The Number Line, Absolute Value, Inequalities, and Properties of R

The Number Line

To draw a number line we draw a line with several dashes in it and ordered numbers below the line, both positive and negative.  The number corresponding to the point on the number line is called the coordinate of the number line. 

Absolute Values

Absolute value signs make the inside positive:

        |-3|  =  3        |2| = 2        |4 - 1|  =  |3|  =  3

We can write

We can read this as "Multiply by (-1) if x is negative, and leave it alone if x is positive."

Inequalities

Recall the four inequalities:

Symbol	Meaning
<	Less Than
>	Greater Than
<	Less Than or Equal to
>	Greater Than or Equal to

When we graph an inequality on a number line we use "["  or "]" to include the point and "(" or ")" to not include the point.  For example, [1, 3)  means all, the points between 1 and 3 including 1 but not including 3.  We can write this on the number line as

The link below will help you investigate the number line and interval notation:
The Number Line and Inequalities

Exercises

Graph the following on a number line

1. {x| x < 3}
   
   

2. {x| x > 2}
   
   

3. {x| 3 < x < -5}
   
   

4. {x| |x| > 4}

Properties of Addition and Multiplication

1. Commutative Property
   

   • Addition:  a + b = b + a
     

   • Multiplication:  ab = ba
     
     

2. Associative Property
   

   • Addition:  (a + b) + c = a + (b + c)
     

   • Multiplication:  (ab)c = a(bc)
     
     

3. Identity Property
   

   • Addition:  there is a 0 such that a + 0 = 0 + a = a
     

   • Multiplication:  there is a 1 such that a1 = 1a = a
     
     

4. Inverse Property
   

   • Addition:  for any a, there is a -a with a + -a = 0
     

   • Multiplication:  for any a not 0, there is a 1/a with a(1/a) = 1
     
     

5. Distributive Property
   

   • a(b + c) = ab + ac
     

   • (a + b)c = ac + bc 
     
     

6. Trichotomy
   
   If a and b are real numbers then one of the three must hold

    

   1. a < b
      

   2. a > b
      

   3. a = b
      
      

7. Transitivity
   
    If a, b, and c are real numbers and 
   
           a  <  b    and    b  <  c 
   
   then 
   
           a  <  c

Examples

        (2 + 3) + 4  =  2 + (3 + 4)  (Associative Property of Addition)

        (x - y)(x + y)  =  (x + y)(x - y) (Commutative Property of Multiplication)

                     1
        (2 - y)             =  1  (Multiplicative inverse)
                   2 - y

Exercises:  

Complete the following.  (If you hold your mouse on the yellow rectangle, you will see the solution.)

1. x - z  =  ________  (Commutative)                               
   
   

2. w (0)  =  _________  (Multiplication property of 0)        
   
   

3. 3 (xy)  =  __________ (Associative)                              
   
   

4. x (y - 3)  =  _________(Distributive)                             

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