve a lot of complex expressions or rational numbers in mathematical settings; nevertheless, that is something we will work on another day.
The Derivative of Absolute Value Functions
The absolute value is a constant function, meaning it is differentiable everywhere. The ensuing 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 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 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 uniform. The left-hand limit is stated 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 zero (0).
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