Algebra: Equations & Inequalities
Miguel Pérez Fontenla
November, 2010
Algebra: Equations & Inequalities
What is an equation?
   
22
22 1
2 3
4 9
yx
x
x

   Example:
An equation is a m...
Algebra: Equations & Inequalities
Properpies of equations
Property 1 - Adding or Subtracting a Number
An equation is not c...
Algebra: Equations & Inequalities
Types of equations?
4 1 5
3 2 2
x x b
ax b x
a
 
     
Linear equations
Quadra...
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...
Algebra: Equations & Inequalities
Solving quadratic equations
2
1 5 2 2
2 6 3 3
x x x 
  1. No parenthesis
2. No frac...
Algebra: Equations & Inequalities
What is an inequality?
SIMBOLS
= Equal to
< Less than
> Greater than
Less than or equal
...
Algebra: Equations & Inequalities
What is an inequality?
   
22
22 1
2 3
4 9
yx
x
x

   Example:
Algebra: Equations & Inequalities
Properpies of inequalities
Property 1 - Adding or Subtracting a Number
The sense of an i...
Algebra: Equations & Inequalities
Solving Linear inequalities
1 1 5 14 2 9 7
4 8 4 5 2 8
x x x x    
    
 
...
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
    ...
Algebra: Equations & Inequalities
Solving quadratic inequalities
2
1 5 2 2
2 6 3 3
x x x 
  1. No parenthesis
2. No f...
Algebra: Equations & Inequalities
Quadratic inequalities: Graphic Solution
  
 
1 1 5 2
1
2 6 3
x x x
x
  
  
...
Algebra: Equations & Inequalities
Puting into a Graph
A linear equation with two variables can
be represented by a straigh...
Algebra: Equations & Inequalities
Solving simultaneous linear inequalities
1 1
2 2 2 2
y x y x
y x y x
    
 
  ...
Algebra: Equations & Inequalities
Solving simultaneous inequalities
2 2
1 1
6 6
y x y x
y x x y x x
    
 
   ...
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
...
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
   
...
Algebra: Equations & Inequalities
Algebra: Equations & Inequalities
Algebra: Equations & Inequalities
Algebra equations & inequalities
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Algebra equations & inequalities

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Algebra equations & inequalities

  1. 1. Algebra: Equations & Inequalities Miguel Pérez Fontenla November, 2010
  2. 2. Algebra: Equations & Inequalities What is an equation?     22 22 1 2 3 4 9 yx x x     Example: An equation is a mathematical statement
  3. 3. 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
  4. 4. 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   
  5. 5. 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                    
  6. 6. 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      
  7. 7. Algebra: Equations & Inequalities What is an inequality? SIMBOLS = Equal to < Less than > Greater than Less than or equal Greater than or equal  
  8. 8. Algebra: Equations & Inequalities What is an inequality?     22 22 1 2 3 4 9 yx x x     Example:
  9. 9. 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)
  10. 10. 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                                     
  11. 11. 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              
  12. 12. 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        
  13. 13. 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                  
  14. 14. 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.
  15. 15. Algebra: Equations & Inequalities Solving simultaneous linear inequalities 1 1 2 2 2 2 y x y x y x y x             
  16. 16. Algebra: Equations & Inequalities Solving simultaneous inequalities 2 2 1 1 6 6 y x y x y x x y x x              
  17. 17. 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                   
  18. 18. 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                       
  19. 19. Algebra: Equations & Inequalities
  20. 20. Algebra: Equations & Inequalities
  21. 21. Algebra: Equations & Inequalities

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