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 wrong, but it's not the entire story.
In mathematics, an absolute value is the magnitude of a real number without regard to its sign. So the absolute value is at all time a positive number or zero (0). Let's check at what absolute value is, how to calculate absolute value, several examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a number is constantly positive or zero (0). It is the magnitude of a real number without regard to its sign. This refers that if you possess a negative figure, the absolute value of that number is the number disregarding the negative sign.
Meaning of Absolute Value
The last explanation means that the absolute value is the length of a figure from zero on a number line. So, if you think about it, the absolute value is the length or distance a number has from zero. You can see it if you check out a real number line:
As you can see, the absolute value of a figure is the distance of the number is from zero on the number line. The absolute value of -5 is 5 reason being it is five units apart from zero on the number line.
Examples
If we graph -3 on a line, we can see that it is three units away from zero:
The absolute value of negative three is 3.
Presently, let's look at another absolute value example. Let's suppose we posses an absolute value of sin. We can plot this on a number line as well:
The absolute value of six is 6. Hence, what does this mean? It tells us that absolute value is always positive, regardless if the number itself is negative.
How to Calculate the Absolute Value of a Figure or Expression
You should be aware of a handful of things before going into how to do it. A few closely linked characteristics will support you comprehend how the figure inside the absolute value symbol works. Thankfully, what we have here is an meaning of the ensuing four fundamental 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 expression itself. Otherwise, the absolute value of a negative number is the non-negative value of that same expression.
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 equal to the product of absolute values.
With above-mentioned 4 basic characteristics in mind, let's look at two more beneficial characteristics 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 equal to the absolute value of the sum of their absolute values.
Considering that we learned these characteristics, we can finally initiate learning how to do it!
Steps to Find the Absolute Value of a Number
You have to obey a handful of steps to discover the absolute value. These steps are:
Step 1: Note down the figure whose absolute value you desire to discover.
Step 2: If the number is negative, multiply it by -1. This will change it to a positive number.
Step3: If the number is positive, do not convert it.
Step 4: Apply all properties significant to the absolute value equations.
Step 5: The absolute value of the number is the figure you get subsequently steps 2, 3 or 4.
Remember that the absolute value sign is two vertical bars on either side of a figure or number, like this: |x|.
Example 1
To set out, let's presume an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To figure this out, we have 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 are provided with the equation |x+5| = 20, and we are required to find the absolute value inside the equation to find x.
Step 2: By utilizing the fundamental properties, we know that the absolute value of the sum of these two figures is as same as the sum 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 calculate for x: x = 20-5, x = 15
As we see, x equals 15, so its length from zero will also equal 15, and the equation above is true.
Example 2
Now let's try another absolute value example. We'll utilize the absolute value function to find a new equation, similar to |x*3| = 6. To do this, we again have to obey the steps:
Step 1: We hold the equation |x*3| = 6.
Step 2: We need 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 possible answers: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can include many complex figures 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 distinguishable at any given point. The ensuing formula gives the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the area is all real numbers except zero (0), and the range 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 differentiable 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 offered as:
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 distinctable at 0.
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