1.5 Intersection, Union, & Compound Inequalities

• The intersection of two sets A and B is the set of all members that are common to A and B.

A conjunction

A ∩ B

Read as: “A and B”

Examples

1) {1,2,3,4} ∩ {2,3,7,8}

= {2,3}

2) {x / x > -2} ∩ {x / x < 1}

= {x / -2 < x < 1}

3) (-∞,-2] ∩ (4, ∞)

= { }

• When two sets have no elements in common, they are said to be disjoint. The solution is an empty set, { } or

• The union of two sets A and B is the collection of all elements belonging to A and/or B.

A disjunction

A U B

Read as “A or B”

Examples

1) {1,2,3,4} U {-1,0,1,2} = {-1,0,1,2,3,4}

2) {x / x ≤ -2} U {x / x > 4} = {x / x ≤ -2 or x > 4}

3) (-∞,2] U (-4, ∞) = (-∞,∞)

1) 1 ≤ x and x < 3

Graph:

Solution [1,3)

Solving compound inequalities

2) -1 ≤ 2x + 5 < 13

-5 - 5 -5

-6 ≤ 2x < 8

2 2 2

-3 ≤ x < 4 Graph

Solution [-3,4)

3) 2x – 5 ≥ -3 and 5x + 2 ≥ 17

+5 +5 -2 -2

2x ≥ 2 5x ≥ 15

2 2 5 5

x ≥ 1 x ≥ 3

Graph

Solution: [3, ∞)

4) x – 4 < -3 or x – 3 ≥ 3

+ 4 +4 + 3 +3

x < 1 or x ≥ 6

Graph:

Solution: (-∞, 1) U [6, ∞)

5) 3x – 11 < 4 or 4x + 9 ≥ 1

+ 11 +11 - 9 -9

3x < 15 4x ≥ -8

x < 5 x ≥ -2

Graph

Solution: (-∞,∞)

6) - 3x - 7 < -1 or x + 4 < -1

+ 7 +7 - 4 - 4

-3x < 6 x < - 5

-3 -3

x > -2

Graph

Solution: (-∞,-5) U (-2, ∞)