Absolute value

earning objectives

  •  Find the absolute value of numbers and expressions.
  •  Represent absolute values ​​with numerical declarations and on the number line.

Introduction

Algebra usually requires us to be careful not only about size and value but also about the sign. It is not the same -10 as 10. 3 + 7 gives us a different result than 3 + (-7). But there are circumstances in which the sign does not matter, in mathematics and in everyday life. Have you ever stumbled when going down an escalator? It doesn’t matter so much if you are moving faster or slower than the ground, it is the magnitude of the difference that makes you lose your balance. Or think of a long walk in the countryside, your feet will hurt no matter if you go north or south. Direction doesn’t matter, only distance.In math, there is a concept for dealing with situations where size matters more than sign. It is called an absolute value . The absolute value of a number consists of its value, regardless of its sign.

Absolute value is a useful concept when we are only interested in the size of the difference between two numbers. The absolute value gives us the distance, but discards the information regarding the direction. Since the address is ignored, the absolute value of a number can only be positive or zero, never negative. When the value of an expression is positive or zero, its absolute value is the same as the original value. When the value of an expression is negative, its absolute value is the same value but without the negative sign.

Absolute value of a number

Defining the absolute value or modulus of a number

The absolute value or modulus of a number x, represented by |x| equals xif the number is positive or 0 and is equal to xif the number is negative. The “-” sign operates onxchanging it to positive.
We write this as follows

Definition

|x|={x,– ,if  ≥ 0if  0


x |



  

En algunos números no es tan evidente su signo. Es por eso que hay que tomar especial cuidado al tomar valor absoluto.

Ejemplo Escriba |3π|

3π


To evaluate numerical expressions with absolute values, we will take into account that they behave like grouping signs. So we will first determine the value of the expressions between the bars, following the hierarchy of the combined operations.

Example Simplify

– ⋅ 6

Solution Click to see the development of the step.
 Determine the value of the expression between the bars of the absolute value.  Apply the definition of the absolute value to remove it. Determine the value of the resulting expression.



Many calculators have the Abs key to evaluate absolute values, also many Math programs work with the Abs () function .

You can see on the button how the numerical expression of the previous example would be encoded for many programs. 


To eliminate the symbol of the absolute value when there is an expression in a variable between the bars, it is necessary to take into account the values ​​of the variable that make the expression positive and the values ​​of this literal in which the expression is negative.

Example
a)
 Determine all the values ​​of the variable for which |2x+6|=(2x+6).
Solution We
want to determine the values ​​of the variable in which the absolute value changes the sign of the expression 2x+6. This occurs if and only if the expression between the bars is negative.
 when the expression between the bars is negative, in terms of an inequality. Solve the proposed inequality.  Reply. b) For which values ​​of the variable do we have to |2x+6|=2x+6

?
 Exercise Write |3x|

without using absolute value.
Solution We will
express |3x| in the form of the given definition of absolute value.
 Determine the values ​​of the variable, x, which make the expression between the bars negative.  Specify the values ​​of the variable,
x, which make the expression between the bars positive or 0.  Write |3x|
 in the form of the definition.


Like x2is a positive number, regardless of the sign of xand  represents the positive square root, we have the following algebraic expression to define the absolute value

Alternative definition

x | =x2



Examples
Hover over the expression to verify that the definitions match in the examples.
This way of writing the definition is particularly useful for demonstrating some properties of the absolute value.   


Absolute value properties

Immediate properties

We first establish some elementary properties that follow from the definition.

Properties
  |to|=00
  |to|0
  |to|=|to|


From property 3 applied to ab we have the following

Property   |ba|=|ab|

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Properties of absolute value on elementary operations

Let’s look at some properties of how absolute value behaves against elementary operations.

Proposition
  |ab|=|to||b|     Multiplicative property
  |tob=to |b |≠ 0
  |ton= | to |n,for n positive integer or zero.

 You can test 6 in a similar way to 5.  We say that the absolute value preserves the operations of multiplication, division and empowerment. But this behavior does not have it against the addition and subtraction.

Triangular inequality

a+b |  | to | + | b |

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Properties that help solve equations and inequalities with absolute value

The idea of ​​solving equations and inequalities with absolute values ​​is to transform the problem into solving equations without absolute value. In certain equations and inequalities we can apply any of the following properties to remove them.

Theorem     for c>0 we have

x | =c– corc

It follows from the definition of absolute value itself.

Theorem    

x | = | and|yor– y

The absolute values ​​of two numbers are equal if and only if the numbers are equal or one is the opposite of the other.

Theorem
    for c>0 we have to

))x | c– ≤ ≤ cx | <c– c

Demonstration to  If we work with complements we can use the previous theorem to prove the next one

Theorem
    for c>0 we have to

))x | c≤ – cor≥ cx | >c– corc

Theorem    

x | < | and|x2and2

 

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