GRE - Algebra
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This document provides an overview of topics related to algebra, including quadratic equations, inequalities, absolute value, and modulus. It discusses how to form quadratic equations given different conditions on the roots. Methods for finding roots such as splitting the middle term and using the quadratic formula are presented. Properties and techniques for solving different types of inequalities and absolute value equations are outlined. Maximum and minimum values of functions involving modulus are also addressed.
![Solving rational Inequalities
Solve for x
2𝑥 + 7
𝑥 − 4
≥ 3
3/17/2017
33
www.georgeprep.com
Answer: 4 < t ≤ 19 - in interval notation: (4; 19]](https://image.slidesharecdn.com/algebra-vizag-170317033739/85/GRE-Algebra-33-320.jpg)
GRE - Algebra
- 2. Topics Quadratic Equations Inequalities Absolute Value www.georgeprep.com
- 4. Quadratic function and its Graph www.georgeprep.com 4
- 5. • Forming a Quadratic Equation www.georgeprep.com 5
- 6. Quadratic expression and quadratic equation www.georgeprep.com 6
- 7. Forming a quadratic equation www.georgeprep.com 1. When roots are given 2. When sum of the roots and the product of the roots are given 3. When the roots are related to the roots of another quadratic equation 7
- 8. Forming a quadratic equation www.georgeprep.com 1. When roots are given Form a quadratic equation whose roots are 1 and 2 8
- 9. Forming a quadratic equation www.georgeprep.com 2. When sum of the roots and the product of the roots are given Form a quadratic equation such that the sum of the roots is 4 and the product of the roots is 3 9
- 10. Forming a quadratic equation www.georgeprep.com 3. When the roots are related to the roots of another quadratic equation Changed roots Changed Q.Eq. 1 α + p and β + p a (x - p)2 + b (x - p) + c =0 2 α - p and β - p a (x + p)2 + b (x + p) + c =0 3 αp and βp a (x / p)2 + b (x / p) + c =0 4 α/p and β/p a (x p)2 + b (x p) + c =0 5 1/ α and 1/ β a (1/x )2 + b (1/x ) + c =0 6 -α and -β a (-x )2 + b (-x ) + c =0 a x 2 - b x + c =0 7 α2 and β2 ax + b root x + c =0 8 αn and βn ax2/n + bx1/n + c=0 Form a quadratic equation whose roots are two more than the roots of the equation x2 -3x +2 = 0 10
- 11. • Roots of a Quadratic Equation www.georgeprep.com 11
- 12. What is a root of a quadratic equation? www.georgeprep.com 12
- 13. Finding roots of a quadratic equation www.georgeprep.com 1. Splitting the middle term 2. Quadratic formula 13
- 14. Finding roots of a quadratic equation www.georgeprep.com Find the roots of the quadratic equation x2 + 5x + 6 = 0 Find the roots of the quadratic equation 6x2 - 5x - 6 = 0 14
- 15. Finding roots of a quadratic equation www.georgeprep.com Find the roots of the quadratic equation x2 + 6x + 10 = 0 Quadratic Formula Roots = −𝑏 ± 𝑏2−4𝑎𝑐 2𝑎 15
- 17. www.georgeprep.com Solve for x in x2 – 5x +6 >0 Solve for x in x2 – 5x +6 <0 Solve for x in x2 – 5x +6 ≥0 Solve for x in x2 – 5x +6 ≤0 17
- 18. www.georgeprep.com Solve for x in 𝑥2 – 5𝑥 + 6 (𝑥 + 2)(4𝑥 − 1) > 0 18
- 19. www.georgeprep.com Solve for x in x2 – 5x +6 >0 Solve for x in x2 – 5x +6 <0 Solve for x in x2 – 5x +6 ≥0 Solve for x in x2 – 5x +6 ≤0 19
- 20. www.georgeprep.com Solve for x in 𝑥2 – 5𝑥 + 6 (𝑥 + 2)(4𝑥 − 1) > 0 20
- 22. Inequalities - Basics 3/17/2017 22 www.georgeprep.com Sense of the Inequality The < and > signs define what is known as the sense of the inequality (indicated by the direction of the sign). Two inequalities are said to have (a) the same sense if the signs of inequality point in the same direction; and (b) the opposite sense if the signs of inequality point in the opposite direction. Trichotomy Property For any two real numbers a and b , exactly one of the following is true: a < b , a = b , a > b The expression a<b is read as a is less than b The expression a>b is read as a is greater than b.
