Math 010 Unit 6

Lesson 4

Objectives:

• to write a set in roster notation

• to write a set in set-builder notation

• to graph an inequality on the number line

Sets - the roster method

The roster method of writing a set encloses a list of the elements in braces.

Examples:

The set of the last three letters in the alphabet.

The set of integers between 0 and 10

1.

2.

3. The set of integers greater than or equal to 4

{x, y, z}

{1, 2, 3, 4, 5, 6, 7, 8, 9}

{4, 5, 6, . . .}

Other definitions

The empty set is the set that contains no elementsThe symbol for the empty set is { } or

The union of two sets, written A B is the set that contains the elements of A and the elements of B

The intersection of two sets, written A B is the set that contains the elements that are common to both A and B

A = {3, 5, 7, 9, 11} B = { 4, 6, 8, 10, 12} C = {2, 8, 10, 14}

A B =

{3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

B C =

{8, 10}

A C =

Another method of representing sets is called set-builder notation.

{x|x < 10, x positive integers}

The set of all x such that x is less than 10 and x is an element of the positive integers.

Read the following set:

{x| x > 5, x integers}

The set of all x such that x is greater than 5 and x is an element of the integers.

Write the following in set builder notation

The set of negative integers greater than -100

The set of real numbers less than 3

{x | x > -100, x negative integers}

{x | x < 3 , x real numbers}

0 5 10- 5-10

Graph the inequality: {x | x > 1, x real numbers}

(

The parentheses indicate that 1 is not included in the graph.

A bracket, [ , would indicate that a number is included in the graph.

Another method of graphing the same inequality is shown below.

0 5 10- 5-10

0 5 10- 5-10

Graph the following: {x | x 2}

Graph the following: {x | x -1} {x | x > 2}

0 5 10- 5-10

Remember: The union of two graphs are the points that are in one graph or the other.

Graph the following: {x | x 5} {x | x > -1}

0 5 10- 5-10

The intersection of the two graphs is the set of points they have in common

Remember:

Solving and

Graphing Inequalities

Linear inequalities are solved like linear equations

• with one exception

Simplify each side of the inequality

Collect terms

Divide by the coefficient of the variable

• switch the inequality sign if dividing or multiplying both sides of the inequality by a negative number

Solve the following inequalities:

x – 5 > -2

x > -2 + 5

x > 3

5x – 9 < 4x + 3

5x – 4x < 3 + 9

x < 12

6x – 5x – 1 3

1 2

6

36x – 2 30x – 3

36x – 30x -3 + 2

6x -1

x 1 6

-

6x – 5x – 1 3

1 2

Solve the following inequalities:

-3x > -9

x < 3

3x – 9 < 8x + 11

3x – 8x < 11 + 9

x > -4

x 4 5

7 6

30

-24x < 35

x 35 24

-

-5x < 20

-

x 4 5

7 6

-

Words and Symbols

at least

at most

A basketball team must win at least 60% of their remaining games to qualify for the playoffs. They have 17 games left. How many must they win?

Let x = games they must win

x .6(17) x 10.2

The team must win at least 11 games

Solve the inequality and graph the solution

-8x > 8

x < -1

0 5 10- 5-10)

0 5 10- 5-10

or