Union, Intersection, and Compound Inequalities

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

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

Intersection of Sets

Intersection of Sets • Intersection (∩) [of 2 sets]:the elements common to both sets. • Usually easier to start with the set containing the least number of elements

Intersection of Sets (Example) Ex 1: Given A = {x | x is a whole number}, B = {x | -2 ≤ x < 5}, find A ∩ B

Intersection of Sets (Example) Ex 2: Given A = {x | xis a natural number}, B = {-3, -2, -1, 0, 1, 2, 3}, find A ∩ B

Intersection of Sets (Examples) Ex 3: Given A = {x | x≤ -7}, B = {x | x < -2}, find A ∩ B

Solving Compound Inequalities Using Intersection

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

Compound Inequalities Separated by and • Solve each linear inequality as normal • Graphing is somewhat trickier: • 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 • The intersection is the area between the left ( or [ and the right ) or ] • Obtain the interval notation from the intersection

Compound Inequalities Separated by and (Example) Ex 4: Solve, graph, and put into interval notation: 2x – 3 ≤ 11 and 2x < 3x – 4

Compound Inequalities Separated by and (Example) Ex 5: Solve, graph, and put into interval notation: 2(x – 3) – 3x≤ 3(x + 1) and 8x – 2(x – 3) > 24

Compound Inequalities with Two Inequality Symbols • Most common way to see an intersection compound inequality • Somewhat trickier to solve • 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

Compound Inequalities with Two Inequality Symbols (Example) Ex 6: Solve, graph, and put into interval notation: -9 < 2x – 7 ≤ 7

Compound Inequalities with Two Inequality Symbols (Example) Ex 7: Solve, graph, and put into interval notation:

Union of Sets

Union of Sets • Union (U) [of 2 sets]:the distinct elements from both sets • In other words, dump the elements of both sets together and remove the duplicates

Union of Sets (Example) Ex 8: Given A = {-2, 0, 1, 3, 4} and B = {1, 2, 3, 4, 5}, find A U B

Union of Sets (Example) Ex 9: Given A = {x | x≥ 0} and B = {x | x≥ 5}, find A U B

Solving Compound Inequalities Using Union

Solving Compound Inequalities Using Union • Two inequalities separated by the word or • Solve each linear inequality as normal • Graphing is somewhat trickier: • 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 • Remove any parentheses or brackets that have shading to the left and right • Obtain the interval notation from the union

Solving Compound Inequalities Using Union (Example) Ex 10: Solve, graph, and put into interval notation: 6(x – 3) – 5(x – 2) > -4 or 3(1 – x) – 6(2 – x) ≥ 0

Solving Compound Inequalities Using Union (Example) Ex 11: Solve, graph, and put into interval notation:

Summary • After studying these slides, you should know how to do the following: • Find the intersection of [2] sets • Solve compound inequalities involving intersection when: • The keyword and is used • The statement contains two inequalities • Find the union of [2] sets • Solve compound inequalities involving union

Union, Intersection, and Compound Inequalities