- 23. Inequalities - Properties 3/17/2017 23 www.georgeprep.com If a < b and b < c , then a < c . If a > b and b > c , then a > c . Note: These properties also apply to "less than or equal to" and "greater than or equal to": If a ≤ b and b ≤ c , then a ≤ c . If a ≥ b and b ≥ c , then a ≥ c . Transitive Property Reversal Property We can swap a and b over, if we make sure the symbol still "points at" the smaller value. If a > b then b < a If a < b then b > a
- 24. Inequalities - Properties 3/17/2017 24 www.georgeprep.com Properties of Addition and Subtraction Addition Properties of Inequality: If a < b , then a + c < b + c If a > b , then a + c > b + c Subtraction Properties of Inequality: If a < b , then a - c < b - c If a > b , then a - c > b - c These properties also apply to ≤ and ≥ : If a ≤ b , then a + c ≤ b + c If a ≥ b , then a + c ≥ b + c If a ≤ b , then a - c ≤ b - c If a ≥ b , then a - c ≥ b - c Adding and Subtracting Inequalities If a < b and c < d, then a + c < b + d If a > b and c > d, then a + c > b + d
- 25. Inequalities - Properties 3/17/2017 25 www.georgeprep.com Properties of Multiplication and Division When we multiply both a and b by a positive number, the inequality stays the same. But when we multiply both a and b by a negative number, the inequality swaps over! Here are the rules: •If a < b, and c is positive, then ac < bc •If a < b, and c is negative, then ac > bc (inequality swaps over!)
- 26. Inequalities - Properties 3/17/2017 26 www.georgeprep.com Inverses Additive Inverse As we just saw, putting minuses in front of a and b changes the direction of the inequality. This is called the "Additive Inverse": If a < b then -a > -b If a > b then -a < -b Multiplicative Inverse Taking the reciprocal (1/value) of both a and b When a and b are both positive or both negative can change the direction of the inequality. If a < b then 1/a > 1/b If a > b then 1/a < 1/b When a and b are of opposite signs, the inequality remains as it is.
- 27. Problems 3/17/2017 27 www.georgeprep.com Which of the following is greater? 11^12 or 12^11 Generalisation If a < b < e then , ab < ba If e < a < b then ab > ba
- 28. Problems 3/17/2017 28 www.georgeprep.com Which of the following is greater? 11^12 or 12^11 Generalisation If a < b < e then , ab < ba If e < a < b then ab > ba Where e = 2.71
- 29. Inequalities – for comparison of values 3/17/2017 29 www.georgeprep.com When the signs of two numbers a and b are known When both are positive If a/b > 1 , then a > b If a/b < 1 , then a < b If a/b = 1 , then a = b When both are negative If a/b > 1 , then a < b If a/b < 1 , then a > b If a/b = 1 , then a = b When the signs of two numbers a and b are NOT known If a - b > 0 , then a > b If a - b < 0 , then a < b If a - b = 0 , then a = b
- 30. Relation between AM, GM and HM 3/17/2017 30 www.georgeprep.com Basic AM-GM Inequality For positive real numbers a, b 𝑎 + 𝑏 2 ≥ 𝑎 ∗ 𝑏 Proof: Squaring, this becomes (a + b) 2 ≥ 4ab, which is equivalent to (a − b) 2 ≥ 0. Equality holds if and only if a = b. In general, 𝐴𝑀 ≥ 𝐺𝑀 ≥ 𝐻𝑀
- 31. Problems involving AM ≥ GM If 𝑥 + 𝑦 + 𝑧 = 19 , what is the maximum value of (𝑥 – 2) (𝑦 – 3) (𝑧 – 2)? 1. 64 2. 16 3. 125 4. None of these 3/17/2017 31 www.georgeprep.com
- 32. Solving quadratic Inequalities Solve 𝑥2 − 7𝑥 + 10 > 0 3/17/2017 32 www.georgeprep.com
