Union, Intersection, and Compound Inequalities

MATH 017

Intermediate Algebra

S. Rook

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

Intersection of Sets

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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

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Intersection of Sets (Example)

Ex 1: Given A = {x | x is a whole number},

B = {x | -2 ≤ x < 5}, find A ∩ B

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Intersection of Sets (Example)

Ex 2: Given A = {x | x is a natural number}, B = {-3, -2, -1, 0, 1, 2, 3}, find A ∩ B

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Intersection of Sets (Examples)

Ex 3: Given A = {x | x ≤ -7}, B = {x | x < -2}, find A ∩ B

Solving Compound Inequalities Using Intersection

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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

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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

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Compound Inequalities Separated by and (Example)

Ex 4: Solve, graph, and put into interval notation: 2x – 3 ≤ 11 and 2x < 3x – 4

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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

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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

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Compound Inequalities with Two Inequality Symbols (Example)

Ex 6: Solve, graph, and put into interval notation: -9 < 2x – 7 ≤ 7

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Compound Inequalities with Two Inequality Symbols (Example)

Ex 7: Solve, graph, and put into interval notation:

38

1

42

x

Union of Sets

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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

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Union of Sets (Example)

Ex 8: Given A = {-2, 0, 1, 3, 4} and

B = {1, 2, 3, 4, 5}, find A U B

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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

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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

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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

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Solving Compound Inequalities Using Union (Example)

Ex 11: Solve, graph, and put into interval notation:

12144

5

3

32

2

xorxx

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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