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# Algebra equations & inequalities

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Presentación para una introducción a las inequaciones.

Presentación para una introducción a las inequaciones.

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• 1. Algebra: Equations &amp; Inequalities Miguel P&#xE9;rez Fontenla November, 2010
• 2. Algebra: Equations &amp; Inequalities What is an equation? &#xF028; &#xF029; &#xF028; &#xF029; 22 22 1 2 3 4 9 yx x x &#xF02D; &#xF02D; &#xF03D; &#xF02B; &#xF02D;Example: An equation is a mathematical statement
• 3. Algebra: Equations &amp; 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) &#x21D4; 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) &#x21D4; 2A = 2B A = B (Dividing both sides by 3 gives) &#x21D4; A/3 = B/3
• 4. Algebra: Equations &amp; Inequalities Types of equations? 4 1 5 3 2 2 x x b ax b x a &#xF02B; &#xF02D; &#xF02D; &#xF03D; &#xF0DE; &#xF03D; &#xF0DE; &#xF03D; &#xF0A7;Linear equations &#xF0A7;Quadratic equations &#xF0FC;Biquadratic &#xF0A7;Simultaneous equations &#xF0FC;Linear &#xF0FC;Quadratic &#xF0A7;Rational equations &#xF0A7;Irrational equations &#xF0A7;Other types 2 2 4 0 2 b b ac ax bx c x a &#xF02D; &#xF0B1; &#xF02D; &#xF02B; &#xF02B; &#xF03D; &#xF0DE; &#xF03D; 2 4 2 2 4 0 2 b b ac ax bx c x a &#xF02D; &#xF0B1; &#xF02D; &#xF02B; &#xF02B; &#xF03D; &#xF0DE; &#xF03D; 5 8 19 2 2 10 x y x y &#xF02D; &#xF03D; &#xF0FC; &#xF0FD; &#xF02D; &#xF03D; &#xF0FE; 2 2 3 8 8 9 28 x y x y &#xF02D; &#xF03D;&#xF0EC; &#xF0FC; &#xF0ED; &#xF0FD; &#xF02D; &#xF03D;&#xF0EE; &#xF0FE; 2 2 3 4 1 4 2 2 x x x x x &#xF02D; &#xF02B; &#xF02B; &#xF03D; &#xF02D; &#xF02B; &#xF02D; 2 15 2 4x x&#xF02B; &#xF03D; &#xF02B; &#xF02B; 3 2 3 4 12 0x x x&#xF02B; &#xF02D; &#xF02D; &#xF03D;
• 5. Algebra: Equations &amp; Inequalities Solving linear equations 1 1 5 14 2 9 7 4 8 4 5 2 8 x x x x&#xF02D; &#xF02D; &#xF02D; &#xF02D;&#xF0E6; &#xF0F6; &#xF02D; &#xF02D; &#xF03D; &#xF02D;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8; 1 5 14 2 9 7 4 32 40 2 8 x x x x&#xF02D; &#xF02D; &#xF02D; &#xF02D; &#xF02D; &#xF02B; &#xF03D; &#xF02D; 1. No parenthesis 2. No fractions 3. Isolate x to side one 4. Obtain x 5. Check your work &#xF07B; &#xF07D;4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x&#xF03D; &#xF0DE; &#xF02D; &#xF02D; &#xF02B; &#xF02B; &#xF02D; &#xF03D; &#xF02D; &#xF02D; 27 80 860 41 53 901x x x&#xF02D; &#xF03D; &#xF02D; &#xF02D; &#xF0DB; &#xF02D; &#xF03D; &#xF02D; 53 901 901 53 901 17 53 53 53 x x x &#xF02D; &#xF02D; &#xF02D; &#xF03D; &#xF02D; &#xF0DB; &#xF03D; &#xF0DB; &#xF03D; &#xF03D; &#xF02D; &#xF02D; &#xF05B; &#xF05D; 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 &#xF02D; &#xF02D; &#xF02D; &#xF0D7; &#xF02D;&#xF0E6; &#xF0F6; &#xF02D; &#xF02D; &#xF03D; &#xF02D; &#xF0DB; &#xF02D; &#xF02B; &#xF03D; &#xF02D; &#xF0DB; &#xF03D;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8;
