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Chapter 1

Chapter 1. Inequalities. Section 1.1. Introduction. The Set of Real Numbers. The set of real numbers satisfies several important properties under addition and multiplication: e.g., closure, commutativity, associativity, distributive property.

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Chapter 1

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  1. Chapter 1 Inequalities

  2. Section 1.1 Introduction

  3. The Set of Real Numbers • The set of real numbers satisfies several important properties under addition and multiplication: e.g., closure, commutativity, associativity, distributive property. • The sum and product of two positive numbers is positive. • The real number a is negative if and only if –a is positive. • The set of real numbers is an ordered set.

  4. Definition • A statement that two numbers are not equal is called an inequality. • For any two real numbers a,b, a > b if and only if a – b > 0. We say that b > a if and only if b – a > 0. • Let a,b be real numbers. We write a  b if either a > b or a = b. We write a  b if either a < b or a = b.

  5. Theorem (Law of Trichotomy) • For any real numbers a and b, exactly one of the following is true: a = b, a > b, a < b.

  6. Theorem* • Let a and b be real numbers such that a<b. • If c is any real number, then a+c < b+c. • If c is a positive number, then ac<bc. If c is a negative number, then ac > bc.

  7. Theorem* (Transitive Property) • Let a, b, c be real numbers such that a < b and b < c. Then a < c.

  8. Example 1 • Let a and b be real numbers such that a < b. Show that

  9. Example 2 • Let a and b be real numbers with a > b > 0. Prove that a(2a + b) > b2.

  10. Example 3 • Let a and b be real numbers such that a < b < 1. Prove that a + ab < a2 + b.

  11. Example 4 • Let a and b be positive numbers. Prove that a > b if and only if a2 > b2.

  12. Theorem* • If a is any real number, a2 0.

  13. Example 1 • Let a and b be real numbers. Show that

  14. Example 2 (AM-GM Inequality) • Let a and b be nonnegative real numbers. Prove that

  15. Example 3 • Let a, b, x, y be positive real numbers such that x2 + y2 = 1 and a2 + b2 = 1. Prove that ax + by  1.

  16. Exercises / Assignment Items • Prove the following statements: • If a > b and c > d, then a + c > b + d. • If 0 < a < 1, then a2 < a. • If a < b < c, then . • For any real numbers a, b, c, and d, (ac + bd)2 (a2 + b2)(c2 + d2). • For 0 < a < b, let h be defined by . Then a < h < b.

  17. Section 1.2 Polynomial and Rational Inequalities

  18. Definition • The domain of a variable in an inequality is the set of real numbers for which both sides of the inequality is defined.

  19. Examples • The inequality 3x3 4x2 + 7x has the set of all real numbers as its domain. • The inequality has the set R \ {2008} as domain.

  20. Remark • There are cases when all elements in the domain of a variable satisfy the inequality. • x + 2 < x + 5 • x2 + 5  0 • However, in general, not all members in the domain of a variable in an inequality yields a true inequality when substituted to the variable.

  21. Definition • Any member in the domain of a variable for which the inequality is true after substitution into the variable is a solution of the inequality. • The set of all solutions is called the solution set of the inequality.

  22. Types of Inequalities • If the solution set of an inequality is exactly the same as the domain of the variable, then we say the inequality is absolute. • A conditional inequality is one for which there is at least one member in the domain of the variable that is not in the solution set of the inequality.

  23. Writing Sets of Real Numbers • Set builder notation Example: {x|x1} • Interval notation Example: [-1,+)

  24. Bounded Intervals

  25. Unbounded Intervals

  26. Examples • Solve the following inequalities, and express the final answer in interval notation: (1) 5x + 6  x + 2 (2) (3) x2 (x + 2) > (4x + 5)(x + 2) (4) 2x2 + 3x – 1 < 6 – 2x  3x + 2

  27. (5) (6) (7)

  28. Assignment Items • Solve the ff. inequalities, and express the final answer in interval notation: (1) (2) (3) (3 – 4x)2006(x + 1)2007(7 – 2x)2008 0 (4) (5)

  29. Section 1.3 Equations and Inequalities Involving the Absolute Value

  30. Definition • The absolute value of a real number is the distance between 0 and the number on the real line. • If x is a real number, then if x  0 if x < 0 • Note that this is the same as the definition of the principal square root of x2. That is, .

  31. Theorem • Let a and b be real numbers. Then • |ab| = |a||b| • if b  0

  32. Remark • Although we can “split” the absolute value of a product or quotient, the same cannot be said for the sum or difference of real numbers. That is, |a  b|  |a|  |b|.

  33. Examples • Find the solution set: (1) |3 – 8x| = 13 (2) 2|3 – 2x| = 5|x + 1| (3) |4 – |6 – 7x|| = 9 (4) |x – 3| + |x – 2| + |1 – x| = 3 (5) |5 – 3x – |3x + 1|| – 4 = –2x

  34. Theorem Let a > 0. Then • |x| < a if and only if –a<x<a. • |x| > a if and only if x<–a or x>a. • |x|  a if and only if –axa. • |x|  a if and only if x–a or xa.

  35. Examples • Solve the following inequalities: (1) (2) (3) (4)

  36. Lemma* • For any real number x, the following inequality is true:

  37. Theorem* (Triangle Inequality) • If a and b are real numbers, then |a + b|  |a| + |b|

  38. Example • Suppose that x and y are real numbers such that |x – 1|  3 and |y + 2|  1. Prove that |3y – 2x|  17.

  39. Exercises / Assignment Items • Find the solution set. For inequalities, express your final answer in interval notation. (1) |4 – 11x| = |5x – 28| (2) |3 – 2x| – |x + 5| – |4 – x| = -8 (3) (5) (4) (6) • Suppose that |x + 5|  4 and |y – 2|  7. Show that |x + 2y|  19.

  40. Chapter 2 Circles and Lines

  41. Section 2.1 The Rectangular Coordinate System

  42. Definition • An ordered pair (x,y) of real numbers has x as its first member and y as its second member. • The model of representing ordered pairs is called the rectangular coordinate system or the cartesian plane. It is developed by considering two real lines intersecting at right angles.

  43. The Cartesian Plane

  44. Definition • Each point in the plane is identified by an ordered pair (x,y) of real number x and y, called the coordinates of the point. • The first coordinate is the x-coordinate or abscissa and the second coordinate is the y-coordinate or ordinate.

  45. Problem • What is the distance between two points (x1, y1) and (x2, y2) in the plane?

  46. Distance Between Two Points • If the points lie on a horizontal line, y1 = y2, and the distance between the points is |x2 – x1|. • If the points lie on a vertical line, x1 = x2, and the distance between the points is |y2 – y1|. • If the two points do not lie on a horizontal or vertical line, they can be used to form a right triangle.

  47. Theorem (Distance Formula) • The distance d between the points A(x1,y1) and B(x2,y2) in the plane is given by

  48. Example 1 • Show that the points A(2,1), B(4,0), and C(5,7) form the vertices of a right triangle.

  49. Example 2 • Find the point on the y-axis that is equidistant from (-5,-2) and (3,2).

  50. Theorem (Triangle Inequality) • If P1, P2, P3 are any three points on the plane, then Moreover, equality is satisfied if and only if P2 is a point on the line segment .

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