Download - Math 010 Unit 6 Lesson 4
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