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# Algebra equations & inequalities

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Presentación para una introducción a las inequaciones.

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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