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 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 elements The 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}

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

Graph the following: {x | x  2} Graph the following: {x | x  -1}  {x | x > 2} 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} 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 > x > 3 5x – 9 < 4x + 3 5x – 4x < x < 12 6x –  5x – x – 2  30x – 3 36x – 30x  x  -1 x  x –  5x –

Solve the following inequalities: -3x > -9 x < 3 3x – 9 < 8x x – 8x < x > -4 x  x < 35 x  x < 20 - x 

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 < ) or