• 6. Algebra: Equations &amp; Inequalities Solving quadratic equations 2 1 5 2 2 2 6 3 3 x x x&#xF02D; &#xF02D; &#xF02D; &#xF03D; &#xF02B;1. No parenthesis 2. No fractions 3. Isolate everything to side one 4. Obtain x 5. Check your work &#xF07B; &#xF07D; 2 2;6;3 6 3 3 5 4 4mcm x x x&#xF03D; &#xF0DE; &#xF02D; &#xF02D; &#xF02B; &#xF03D; &#xF02B; 2 3 5 2 0x x&#xF02D; &#xF02D; &#xF03D; 2 5 7 2 ( 5) ( 5) 4 3 ( 2) 5 49 6 5 7 12 3 6 6 3 x x &#xF02B; &#xF03D; &#xF02D; &#xF02D; &#xF0B1; &#xF02D; &#xF02D; &#xF0D7; &#xF0D7; &#xF02D; &#xF0B1; &#xF03D; &#xF03D; &#xF0DE; &#xF03D; &#xF02D;&#xF0D7; &#xF03D; &#xF02D; &#xF028; &#xF029;&#xF028; &#xF029; &#xF028; &#xF029; 2 1 2 1 2 5 2 3 3 6 3 1 2 1 2.... 2 6 3 2 6 3 2 2 &#xF02D; &#xF02B; &#xF02D; &#xF02D; &#xF02D; &#xF03D; &#xF02B; &#xF0DB; &#xF02D; &#xF03D; &#xF0DB; &#xF02B; &#xF03D; &#xF028; &#xF029;&#xF028; &#xF029; &#xF028; &#xF029; 1 1 5 2 1 2 6 3 x x x x &#xF02D; &#xF02B; &#xF02D; &#xF02D; &#xF03D; &#xF02B;
• 7. Algebra: Equations &amp; Inequalities What is an inequality? SIMBOLS = Equal to &lt; Less than &gt; Greater than Less than or equal Greater than or equal &#xF0A3; &#xF0B3;
• 8. Algebra: Equations &amp; Inequalities What is an inequality? &#xF028; &#xF029; &#xF028; &#xF029; 22 22 1 2 3 4 9 yx x x &#xF02D; &#xF02D; &#xF0A3; &#xF02B; &#xF02D;Example:
• 9. Algebra: Equations &amp; 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 &gt; 6 (adding 4 to both sides gives) &#x21D4; 9 + 4 &gt; 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 &lt; 15 (Multiplying both sides by 2 gives) &#x21D4; 8 &#xD7; 2 &lt; 15 &#xD7; 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 &gt; &#x2212;2 (Multiplying both sides by -3 gives) &#x21D4; 4 &#xD7; &#x2212;3 &lt; &#x2212;2 &#xD7; &#x2212;3 &#x21D4; &#x2212;12 &lt; 6 (Note the change in the sign used)
• 10. Algebra: Equations &amp; Inequalities Solving Linear inequalities 1 1 5 14 2 9 7 4 8 4 5 2 8 x x x x&#xF02D; &#xF02D; &#xF02D; &#xF02D;&#xF0E6; &#xF0F6; &#xF02D; &#xF02D; &#xF0B3; &#xF02D;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8; 1 5 14 2 9 7 4 32 40 2 8 x x x x&#xF02D; &#xF02D; &#xF02D; &#xF02D; &#xF02D; &#xF02B; &#xF0B3; &#xF02D; 1. No parenthesis 2. No fractions 3. Isolate x to side one 4. Obtain x 5. Check &#xF07B; &#xF07D;4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x&#xF03D; &#xF0DE; &#xF02D; &#xF02D; &#xF02B; &#xF02B; &#xF02D; &#xF0B3; &#xF02D; &#xF02D; 27 80 860 41 53 901 53 901 ...x x x x&#xF02D; &#xF0B3; &#xF02D; &#xF02D; &#xF0DB; &#xF02D; &#xF0B3; &#xF02D; &#xF0DB; &#xF0A3; &#xF0DB; 901 ... 17 53 x&#xF0DB; &#xF0A3; &#xF03D; 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 &#xF02D; &#xF02D; &#xF02D; &#xF0D7; &#xF02D; &#xF02D; &#xF02D; &#xF02D;&#xF0E6; &#xF0F6; &#xF0E6; &#xF0F6; &#xF03D; &#xF0DE; &#xF02D; &#xF02D; &#xF0B3; &#xF02D; &#xF0DB; &#xF02D; &#xF0B3; &#xF02D; &#xF0DB;&#xF0E7; &#xF0F7; &#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8; &#xF0E8; &#xF0F8; &#xF02D; &#xF02D; &#xF02D; &#xF0DB; &#xF02B; &#xF0B3; &#xF0DB; &#xF0B3;
