Day 1: Representing Inequalities

Warm-Up Barry makes $100 a week plus a 8% commission on his sales. If his sales were $750 this week, calculate his total pay for the week.

Vocabulary inequality set-builder notation solution to an inequality interval notation

The Basics

An inequality is any statement that two quantities are not equal. The quantities are compared using the following signs:

A solution to an inequality is any value that makes the inequality true. Often, an inequality has too many solutions to list individually, so we use a graph. Example List 3 solutions and 3 non-solutions to the inequality x < 5.

Solutions Nonsolutions

Graph the solution set of x < 5:

1 Representing Solutions to Inequalities

The solution to an inequality can be represented in four ways: 1) As an Inequality

Using the symbols >, <, ≤, ≥ Examples x is greater than 5 x is less than -4 x is greater than or equal to 7 x is less than or equal to -2

Note: If an inequality is inverted, we turn it around so that we can read it with the variable first. 5 > 3 is the same as 3 < 5. 3 > x is the same as x < 3.

2) Set-builder Notation We can write the solution to an inequality as a set of all numbers that fit a certain description. Inequality Set-builder notation

This is read “the set of all x such that x is less than 5.”

Model Problem Exercise Write each inequality in set-builder notation. Write each inequality in set-builder notation. 1) ______1) ______

2) ______2) ______

3) ______3) ______

2 3) Using a Graph (Number Line) Examples

4) Interval Notation An interval is a space between points, called endpoints. Interval notation represents a set of numbers using the endpoints and indicates whether the endpoints themselves are included in a set. An open interval does not include the endpoints. An open interval is indicated by parentheses: ()

A closed interval does include the endpoints. A closed interval is indicated by square brackets: [ ]

An interval can also be half-open, including the endpoints on only one side. When there is no endpoint or one or more sides of an interval, we use the symbols ∞ and – ∞. (Note: these symbols always get parentheses on their side) The symbol ∞means there is no highest number in the interval. The symbol -∞ means there is no lowest number in the interval.

3 Examples

Graph Inequality Interval Notation

Exercise Write the inequality indicated by each graph. Then write it in interval notation.

______

Summary

 Inequalities can be expressed in a number of ways:

o As an inequality

o In set-builder notation

o In interval notation

4 o On a graph Smarty-Q Sketch the graph of the solution to the inequality -2x < 6. Give one number that is NOT in the solution set.

Exit Ticket Express the given inequality in the ways indicated.

Inequality Set-builder notation Interval notation Graph

Homework Fill in the missing boxes in the chart below.

Inequality Set-builder notation Interval Notation Graph

2) {m│m ≥ -5}

5) {x│x > 1.5}

9) {b│b < 8}

6 Day 2: Solving One-Variable Inequalities Warm-Up Graph the inequality y < -3 and express in interval notation.

Solving Inequalities Using Addition and Subtraction

Model Problem Find the solution set of each inequality. Graph and express in interval notation. 1)

Interval Notation:

Exercise Find the solution set of each inequality. Graph and express in interval notation. 1)

Interval Notation:

7 Solving Inequalities with Multiplication and Division

As you saw, solving inequalities using addition and subtraction is just like solving equations. However, when it comes to multiplication and division, there is one slight difference. This difference involves multiplication and division by negative numbers. Consider this example. What inequality sign belongs in the box?

______

When multiplying or dividing both sides of an inequality by a negative number, the sense of the inequality changes.

The “sense” of the inequality refers to the statement that the inequality is making. Note: We do not change the sense of the inequality when we reverse the entire statement. We simply change its direction. For example:

4 > 3 and 3 < 4 These inequalities are the same. We did not change the sense of the inequality.

4 > x and x < 4 These inequalities are the same. We did not change the sense of the inequality. x < 4 and x > 4 These inequalities are different. One says that x is greater than 4 and one says it is less. We DID change the sense of the inequality.

8 Model Problems Solve and graph each inequality. Write the solution in interval notation. 1)

Interval Notation: ______

Interval Notation:

______

Interval Notation:

______

9 Interval Notation:

______

10 Exercise Solve and graph each inequality. Write the solution in interval notation.

Interval Notation:

______

Interval Notation:

______

11 Interval Notation:

______

More Inequalities

Interval Notation:

______

2) Find all positive integers that satisfy the inequality:

Summary

 Solving inequalities is just like solving equations, except when multiplying or dividing by a negative number.

 Multiplying or dividing by a negative number switches the sense of the inequality. That is, the inequality faces the other number.

Exit ticket

12 What is the smallest whole number in the solution set of 4r - 4.9 > 14.95?

Smarty-Q Express each phrase using an inequality. x is a positive number ______x is a negative number ______x is not a negative number ______

Homework

Write each solution in interval notation.

13 Day 3: Solving Inequalities with the Variable on Both Sides Warm-Up Solve for x and graph. Write the result in interval notation:

Model Problems Solve. Express as in inequality, set-builder notation, and interval notation. Graph each inequality.