- 33. Solving rational Inequalities Solve for x 2𝑥 + 7 𝑥 − 4 ≥ 3 3/17/2017 33 www.georgeprep.com Answer: 4 < t ≤ 19 - in interval notation: (4; 19]
- 34. Absolute Value ( Modulus) 3/17/2017 www.georgeprep.com 34
- 35. If x is the coordinate of a point on a real number line, then the distance of a from the origin is represented by 𝑥 . x is called Absolute value of x or modulus of x. 3/17/2017 35 www.georgeprep.com Modulus and Distance
- 36. Definition of Absolute value 3/17/2017 36 www.georgeprep.com Modulus and Distance
- 37. Properties of Modulus 1. 𝑥 ≥ 0 2. − 𝑥 ≤ 0 3. 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 4. 𝑥 − 𝑦 ≤ 𝑥 − 𝑦 5. 𝑥 ∗ 𝑦 = 𝑥 ∗ 𝑦 6. 𝑥 𝑦 = 𝑥 𝑦 ( as long as y≠0) 7. 𝑥 2 = 𝑥2 3/17/2017 37 www.georgeprep.com Modulus
- 38. 𝑥 = 3 𝑥 < 3 𝑥 > 3 3/17/2017 38 www.georgeprep.com Modulus and Distance
- 39. 3/17/2017 39 www.georgeprep.com Modulus and Distance What if you have to measure a distance from a point other than 0? Distance of a coordinate x from a is given by 𝑥 − 𝑎 Then what does 𝑥 + 𝑎 represent? Distance of x from -a
- 40. 𝑥 − 1 = 3 𝑥 − 1 < 3 𝑥 − 1 > 3 3/17/2017 40 www.georgeprep.com Modulus and Distance
- 41. Find the value of x if 3𝑥 + 2 = 5 Answer : 1 or − 7 3 3/17/2017 41 www.georgeprep.com Modulus
- 44. 3/17/2017 44 www.georgeprep.com Modulus - Problems Solve for x if 2𝑥 − 1 < 3 Answer : (-1, 2) Solve for x if 7 − 3𝑥 > 2 Answer : −∞, 5 3 ∪ (3, ∞)
- 45. 3/17/2017 45 www.georgeprep.com Modulus How about solving for x for the expression below? 𝑥 − 4 = 3𝑥 − 8 Answers: 𝑥 = 2 𝑜𝑟 3
- 46. 3/17/2017 46 www.georgeprep.com Modulus - Problems Solve for x for the expression below? 3𝑥 − 4 = 𝑥 + 5
- 47. 3/17/2017 47 www.georgeprep.com Modulus – Maximum and Minimum Values What is the maximum value of 𝑓 𝑥 = 15 − 9 + 𝑥 ?
- 48. 3/17/2017 48 www.georgeprep.com Modulus – Maximum and Minimum Values Maximum and Minimum values when more than one modulus functions are given. What is the minimum value of the function 𝑥 − 2 + 𝑥 − 9 + 𝑥 + 4 Answer : 13
- 49. 3/17/2017 49 www.georgeprep.com Modulus – Maximum and Minimum Values How many integer solutions are possible for the inequality 𝑥 − 6 + 𝑥 − 8 + 𝑥 + 4 < 11 1. 1 2. 2 3. 0 4. Infinitely many Answer : 0
- 50. 3/17/2017 50 www.georgeprep.com Modulus – Maximum and Minimum Values How many integer solutions are possible for the inequality 𝑥 − 6 ∗ 𝑥 − 7 ≤ 15 1. 6 2. 5 3. 8 4. More than 8 Answer : 8
Editor's Notes
- #36: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #37: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #38: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #39: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #40: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #41: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #42: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #43: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #44: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #45: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #46: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #47: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #48: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #49: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #50: Solution: 2x/5 = 3y/3 X = 5 Answer : 16
- #51: Solution: 2x/5 = 3y/3 X = 5 Answer : 16