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
Two inequalities joined by the word “and” or the word “or” are called compound inequalities.
Jan 03, 2016
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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
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