INTERSECTIONS, UNIONS, AND COMPOUND INEQUALITIES

• Two inequalities joined by the word “and” or the word “or” are called compound

inequalities.

" 3 0"x or x

" 5 3"x and x

INTERSECTIONS OF SETS AND CONJUNCTIONS OF SENTENCES

A B

A

The intersection of two set A and B is the set of all elements that are common in both A and B.

EXAMPLE 1

Find the intersection. {1, 2, 3, 4, 5}

Solution: The numbers 1, 2, 3, are common to both sets, so the intersection is {1, 2, 3}

CONJUNCTION OF THE INTERSECTION

When two or more sentences are joined by the word and to make a compound sentence, the new sentence is called a conjunction of the intersection. The following is a conjunction of inequalities.

A number is a solution of a conjunction if it is a solution of both of the separate parts.

12 anx xd

The solution set of a conjunction is the intersection of the solution sets of the individual sentences.

EXAMPLE 2Graph and write interval notation for the conjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

12 anx xd

{ | 2 }x x ){ | 1}x x

)

)

{ | 2 } { | 1}

{ | 2 1}

x x x x

x x and x

)

( 2, )

( ,1)

( 2,1)

MATHEMATICAL USE OF THE WORD “AND”

The word “and” corresponds to “intersection” and to the symbol ““. Any solution of a conjunction must make each part of the

conjunction true.

EXAMPLE 3Graph and write interval notation for the conjunction

1 2 5 13x SOLUTION: This inequality is an abbreviation for the conjunction true

1 2 5 2 5 13andx x

1 2 5 2 5 13andx x Subtracting 5 from both sides of each inequality

6 2 2 8an xdx Dividing both sides of each inequality by 2

3 4anx xd

EXAMPLE 3Graph and write interval notation for the conjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

3 4anx xd { | 3 }x x

{ | 4}x x

)

)

{ | 3 } { | 4}

{ | 3 4}

x x x x

x x

[

[

[ 3, )

( ,4)

[ 3, 4)

THE STEPS IN EXAMPLE 3 ARE OFTEN COMBINED AS FOLLOWS

1 2 5 13

1 5 5 52 5 13

6 2 8

3 4

x

x

x

x

Subtracting 5 from all three regions

Dividing by 2 in all three regions

Caution: The abbreviated form of a conjunction, like -3 can be written only if both inequality symbols point in the same direction. It is not acceptable to write a sentence like -1 > x < 5 since doing so does not indicate if both -1 > x and x < 5 must be true or if it is enough for one of the separate inequalities to be true

EXAMPLE 4Graph and write interval notation for the conjunction

2 5 3 5 2 17x and x SOLUTION: We first solve each inequality retaining the word and

2 5 3 5 2 17andx x Add 5 to both sides

2 2 5 15andx x

Divide both sides by 2

1 3anx d x Divide both sides by 5

Subtract 2 from both sides

EXAMPLE 4Graph and write interval notation for the conjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

{ | x 1}x

{ | 3}x x

{ | x 1} { | 3}

{ | 3}

x x x

x x

[

[

[1, )

[3, )

[3, )

1 3anx d x

[

EXAMPLE 5

Sometimes there is no way to solve both parts of a conjunction at once

A B When A A and B are said to be disjoint.

A

EXAMPLE 5Graph and write interval notation for the conjunction

2 3 1 3 1 2x and x SOLUTION: We first solve each inequality separately

2 3 1 3 1 2andx x

Add 3 to both sides of this inequality

2 4 3 3andx x

Divide by 2

2 1anx d x Divide by 3

Add 1 to both sides of this inequality

EXAMPLE 5Graph and write interval notation for the disjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

{ | x 2}x

{ | 1}x x

{ | x 2} { | 1}

{ | 2 1}

x x x

x x and x

(2, )

( ,1)

)

) )

)

2 1anx d x

UNIONS OF SETS AND DISJUNCTIONS OF SENTENCES

A B

A

The union of two set A and B is the collection of elements that belong to A and / or B.

EXAMPLE 6

Find the union. {2, 3, 4

Solution: The numbers in either or both sets are 2, 3, 4, 5, and 7, so the union is {2, 3, 4, 5, 7}

DISJUNCTIONS OF SENTENCES

When two or more sentences are joined by the word or to make a compound sentence, the new sentence is called a disjunction of the sentences. Here is an example.

A number is a solution of a disjunction if it is a solution of at least one of the separate parts.

3 3o xx r

The solution set of a disjunction is the union of the solution sets of the individual sentences.

EXAMPLE 7Graph and write interval notation for the conjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

{ | 3}x x )

{ | 3}x x

)

)

{ | 3} { | 3}

{ | 3 3}

x x x x

x x or x

)3 3o xx r

( , 3)

(3, )

( , 3) (3, )

MATHEMATICAL USE OF THE WORD “OR”

The word “or” corresponds to “union” and to the symbol ““. For a number to be a solution of

a disjunction, it must be in at least one of the solution sets of the individual sentences.

EXAMPLE 8Graph and write interval notation for the disjunction

7 2 1 13 5 3x or x SOLUTION: We first solve each inequality separately

Subtract 7 from both sides of inequality

2 8 5 10x or x

Divide both sides by 2

4 2x or x Divide both sides by -5

7 2 1 13 5 3x or x

Subtract 13 from both sides of inequality

EXAMPLE 8Graph and write interval notation for the disjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

{ | 4}x x

{ | 2}x x

){ | 4} { | 2}

{ | 4 2}

x x x x

x x or x

[

[

( , 4)

[2, )

( , 4) [2, )

4 2x or x )

Caution: A compound inequality like:

4 2x or x

As in Example 8, cannot be expressed as because to do so would be to day that x is simultaneously less than -4 and greater than or equal to 2. No number is both less than -4 and greater than 2, but many are less than -4 or greater than 2.

2 4x

EXAMPLE 9Graph and write interval notation for the disjunction

2 5 2 3 10x or x SOLUTION: We first solve each inequality separately

Add 5 to both sides of this inequality

2 3 7x or x

Divide both sides by -2 37

2x or x

Add 3 to both sides of this inequality

2 5 2 3 10x or x

37

2x or x

EXAMPLE 9Graph and write interval notation for the conjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

3{ | }

2x x

{ | 7}x x

)

3{ | 7} { | }

23

{ | 7 }2

x x x x

x x or x

3( , )2

( , 7)

3( , 7) ( , )

2

))

)

EXAMPLE 10Graph and write interval notation for the disjunction

3 11 4 4 9 1x or x SOLUTION: We first solve each inequality separately

Add 11 to both sides of this inequality

3 15 4 8x or x

Divide both sides by 3

5 2x or x

Subtract 9 from both sides of this inequality

3 11 4 4 9 1x or x

Divide both sides by 4

EXAMPLE 10Graph and write interval notation for the disjunction

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

{ | 5}x x

{ | 2}x x

{ | 5} { | 2}

{ | 5 2}

x x x x

x x or x

[

( ,5)

[ 2, )

( , )

)

5 2x or x