Absolute ValueMeaning, How to Find Absolute Value, Examples
Many perceive absolute value as the length from zero to a number line. And that's not wrong, but it's not the complete 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 number or zero (0). Let's check at what absolute value is, how to discover absolute value, some examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is at all times zero (0) or positive. It is the extent of a real number without regard to its sign. This refers that if you hold a negative number, the absolute value of that number is the number ignoring the negative sign.
Meaning of Absolute Value
The previous definition means that the absolute value is the distance of a figure 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 check out a real number line:
As you can see, the absolute value of a figure is the length of the number is from zero on the number line. The absolute value of -5 is 5 due to the fact it is five units away from zero on the number line.
Examples
If we plot negative three on a line, we can observe that it is 3 units away from zero:
The absolute value of -3 is 3.
Now, let's look at another absolute value example. Let's assume 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 tell us? It shows us that absolute value is always positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Expression or Figure
You should know a handful of points before working on how to do it. A couple of closely associated characteristics will assist you grasp how the figure within the absolute value symbol functions. Thankfully, here we have an definition of the ensuing 4 fundamental features of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of ever real number is at all time positive or zero (0).
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 equivalent to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these 4 essential characteristics in mind, let's look at two more beneficial properties of the absolute value:
Positive definiteness: The absolute value of any real number is constantly zero (0) or positive.
Triangle inequality: The absolute value of the variance among two real numbers is less than or equal to the absolute value of the total of their absolute values.
Taking into account that we learned these properties, we can in the end start learning how to do it!
Steps to Find the Absolute Value of a Figure
You need to observe a couple of steps to find the absolute value. These steps are:
Step 1: Jot down the expression whose absolute value you desire to calculate.
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 alter it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the expression is the expression you get following steps 2, 3 or 4.
Bear in mind that the absolute value sign is two vertical bars on either side of a figure or expression, similar to this: |x|.
Example 1
To begin with, let's consider an absolute value equation, such as |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To work this out, we are required to locate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:
Step 1: We are given the equation |x+5| = 20, and we have to find the absolute value inside the equation to find x.
Step 2: By using the basic characteristics, we understand that the absolute value of the addition of these two numbers is the 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 eliminate 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 genuine.
Example 2
Now let's work on one more absolute value example. We'll use the absolute value function to get a new equation, similar to |x*3| = 6. To get there, we again have to follow the steps:
Step 1: We have the equation |x*3| = 6.
Step 2: We are required to find the value of x, so we'll begin by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential solutions: x = 2 and x = -2.
Step 4: Hence, the original equation |x*3| = 6 also has two possible solutions, x=2 and x=-2.
Absolute value can contain several complicated expressions or rational numbers in mathematical settings; however, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, meaning it is distinguishable at any given point. The following formula gives 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 range 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 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 stated as:
I'm →0−(|x|/x)
The right-hand limit is provided 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 distinguishable at zero (0).
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