• 11. Algebra: Equations &amp; Inequalities Linear inequalities: Graphic Solution 1 1 5 14 2 9 7 17 4 8 4 5 2 8 x x x x x &#xF02D; &#xF02D; &#xF02D; &#xF02D;&#xF0E6; &#xF0F6; &#xF02D; &#xF02D; &#xF0A3; &#xF02D; &#xF0DE; &#xF0B3;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8;
• 12. Algebra: Equations &amp; Inequalities Solving quadratic inequalities 2 1 5 2 2 2 6 3 3 x x x&#xF02D; &#xF02D; &#xF02D; &#xF0B3; &#xF02B;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 &#xF07B; &#xF07D; 2 2;6;3 6 3 3 5 4 4mcm x x x&#xF03D; &#xF0DE; &#xF02D; &#xF02D; &#xF02B; &#xF0B3; &#xF02B; 2 3 5 2 0x x&#xF02D; &#xF02D; &#xF0B3; 2 5 7 2 ( 5) ( 5) 4 3 ( 2) 5 49 6 5 7 12 3 6 6 3 x x &#xF02B; &#xF03D; &#xF02D; &#xF02D; &#xF0B1; &#xF02D; &#xF02D; &#xF0D7; &#xF0D7; &#xF02D; &#xF0B1; &#xF03D; &#xF03D; &#xF0DE; &#xF03D; &#xF02D;&#xF0D7; &#xF03D; &#xF02D; &#xF028; &#xF029;&#xF028; &#xF029; &#xF028; &#xF029; 1 1 5 2 1 2 6 3 x x x x &#xF02D; &#xF02B; &#xF02D; &#xF02D; &#xF0B3; &#xF02B; &#xF028; &#xF029; 1 , 2, 3 &#xF0E6; &#xF0F6; &#xF02D;&#xF0A5; &#xF02D; &#xF02B;&#xF0A5;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8;
• 13. Algebra: Equations &amp; Inequalities Quadratic inequalities: Graphic Solution &#xF028; &#xF029;&#xF028; &#xF029; &#xF028; &#xF029; 1 1 5 2 1 2 6 3 x x x x &#xF02D; &#xF02B; &#xF02D; &#xF02D; &#xF0B3; &#xF02B; &#xF028; &#xF029; : 1 , 2, 3 1 / 2 3 Solutions x x x &#xF0E6; &#xF0F6; &#xF02D;&#xF0A5; &#xF02D; &#xF02B;&#xF0A5;&#xF0E7; &#xF0F7; &#xF0E8; &#xF0F8; &#xF0EC; &#xF0FC; &#xF0CE; &#xF0A3; &#xF02D; &#xF0DA; &#xF0B3;&#xF0ED; &#xF0FD; &#xF0EE; &#xF0FE;
• 14. Algebra: Equations &amp; 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. Algebra: Equations &amp; Inequalities Solving simultaneous linear inequalities 1 1 2 2 2 2 y x y x y x y x &#xF02D; &#xF03E; &#xF03E; &#xF02B;&#xF0FC; &#xF0FC; &#xF0FD; &#xF0FD; &#xF02B; &#xF0A3; &#xF0A3; &#xF02D; &#xF02B;&#xF0FE; &#xF0FE;
• 16. Algebra: Equations &amp; Inequalities Solving simultaneous inequalities 2 2 1 1 6 6 y x y x y x x y x x &#xF02D; &#xF03C; &#xF03C; &#xF02B;&#xF0FC; &#xF0FC; &#xF0FD; &#xF0FD; &#xF02D; &#xF0B3; &#xF02D; &#xF0B3; &#xF02D; &#xF02B;&#xF0FE; &#xF0FE;
• 17. Algebra: Equations &amp; 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 &#xF0FC; &#xF0FC;&#xF02B; &#xF0A3; &#xF02D; &#xF0A3; &#xF02D; &#xF02D; &#xF02B;&#xF0EF; &#xF0EF; &#xF0FD; &#xF0FD; &#xF02D; &#xF03E; &#xF02D; &#xF03E; &#xF02D; &#xF02B;&#xF0EF; &#xF0EF;&#xF0FE; &#xF0FE;
• 18. Algebra: Equations &amp; 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 &#xF0B3; &#xF02D; &#xF02B; &#xF0FC; &#xF02B; &#xF0B3; &#xF0FC; &#xF0EF; &#xF0EF; &#xF0EF; &#xF02B; &#xF0A3; &#xF0A3; &#xF02D; &#xF02B;&#xF0FD; &#xF0FD; &#xF0EF; &#xF0EF;&#xF02D; &#xF0A3; &#xF0FE; &#xF0B3; &#xF02D; &#xF0EF;&#xF0FE;
• 19. Algebra: Equations &amp; Inequalities
• 20. Algebra: Equations &amp; Inequalities
• 21. Algebra: Equations &amp; Inequalities