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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

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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

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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

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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

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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}.

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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

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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

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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.

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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

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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.

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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

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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) [ )

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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

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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.

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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

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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

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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

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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

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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.

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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

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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

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

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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, )

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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

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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.

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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 [ )

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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.

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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

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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

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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

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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

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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%

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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

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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

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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 –

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