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Union, Intersection, and Compound Inequalities

2. Overview. Section 2.5 in the textbookIntersection of setsSolving compound inequalities involving intersection:Using the word andHaving two inequalitiesUnion of setsSolving compound inequalities involving union. Intersection of Sets. . 4. Intersection of Sets. Intersection (n) [of 2 sets]:

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Union, Intersection, and Compound Inequalities

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    1. Union, Intersection, and Compound Inequalities MATH 017 Intermediate Algebra S. Rook

    2. 2 Overview Section 2.5 in the textbook Intersection of sets Solving compound inequalities involving intersection: Using the word and Having two inequalities Union of sets Solving compound inequalities involving union

    3. Intersection of Sets

    4. 4 Intersection of Sets Intersection (n) [of 2 sets]: the elements common to both sets Some guidelines when finding the intersection of 2 sets: Usually easier to start with the set containing the least number of elements Write sets expressed in set-builder notation in roster notation

    5. 5 Intersection of Sets (Examples) Ex 1: Find the following: a) Given A = {x | x ? wholes} and B = {x | x ? integers, -2 = x < 5}, find A n B b) Given A = {..., -9, -8, -7} and B = {-11, -10, -9, }, find A n B

    6. Solving Compound Inequalities Using Intersection

    7. 7 Solving Compound Inequalities Using Intersection Can be found in two formats: Two linear inequalities separated by the word and A statement containing two inequality symbols -4 < x < 7

    8. 8 Compound Inequalities Separated by and Solve each linear inequality as normal Graphing is somewhat tricky, but a good method is to: Draw 3 number lines with equal intervals On the first number line, graph the solution to the first inequality On the second number line, graph the solution to the second inequality On the third number line, graph the intersection of the first two number lines i.e. the common shaded areas on the first two graphs Obtain the interval notation from the intersection

    9. 9 Compound Inequalities Separated by and (Example) Ex 2: Solve, graph, and put into interval notation: a) 5x 17 > -7 and 2(x 1) + 2 = 0 b) 3x 5 < -4 and x > 2 c) 2x + 3 = -11 and 2x > 3x + 4

    10. 10 Compound Inequalities with Two Inequality Symbols Most common way to see an intersection compound inequality Somewhat awkward to solve the first few times: Goal is to isolate the variable between the 2 inequalities Perform Algebraic operations on 3 sides instead of 2 Simple to graph Once the variable is isolated, the intersection is obtained Parallel to and notation: e.g. 1 < x < 3 means the same as x > 1 and x < 3

    11. 11 Compound Inequalities with Two Inequality Symbols (Example) Ex 3: Solve, graph, and put into interval notation: a) b) -9 < 2x 7 = 7

    12. Union of Sets

    13. 13 Union of Sets Union (U) [of 2 sets]: the combination of the distinct elements from both sets Some guidelines when finding the union of 2 sets: Dump the elements of both sets together and remove the duplicates Write sets expressed in set-builder notation in roster notation

    14. 14 Union of Sets (Example) Ex 4: Find the following: a) Given A = {x | x ? naturals} and B = {x | x ? integers, -4 = x < 2}, find A U B b) Given A = {-2, 0, 1, 3, 4} and B = {1, 2, 3, 4, 5}, find A U B

    15. Solving Compound Inequalities Using Union

    16. 16 Solving Compound Inequalities Using Union Two inequalities separated by the word or Solve each linear inequality as normal Graphing is somewhat tricky: Draw 3 number lines with equal intervals On the first number line, graph the solution to the first inequality On the second number line, graph the solution to the second inequality On the third number line, lay the first two number lines on top of each other this represents the union Determine the endpoints and remove any parentheses or brackets that are NOT on the endpoints Obtain the interval notation from the union

    17. 17 Solving Compound Inequalities Using Union (Example) Ex 5: Solve, graph, and put into interval notation: a) 6(x 3) 5(x 2) > -4 or -3x 10 < 15 + 2x b)

    18. 18 Summary After studying these slides, you should know how to do the following: Find the intersection or union of [2] sets Solve compound inequalities involving intersection when: The keyword and is used The statement contains two inequalities Solve compound inequalities involving union Additional Practice: Attempt the suggested problems for 2.5 Next Lesson: Section 2.6: Absolute Value Equations

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