Presentación para una introducción a las inequaciones.

1. Algebra: Equations & Inequalities
Miguel Pérez Fontenla
November, 2010

3. Algebra: Equations & Inequalities
What is an equation?
   
22
22 1
2 3
4 9
yx
x
x

   Example:
An equation is a mathematical statement

4. Algebra: Equations & Inequalities
Properpies of equations
Property 1 - Adding or Subtracting a Number
An equation is not changed when the same number is added or subtracted from
both sides of the equality.
Example: A = B (adding 4 to both sides gives) ⇔ A + 4 = B + 4
Property 2 - Multiplying or dividing by a Number
An equation is not changed if both sides are multiplied or divided by the same
number.
Example: A = B (Multiplying both sides by 2 gives) ⇔ 2A = 2B
A = B (Dividing both sides by 3 gives) ⇔ A/3 = B/3

5. Algebra: Equations & Inequalities
Types of equations?
4 1 5
3 2 2
x x b
ax b x
a
 
     
Linear equations
Quadratic equations
Biquadratic
Simultaneous equations
Linear
Quadratic
Rational equations
Irrational equations
Other types
2
2 4
0
2
b b ac
ax bx c x
a
  
    
2
4 2 2 4
0
2
b b ac
ax bx c x
a
  
    
5 8 19
2 2 10
x y
x y
  

  
2 2
3 8 8
9 28
x y
x y
  
 
  
2
2
3 4 1
4 2 2
x x
x x x
 
 
  
2 15 2 4x x   
3 2
3 4 12 0x x x   

6. Algebra: Equations & Inequalities
Solving linear equations
1 1 5 14 2 9 7
4 8 4 5 2 8
x x x x    
    
 
1 5 14 2 9 7
4 32 40 2 8
x x x x   
   
1. No parenthesis
2. No fractions
3. Isolate x to side one
4. Obtain x
5. Check your work
 4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x         
27 80 860 41 53 901x x x       
53 901 901
53 901 17
53 53 53
x
x x
 
       
 
 
17 1 1 17 5 14 2 17 17 9 7 1 7 25 25
4 3 4 4
4 8 4 5 2 8 8 8 8 8
     
           
 

7. Algebra: Equations & Inequalities
Solving quadratic equations
2
1 5 2 2
2 6 3 3
x x x 
  1. No parenthesis
2. No fractions
3. Isolate everything
to side one
4. Obtain x
5. Check your work
  2
2;6;3 6 3 3 5 4 4mcm x x x      
2
3 5 2 0x x  
2
5 7
2
( 5) ( 5) 4 3 ( 2) 5 49 6
5 7 12 3 6
6 3
x x


        
   

 
  
 
2 1 2 1 2 5 2 3 3 6 3 1
2 1 2....
2 6 3 2 6 3 2 2
   
        
  
 
1 1 5 2
1
2 6 3
x x x
x
  
  

8. Algebra: Equations & Inequalities
What is an inequality?
SIMBOLS
= Equal to
< Less than
> Greater than
Less than or equal
Greater than or equal



9. Algebra: Equations & Inequalities
What is an inequality?
   
22
22 1
2 3
4 9
yx
x
x

   Example:

10. Algebra: Equations & Inequalities
Properpies of inequalities
Property 1 - Adding or Subtracting a Number
The sense of an inequality is not changed when the same number is added or
subtracted from both sides of the inequality.
Example: 9 > 6 (adding 4 to both sides gives) ⇔ 9 + 4 > 6 + 4
Property 2 - Multiplying by a Positive Number
The sense of the inequality is not changed if both sides are multiplied or divided by
the same positive number.
Example: 8 < 15 (Multiplying both sides by 2 gives) ⇔ 8 × 2 < 15 × 2
Property 3 - Multiplying by a Negative Number
The sense of the inequality is reversed if both sides are multiplied or divided by the
same negative number.
Example: 4 > −2 (Multiplying both sides by -3 gives) ⇔ 4 × −3 < −2 × −3 ⇔ −12 < 6
(Note the change in the sign used)

11. Algebra: Equations & Inequalities
Solving Linear inequalities
1 1 5 14 2 9 7
4 8 4 5 2 8
x x x x    
    
 
1 5 14 2 9 7
4 32 40 2 8
x x x x   
   
1. No parenthesis
2. No fractions
3. Isolate x to side one
4. Obtain x
5. Check
 4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x         
27 80 860 41 53 901 53 901 ...x x x x          
901
... 17
53
x  
0 1 1 0 5 14 2 0 0 9 7 1 1 51 9 7
If 0
4 8 4 5 4 8 4 8 20 4 8
1 51 25 11 25
4 160 8 160 8
x
          
             
   
  
    

12. Algebra: Equations & Inequalities
Linear inequalities: Graphic Solution
1 1 5 14 2 9 7
17
4 8 4 5 2 8
x x x x
x
    
      
 

13. Algebra: Equations & Inequalities
Solving quadratic inequalities
2
1 5 2 2
2 6 3 3
x x x 
  1. No parenthesis
2. No fractions
3. Isolate everything
to side one
4. Obtain solutions of
the equation
5. Set the intervals
solution
6. Check
  2
2;6;3 6 3 3 5 4 4mcm x x x      
2
3 5 2 0x x  
2
5 7
2
( 5) ( 5) 4 3 ( 2) 5 49 6
5 7 12 3 6
6 3
x x


        
   

 
  
 
1 1 5 2
1
2 6 3
x x x
x
  
  
 
1
, 2,
3
 
   
 

14. Algebra: Equations & Inequalities
Quadratic inequalities: Graphic Solution
  
 
1 1 5 2
1
2 6 3
x x x
x
  
  
 
:
1
, 2,
3
1
/ 2
3
Solutions
x x x
 
   
 
 
     
 

15. Algebra: Equations & Inequalities
Puting into a Graph
A linear equation with two variables can
be represented by a straight line in the
plane.
A quadratic equation with two variables
can be represented by a parabole in the
plane.

16. Algebra: Equations & Inequalities
Solving simultaneous linear inequalities
1 1
2 2 2 2
y x y x
y x y x
    
 
     

17. Algebra: Equations & Inequalities
Solving simultaneous inequalities
2 2
1 1
6 6
y x y x
y x x y x x
    
 
      

18. Algebra: Equations & Inequalities
Solving simultaneous quadratic inequalities
2 2
2 2
3 2 2 3
6 6
y x x y x x
y x x y x x
        
 
       

19. Algebra: Equations & Inequalities
Solving simultaneous inequalities
2 2
2 2
1
2 4 2
2
2
2
y x
y x
x y y x
x y
y x
   
   
 
     
      

20. Algebra: Equations & Inequalities

21. Algebra: Equations & Inequalities

22. Algebra: Equations & Inequalities