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College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson. Prerequisites. P. The Real Number Line and Order. P.3. The Real Line. Introduction. The real numbers can be represented by points on a line, as shown.

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College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

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  1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

  2. Prerequisites P

  3. The Real Number Lineand Order P.3

  4. The Real Line

  5. Introduction • The real numbers can be represented by points on a line, as shown. • The positive direction (toward the right) is indicated by an arrow.

  6. Origin • We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0.

  7. Origin • Given any convenient unit of measurement, • Each positive number x is represented by the point on the line a distance of x units to the right of the origin. • Each negative number –x is represented by the point x units to the left of the origin.

  8. Coordinate • The number associated with the point P is called: • The coordinate of P

  9. Real Line • Then, the line is called any of the following: • Coordinate line • Real number line • Real line

  10. Real Line • Often, we identify the point with its coordinate and think of a number as being a point on the real line.

  11. Order on the Real Line

  12. Order of Numbers • The real numbers are ordered. • We say that a is less than b,and write a <b if b –a is a positive number. • Geometrically, this means that a lies to the left of b on the number line. • Equivalently, we can say that b is greater than a, and write b >a.

  13. Symbol a ≤ b • The symbol a≤ b (or b ≥ a): • Means that either a <b or a =b. • Is read “a is less than or equal to b.”

  14. Inequalities • For instance, these are true inequalities:

  15. E.g. 1—Graphing Inequalities • On the real line, graph all the numbers x that satisfy the inequality: • x < 3 • x≥ –2

  16. Example (a) E.g. 1—Graphing Inequalities • We must graph the real numbers that are smaller than 3. • Those that lie to the left of 3 on the real line.

  17. Example (a) E.g. 1—Graphing Inequalities • The graph is shown here. • Note that the number 3 does not satisfy the inequality. • So, it is indicated with an open dot on the real line.

  18. Example (b) E.g. 1—Graphing Inequalities • We must graph the real numbers that are greater than or equal to–2. • Those that lie to the right of –2 on the real line, including the number–2.

  19. Example (b) E.g. 1—Graphing Inequalities • The graph is shown here. • Note that the number –2 does satisfy the inequality. • So, it is indicated with a solid dot on the real line.

  20. Sets and Intervals

  21. Sets & Elements • A set is a collection of objects. • These objects are called the elements of the set.

  22. Sets • If S is a set, the notation a S means that a is an element of S. • b S means that b is not an element of S. • For example, if Z represents the set of integers, then –3 Z but πZ.

  23. Braces • Some sets can be described by listing their elements within braces. • For instance, the set A that consists of all positive integers less than 7 can be written as: A = {1, 2, 3, 4, 5, 6}

  24. Set-Builder Notation • We could also write A in set-builder notation as: • A = {x | x is an integer and 0 < x < 7} • This is read: “A is the set of all x such that x is an integer and 0 < x < 7.”

  25. Union of Sets • If S and T are sets, then their union S T is: • The set that consists of all elements that are in S or T (or both).

  26. Intersection of Sets • The intersection of S and T is the set S T consisting of all elements that are in both S and T. • That is, S T is the common part of S and T.

  27. Empty Set • The empty set, denoted by Ø, is: • The set that contains no element.

  28. E.g. 2—Union & Intersection of Sets • If S = {1, 2, 3, 4, 5} T = {4, 5, 6, 7} V = {6, 7, 8} • find the sets S T, S T, S V

  29. E.g. 2—Union & Intersection of Sets • S T = {1, 2, 3, 4, 5, 6, 7}(All elements in S or T) • S T = {4, 5}(Elements common to both S and T) • S V = Ø(S and V have no elements in common)

  30. Intervals • Certain sets of real numbers occur frequently in calculus and correspond geometrically to line segments. • These are called intervals.

  31. Open Interval • If a <b, the open interval from a to b consists of all numbers between a and b. • It isdenoted (a, b).

  32. Closed Interval • The closed interval from a to b includes the endpoints. • It is denoted [a, b].

  33. Open & Closed Intervals • Using set-builder notation, we can write: (a, b) = {x | a < x < b} [a, b] = {x | a≤ x ≤ b}

  34. Open Intervals • Note that parentheses ( ) in the interval notation and open circles on the graph in this figure indicate that: • Endpoints are excluded from the interval.

  35. Closed Intervals • Note that square brackets and solid circles in this figure indicate that: • Endpoints are included.

  36. Intervals • Intervals may also include one endpoint but not the other. • They may also extend infinitely far in one direction or both.

  37. Types of Intervals • The table lists the possible types of intervals.

  38. E.g. 3—Graphing Intervals • Express each interval in terms of inequalities, and then graph the interval. • [–1, 2) • [1.5, 4] • (–3, ∞)

  39. Example (a) E.g. 3—Graphing Intervals • [–1, 2) = {x | –1 ≤ x < 2}

  40. Example (b) E.g. 3—Graphing Intervals • [1.5, 4] = {x | 1.5 ≤ x ≤ 4}

  41. Example (c) E.g. 3—Graphing Intervals • (–3, ∞) = {x | –3 < x}

  42. E.g. 4—Finding Unions & Intersections of Intervals • Graph each set. • (a) (1, 3)  [2, 7] • (b) (1, 3)  [2, 7]

  43. Example (a) E.g. 4—Intersection of Intervals • The intersection of two intervals consists of the numbers that are in both intervals. • Therefore, (1, 3)  [2, 7] = {x | 1 < x < 3 and 2 ≤ x ≤ 7} = {x | 2 ≤ x < 3} = [2, 3)

  44. Example (a) E.g. 4—Intersection of Intervals • This set is illustrated here.

  45. Example (b) E.g. 4—Union of Intervals • The union of two intervals consists of the numbers that are in either one of the intervals. • Therefore, (1, 3)  [2, 7] = {x | 1 < x < 3 or 2 ≤ x ≤ 7} = {x | 1 < x ≤ 7} =(1, 7]

  46. Example (b) E.g. 4—Union of Intervals • This set is illustrated here.

  47. Absolute Value and Distance

  48. Absolute Value • The absolute value of a number a, denoted by |a|, is: • The distance from a to 0 on the real number line.

  49. Distance • Distance is always positive or zero. • So, we have: • |a| ≥ 0 for every number a • Remembering that –a is positive when a is negative, we have the following definition.

  50. Absolute Value—Definition • If a is a real number, the absolute value of a is:

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