1. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.2 Compound InEqualities

2. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.1 → Solving Linear InEqualities  Any QUESTIONS About HomeWork §4.1 → HW-11 4.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 3 Bruce Mayer, PE Chabot College Mathematics Compound InEqualities  Two inequalities joined by the word “and” or the word “or” are called compound inequalities  Examples

4. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 4 Bruce Mayer, PE Chabot College Mathematics 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 and B as AB

5. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 5 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 6 Bruce Mayer, PE Chabot College Mathematics 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 < x and x < 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 7 Bruce Mayer, PE Chabot College Mathematics Intersections & Conjunctions  Note that the soln set of a conjunction is the intersection of the solution sets of the individual sentences. -1 3 3

8. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  “anded” InEquality  Given the compound inequality x > −5 and x < 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  “anded” InEquality  SOLUTION → Graph x > −5 & x < 2 ( ) () x > 5x > 5 x < 2 x >  5 and x < 2

10. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 10 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 11 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Solve “&” InEqual  SOLUTION: Write for  Without “and”: −2 ≤ x < 4  Set-builder notation: {x| −2 ≤ x < 4}  Interval notation: [−2, 4) [ )

13. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 13 Bruce Mayer, PE Chabot College Mathematics “and” Abbreviated  Note that for a < b a < x and x < b can be abbreviated a < x < b  and, equivalently, b > x and x > a can be abbreviated b > x > a  So 3 < 2x +1 < 7 can be solved as 3 < 2x +1 and 2x + 1 < 7

14. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 14 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 15 Bruce Mayer, PE Chabot College Mathematics No Conjunctive Solution  Sometimes there is NO way to solve BOTH parts of a conjunction at once. AB  In this situation, A and B are said to be disjoint

16. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 16 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 17 Bruce Mayer, PE Chabot College Mathematics Union of Sets  The union of two sets A and B is the collection of elements belonging to A or B. We denote the union of sets, A or B, by AB

18. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Union of Sets  Find the Union for Sets  SOLUTION: Look for OverLapping (Redundant) Elements  Thus the Union of Sets

19. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 19 Bruce Mayer, PE Chabot College Mathematics 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 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 20 Bruce Mayer, PE Chabot College Mathematics Disjunction of Sets  Note that the solution set of a disjunction is the union of the solution sets of the individual sentences. 8 2 8 2

21. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 21 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Disjunction InEqual  SOLUTION Graph → [ ) [)

23. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 23 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Disjunction InEqual  Solve and Graph →  SOLUTION:       or

25. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 25 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Disjunction InEqual  Solve and Graph →  SOLUTION: 01−1−1 [ )

27. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 27 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 28 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  [10°C, 20°C] → °F  Carry Out 10 ≤ C ≤ 20.  State: the temperature range of 10º to 20º Celsius corresponds to a range of 50º to 68º Fahrenheit

30. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 30 Bruce Mayer, PE Chabot College Mathematics Solving Inequalities Summarized and  “and” type Compound Inequalities 1.Solve each inequality in the compound inequality 2.The solution set will be the intersection of the individual solution sets. or  “or” type Compound Inequalities 1.Solve each inequality in the compound inequality. 2.The solution set will be the union of the individual solution sets

31. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 31 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.2 Exercise Set Toy Prob (ppt), 22, 32, 58, 78  Electrical Engineering Symbols for and & or

32. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 32 Bruce Mayer, PE Chabot College Mathematics P4.2-Toys  Which Toys Fit Criteria More than 40% of Boys OR More than 10% of Girls More than 10% More than 40%

33. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 33 Bruce Mayer, PE Chabot College Mathematics P4.2-Toys  Toys That fit the or Criteria DollHouses Domestic Items Dolls S-T Toys Sports Equipment Toy Cars & Trucks

34. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 34 Bruce Mayer, PE Chabot College Mathematics All Done for Today Spatial Temporal Toy

35. BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 35 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –