Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §4.2 Compound InEqualities Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 4.1 Review § • Any QUESTIONS About • §4.1 → Solving Linear InEqualities • Any QUESTIONS About HomeWork • §4.1 → HW-11

3. Compound InEqualities • Two inequalities joined by the word “and” or the word “or” are called compound inequalities • Examples

4. A B Intersection of Sets • The intersection of two sets A and B is the set of all elements that are common to both A and B. We denote the intersection of sets A andB as

5. Example  Intersection • Find the InterSection of Two Sets • SOLUTION: Look for common elements • The letters a and e are common to both sets, so the intersection is {a, e}.

6. Conjunctions of Sentences • When two or more sentences are joined by the word and to make a compound sentence, the new sentence is called a conjunction of the sentences. • This is a conjunction of inequalities: −1 <xandx< 3. • A number is a soln of a conjunction if it is a soln of both of the separate parts. For example, 0 is a solution because it is a solution of −1 < x as well as x < 3

7. -1 3 -1 3 Intersections & Conjunctions • Note that the soln set of a conjunction is the intersection of the solution sets of the individual sentences.

8. Example  “anded” InEquality • Given the compound inequality x > −5 andx < 2 • Graph the solution set and write the compound inequality without the “and,” if possible. • Then write the solution in set-builder notation and in interval notation.

9. Example  “anded” InEquality • SOLUTION → Graph x > −5 &x < 2 ( x>5 ) x< 2 x>5 andx< 2 ( )

10. Example  “anded” InEquality • SOLUTION → Write x > −5 &x < 2 • x > −5 and x < 2 • Without “and”: −5 < x < 2 • Set-builder notation: {x| −5 < x < 2} • Interval notation: (−5, 2) • Warning:Be careful not to confuse the interval notation with an ordered pair.

11. Example  Solve “&” InEqual • Given InEqual → • Graph the solution set. Then write the solution set in set-builder notation and in interval notation. • SOLUTION: Solve each inequality in the compound inequality and

12. Example  Solve “&” InEqual • SOLUTION: Write for ) [ • Without “and”: −2 ≤x < 4 • Set-builder notation: {x| −2 ≤x < 4} • Interval notation: [−2, 4)

13. “and” Abbreviated • Note that for a < b • a < xand x < b can be abbreviated a < x < b • and, equivalently, • b >xand x > a can be abbreviated b > x > a • So 3 < 2x +1 < 7 can be solved as 3 < 2x +1and2x + 1 < 7

14. Mathematical use of “and” • The word “and” corresponds to “intersection” and to the symbol ∩ • Any solution of a conjunction must make each part of the conjunction true.

15. A B No Conjunctive Solution • Sometimes there is NO way to solve BOTH parts of a conjunction at once. • In this situation, A and B are said to be disjoint

16. Example  DisJoint Sets • Solve and Graph: • SOLUTION: • Since NO number is greater than 5 and simultaneously less than 1, the solution set is the empty set Ø • The Graph: 0

17. A B Union of Sets • The union of two sets A and B is the collection of elements belonging to AorB. We denote the union of sets, A or B, by

18. Example  Union of Sets • Find the Union for Sets • SOLUTION: Look for OverLapping (Redundant) Elements • Thus the Union of Sets

19. DisJunction of Sentences • When two or more sentences are joined by the word or to make a compound sentence, the new sentence is called a disjunction of the sentences • Example  x < 2 orx > 8 • A number is a solution of a disjunction if it is a solution of at least one of the separate parts. For example, x = 12 is a solution since 12 > 8.

20. 8 2 2 8 Disjunction of Sets • Note that the solution set of a disjunction is the union of the solution sets of the individual sentences.

21. Example  Disjunction InEqual • Given Inequality → • Graph the solution set. Then write the solution set in set-builder notation and in interval notation • SOLUTION: First Solve for x or

22. Example  Disjunction InEqual • SOLUTION Graph → [ ) ) [

23. Example  Disjunction InEqual • SOLN Write → • Solution set: x < −1 or x≥ 1 • Set-builder notation: {x|x < −1 or x≥ 1} • Interval notation: (−, −1 )U[1, )

24.          Example  Disjunction InEqual • Solve and Graph → • SOLUTION: or

25. Mathematical use of “or” • The word “or” corresponds to “union” and to the symbol  ( or sometimes “U”) for a number to be a solution of a disjunction, it must be in at least one of the solution sets of the individual sentences.

26. Example  Disjunction InEqual • Solve and Graph → • SOLUTION: [ ) −1 0 1

27. Example  [10°C, 20°C] → °F • The weather in London is predicted to range between 10º and 20º Celsius during the three-week period you will be working there. • To decide what kind of clothes to bring, you want to convert the temperature range to Fahrenheit temperatures.

28. Example  [10°C, 20°C] → °F • Familiarize: The formula for converting Celsius temperature C to Fahrenheit temperature F is • Use this Formula to determine the temperature we expect to find in London during the visit there

29. Example  [10°C, 20°C] → °F 10 ≤ C ≤ 20. • Carry Out • State: the temperature range of 10º to 20º Celsius corresponds to a range of 50º to 68º Fahrenheit

30. Solving Inequalities Summarized • “and” type Compound Inequalities • Solve each inequality in the compound inequality • The solution set will be the intersection of the individual solution sets. • “or” type Compound Inequalities • Solve each inequality in the compound inequality. • The solution set will be the union of the individual solution sets

31. WhiteBoard Work • Problems From §4.2 Exercise Set • Toy Prob (ppt), 22, 32, 58, 78 • Electrical Engineering Symbols for and & or

32. P4.2-Toys More than10% • Which Toys Fit Criteria • More than 40% of Boys • OR • More than 10% of Girls More than40%

33. P4.2-Toys • Toys That fit the or Criteria • DollHouses • Domestic Items • Dolls • S-T Toys • Sports Equipment • Toy Cars & Trucks

34. All Done for Today SpatialTemporalToy

35. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –