Union, Intersection, and Compound Inequalities MATH 017 Intermediate Algebra S. Rook.
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Transcript of 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
4
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
5
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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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
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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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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