1) y ≤ 4y + 18

Inequality: ______Set-builder Notation: ______Interval Notation: ______

2) 4m – 3 < 2m + 6

3) 2(k – 3) > 6 + 3k – 3

14 Exercise Solve for the value of the variable. Express as in inequality, set-builder notation, and interval notation. Graph each inequality.

1) 4x ≥ 7x + 6

2) 5t + 1 < –2t – 6

3) 0.9y ≥ 0.4y – 0.5

4) 5(2 – r) ≥ 3(r – 2)

Inequality: ______Set-builder Notation: ______Interval Notation: ______15 Identities and Contradictions

Model Problems Solve for x. Tell the solution set.

Exercise Solve for the value of each variable.

16 Homework Solve for the value of the variable. Express in each notation shown. Graph.

Inequality: ______Inequality: ______

Set-builder: ______Set-builder: ______

Interval: ______Interval: ______

Inequality: ______Inequality: ______Set-builder: ______Set-builder: ______Interval: ______Interval: ______

Interval: ______Interval: ______17 Day 4: Solving Compound Inequalities Warm Up

Vocabulary compound inequality disjunction conjunction

A compound inequality is a statement that combines two simple inequalities using AND or OR. A statement that combines two inequalities using AND is called a conjunction. A statement that combines two inequalities using OR is called a disjunction.

Conjunctions

In this diagram, oval A represents some integer solutions of x < 10 and oval B represents some integer solutions of x > 0. The overlapping region represents numbers that belong in both ovals. Those numbers are solutions of both x < 10 and x > 0.

We write this solution set as: ______or as ______We say, “ ______”

18 You can graph the solutions of a compound inequality involving AND by using the idea of an overlapping region. The overlapping region is called the intersection and shows the numbers that are solutions of both inequalities.

Model Problems Solve and graph each inequality. Express in each notation indicated.

Set-builder notation: ______Interval Notation: ______

Set-builder notation: ______Interval Notation: ______19 3)

Exercises Solve and graph each inequality. Express in each notation indicated.

20 Disjunctions

In this diagram, circle A represents some integer solutions of x < 0, and circle B represents some integer solutions of x > 10. The combined shaded regions represent numbers that are solutions of either x < 0 or x >10.

You can graph the solutions of a compound inequality involving OR by using the idea of combining regions. The combine regions are called the union and show the numbers that are solutions of either inequality.

Model Problems Solve and graph each inequality. Express in each notation indicated. 1)

21 Exercise Solve and graph each inequality. Express in each notation indicated.

1) 4x ≤ 20 OR 3x > 21

Summary

22 Model Problems Write the compound inequality shown by each graph.

Inequality: ______

Exercise Write the compound inequality shown by each graph.

Inequality: ______

Exit ticket

Smarty-Q

23 Homework

Regents Review

24 Day 5: Sets and Set Notation Warm-Up Write the compound inequality given by the graph:

Vocabulary Set Subset Element of a set Complement

What is a set?

When we solve equa

Exercise In roster form, list all the elements that belong to each set.

25 What is a subset?

We write: A  B to mean “A is a subset of B” Exercise Consider the set A defined as A = {1, 2, 3}. List all 8 subsets of A.

What is the complement of a subset?

Suppose a set U = {1, 2, 3, 4, 5, 6} represents a “given universe” for a certain situation. If set A = {1, 2, 3}, then A is a subset of U, and we can write A U. However, the set {4, 5, 6} are the elements that are in the universe, but are not in subset A. We call the set {4, 5, 6} the complement of set A. The complement of a subset consists of all the elements in a “given universe” that are not in the subset.

“The complement of A” can be written: , , or ~A.

Model Problem The universe and the elements of set A are given. Find , the complement of A.

1) U = {red, orange, yellow, green, indigo, blue, violet} A = {red, blue, violet} = ______

2) U = {0, 2, 4, 6, 8, 16} A = {4, 8} = ______

3) U = {circle, triangle, square} = ______A = { } 26 4) U = {a│0 ≤ a 4, where a is an integer} A = {2} = ______Exercise 1) U = {M, A, D, I, S, O, N} A = {S, O, N} = ______

2) U = {3, 4, 7, 10, 14, 25, 32} A = {4, 7, 10, 25} = ______

3) U = {Nina, Pinta, Santa Maria} A = {Nina, Pinta, Santa Maria} = ______

4) Let the universe be the set of all integers between -3 and 6 inclusive. A subset of this universe is the positive factors of 3. What is the complement of this subset? ______

Identifying the Complement from a Graph

Model Problems 1) Write the inequality shown in each graph. 2) Graph the complement of each subset within the universe of real numbers. 3) Write an inequality for the complement.

1) Inequality: ______

Complement: ______

27 2) Inequality: ______Interval Notation: ______

Complement: ______

Exercise 1) Write the inequality shown in each graph. 2) Graph the complement of each subset within the universe of real numbers. 3) Write an inequality for the complement.

