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# Absolute Values

In absolute values, the value is referring to the distance of a number or point from the origin or zero location of a number line. Geometrically the absolute values of |x| are a real number of x, and it is a geometric value without regard the sign. For example, a number 3 is the absolute values of both 3 and -3. The absolute values are represented by two vertical lines, such as | x | (read by modulus of x).

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## What is Absolute Value?

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Absolute value in math is nothing but if we represent a number as absolute value the result will be positive. If the number is positive or negative the result of the absolute value is math is positive.
For example |-2| = +2 and |+ 2| = +2.

## Absolute Value Definition

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The absolute values are the distance from the origin to the number. It is used to connect the complex numbers absolute value and the magnitudes of the vector.
We can define the absolute values like the following

{ x if x ≥ 0 }
|x| =
{ -x if x < 0 }
This is the main definition we will follow when we are writing absolute values.

### Absolute Value of 0 (Zero)

Absolute zero, is not a absolute value for 0. As we know absolute value changes the sign of the numbers in to positive.

## Absolute Value Symbol

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Absolute value represents the distance of a number from ‘0’ on number line. And absolute value sign represents the polarity of the absolute value.i.e, whether it is positive or negative. As it was told that, absolute value represents the distance, distance can never be negative. So, simply we can say that the absolute value sign is always positive.
Absolute value symbol is ' | | ' , we use | (pipe) to represent symbol for absolute value. Absolute value is represented as |A|, where A is the number whose absolute value has to be determined.

## Absolute Value Properties

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Let us take, a and b are real numbers and then the absolute values are satisfying the following properties,

1.Non-negativity :

| a | 0

2.Positive-definiteness:

| a | = 0 a = 0

3. Multiplicativeness:

| a × b| = |a | × |b|

4. Subadditivity:

| a + b| | a | + | b |

5. Symmetry :

| - a | = | a |

6.Identity of indiscernible (equivalent to positive-definiteness) :

| a - b | = 0 a = b

7. Triangle inequality Triangle inequality (equivalent to subadditivity ) :

| a - b | | a - c | + | c - a |

8.Preservation of division (equivalent to multiplicativeness) :

| a / b| = | a | / | b |

9. Equivalent to subadditivity :

| a - b | | | a | - | b | |

## Absolute Value of a Real Number

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In definition of absolute values, for all real number ‘x’ the absolute values, satisfy the following conditions

| x | = x , if x 0

| x | = - x , if x < 0.

In number line, the representations of the absolute values of a real number are the relative number's distance from zero or origin. For example, |2| is the distance of 2 from 0(zero). Here, both 2 and -2 are single distance of 2 units from zero (0). |2| = |-2| = 2. In mathematics, the measurement of any distance is not negative values.

The absolute values are also called the positive square roots; it is represented by the square-root symbol without sign. Such as,

| x | = ( x2)

## Absolute Value Problems

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Below you could see the absolute value problems

### Solved Examples

Question 1: Arrange the order of small number to larger number (ascending)
-|-15|, |12|, |7|, |-99|, |-5|, |-8|, |-65|, |6|
Solution:
Initially we solve the absolute value (or modulus) symbol
-15, 12, 7, 99, 5, 8, 65, 6
After to arrange the given small number to the larger number (ascending order)
-15, 5, 6, 7, 8, 12, 65, 99

Question 2: Arrange the order of small number to larger number (ascending)
-|-25|, |22|,|17|,|-109|,|-15|,|-18|, |-75|, |16|
Solution:
Initially we solve the absolute value (or modulus) symbol
-25, 22, 17, 109, 15, 18, 75, 16
After to arrange the given small number to the larger number (ascending order)
-25, 15, 16, 17,18, 22, 75, 109

## Graphing Absolute Value

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Absolute value graphs are nothing but graph of the absolute values of numbers and functions. We know absolute value of any number is if it is positive or negative the absolute value will be positive. So the absolute value of any number or functions graph will lies on the positive side.

Below you could see the example for absolute value graph

Example : Graph the absolute value of the number -9.

Solution: Given number is -9.

Absolute value of |-9| is +9.

So the graph of the number -9 will be like the following More topics in Absolute Values Absolute Value of a Number Absolute Value of a Complex Number
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