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

4.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §4.1 → Solving Linear InEqualities Any QUESTIONS About HomeWork §4.1 → HW-08

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

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 A B

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}.

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

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

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.

Example  “anded” InEquality SOLUTION → Graph x > −5 & x < 2 x > 5 ( x < 2 ) x > 5 and x < 2 ( )

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.

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

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

“and” Abbreviated Note that for a < b and, equivalently, 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

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.

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

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:

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 A B

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

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 or 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.

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

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

Example  Disjunction InEqual SOLUTION Graph → [ ) ) [

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, )

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

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.

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.

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

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

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

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

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

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

All Done for Today Spatial Temporal Toy

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

Example  Disjunction InEqual Solve and Graph → SOLUTION: ) [ −1 1