Complement: ______

28 Complement: ______

Interval Notation: ______Summary

 A set is a collection of objects called elements.

 A subset is any part of a set, including all or none of it.

 The complement of a subset is all the elements that are not in the subset, but in the “given universe.”

 We can find the complement of an infinite set or a finite set.

Exit ticket Consider all the positive integers from 1 to 10, exclusive. A subset of this universe are the prime numbers less than 10. What is the complement of this subset? ______

29 Homework 1) Set A defined as A = {2, 3, 5, 7, 11, 13} 7) Write an inequality for each graph. Set B defined as B = {2, 5, 7}. Graph the complement of the set shown in the Determine the complement of set B within set A. universe of real numbers. Write an inequality for each complement.

a. 2) Set A defined as A = {2, 4, 6, 8} Set B defined as B = {2, 6}. Determine the complement of set B within set A.

b. 3) Set A defined as A = {–1, -3, -6, -9} Set B defined as B = {-3, -9}. Determine the complement of set B within set A.

8) What is the complement of the set denoted by:

4) Set A defined as A = {H, E, A, R, T} a. [-6, 10) Set B defined as B = {A, R, T}. Determine the complement of set B within set A. b. [-

5) Consider the set of integers greater than -4 and 9) Can a set be its own complement? If so, less than 8. A subset of this set is the positive give an example. If not, explain why not. factors of 5. What is the complement of this subset?

6) Consider the set of integers greater than -18 and less than -1. A subset of this set is the negative factors of 3. What is the complement of this subset?

30 Day 6: Union and Intersection of Sets Warm-Up Given A = {3, 4, 5, 6, 7, 8} and a subset B = {6, 8}. What is the complement of set B in the universe of A?

Vocabulary union of sets intersection of sets

Set Operations

Definitions The union of sets A and B is the set of all elements that are in either A or B. We write: A B

The intersection of sets A and B is the set of all elements that are in both A and B. We write: A B

Model Problems The elements of sets A and B are given. Find A B and A B.

(c) 31 Exercise Determine the Union and the Intersection of the following sets.

1) A = {2, 4, 6, 8, 10} B = {2, 3, 4, 5, 6}

A B = A B =

2) A = {G, O, A, T} B = {P, O, P, E}

A B = A B =

3) A = {apples, bananas} B = {orange, plums}

A B = A B =

4) A = {–3, –6, –9} B = {0, –2, –4, –6}

A B = A B =

5) A = {H, A, I, R} B = {B, R, U, S, H}

A B = A B =

6) 32 (hint: remember is a symbol for the empty set)

The Union and Intersection of Infinite Sets

When sets have an infinite number of elements, we can use a number line to describe them. We use a procedure similar to the one to find the complement of an infinite set.

Model Problems Describe each set using an inequality. Find and graph and .

Inequality: ______

33 Inequality: ______

Inequality: ______

Exercise Describe each set using an inequality. Find and graph and .

Inequality: ______

Summary

34 The union of two sets A and B is the set of all elements in either A OR B. The intersections of two sets A and B is the set of all elements in BOTH A and B.

Smarty-Q

Challenge!

35 Homework

Find A B and A B .

1. A = {cats, dogs} B = {owls, snakes}

A B = A B =

2. A = {rings, necklaces, bracelets} B = {earrings}

A B = A B =

3. A = {10, 12, 14} B = {–10, –12, –14}

A B = A B =

Complete the following table.

36 The graphs of two sets, A and B, are shown below. Graph Then write an inequality that describes each graph.

37 Regents Review

38 Day 7: Review of Inequalities and Set Theory

1) SWBAT: Solve multi-step inequalities and graph the result on a number line.

Solve and graph each inequality on a number line.

3) 4) -3x + 1 ≥ 10

39 2) SWBAT: Use interval notation and/or set-builder notation to express the elements of a set. Solve each inequality and graph the results on a number line. Then express the results in interval notation.

Graph: Graph:

Interval Notation: Interval Notation:

Complete the following chart:

Inequality Graph Interval Notation

9) [-1, 7]

40 3) SWBAT: Find the intersection of sets and/or the union of sets.

Find the intersection and the union of each set. Make sure your answer uses proper set notation. For # 13 and 14, make sure to indicate you answer both graphically and in set notation.

Given Sets Union Intersection

L = {1, 3, 5, 7, 9, 11}

K = {3, 4, 5, 6}

A = {Fran, Karen}

B = {Kevin, Kim}

G = {x │x > 6}

H = {x │ 0 ≤ x ≤ 8}

G H = G H =

S = {x │0< x ≤ -1}

T = {x │- 1 ≤ x < 5}

S T = S T = 41 4) SWBAT: Find the complement of a subset of a given set, within a given universe.

Directions: Let U = {2, 4, 6, 8, 10, 12} and subsets A = {4, 8, 10} and B = {6, 8, 10}

Find: 15)

17) (the complement of A within the universe U)

42 Chapter 3: Inequalities and Set Theory

Mrs. Steptoe