This document provides an overview of equations, inequalities, and their properties and solutions. It defines what an equation and inequality are using examples. It describes properties of equations and inequalities such as adding/subtracting/multiplying numbers on both sides. The document outlines different types of equations like linear, quadratic, and simultaneous equations. It also discusses solving linear and quadratic equations and inequalities using step-by-step processes. Finally, it touches on representing equations and inequalities graphically and solving simultaneous systems of linear and quadratic inequalities.

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