Download Representing Solutions to Inequalities

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)
2)
3)
___________________
___________________
___________________
1)
2)
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
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.
4
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
1)
2)
{m│m ≥ -5}
3)
4)
5)
{x│x > 1.5}
6)
7)
5
8)
9)
{b│b < 8}
10)
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:
2)
Interval Notation:
Exercise
Find the solution set of each inequality. Graph and express in interval notation.
1)
Interval Notation:
2)
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:
____________________
2)
Interval Notation:
_________________
3)
Interval Notation:
_________________
4)
Interval Notation:
_________________
9
Exercise
Solve and graph each inequality. Write the solution in interval notation.
1)
Interval Notation:
____________________
2)
Interval Notation:
_________________
3)
Interval Notation:
_________________
4)
Interval Notation:
_________________
10
More Inequalities
1)
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
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 _____________
11
Homework

Write each solution in interval notation.
12
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
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
3) 2(k – 3) > 6 + 3k – 3
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
13
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
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
2) 5t + 1 < –2t – 6
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
3) 0.9y ≥ 0.4y – 0.5
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
4) 5(2 – r) ≥ 3(r – 2)
Inequality: ________________ Set-builder Notation: _________________ Interval Notation: ______________
14
Identities and Contradictions
Model Problems
Exercise
Solve for x. Tell the solution set.
Solve for the value of each variable.
15
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: _____________
Inequality: ___________
Set-builder: __________
Interval: _____________
Inequality: ___________
Set-builder: __________
Interval: _____________
Inequality: ___________
Inequality: ___________
Set-builder: __________
Set-builder: __________
Interval: _____________
Interval: _____________
Inequality: ___________
Inequality: ___________
Set-builder: __________
Set-builder: __________
Interval: _____________
Interval: _____________
16
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, “ ________________________________________________________________”
17
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.
1)
Set-builder notation: _______________________ Interval Notation: __________________________
2)
Set-builder notation: _______________________ Interval Notation: __________________________
18
3)
Set-builder notation: _______________________ Interval Notation: __________________________
Exercises
Solve and graph each inequality. Express in each notation indicated.
1)
Set-builder notation: _______________________ Interval Notation: __________________________
2)
Set-builder notation: _______________________ Interval Notation: __________________________
3)
Set-builder notation: _______________________ Interval Notation: __________________________
19
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)
Set-builder notation: _______________________ Interval Notation: __________________________
2)
Set-builder notation: _______________________ Interval Notation: __________________________
20
Exercise
Solve and graph each inequality. Express in each notation indicated.
1) 4x ≤ 20 OR 3x > 21
Set-builder notation: _______________________ Interval Notation: __________________________
2)
Set-builder notation: _______________________ Interval Notation: __________________________
Summary
21
Model Problems
Write the compound inequality shown by each graph.
Inequality: _____________________
Inequality: _____________________
Exercise
Write the compound inequality shown by each graph.
Inequality: _____________________
Inequality: _____________________
Exit ticket
Smarty-Q
22
Homework
Regents Review
23
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.
24
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={ }
25
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: _______________
Interval Notation:
____________
Complement: _____________
Interval Notation: _____________
26
2)
Inequality: _______________
Interval Notation:
____________
Complement: _____________
Interval Notation: _____________
3)
Inequality: _______________
Interval Notation:
____________
Complement: _____________
Interval Notation: _____________
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.
1)
Inequality: _______________
Interval Notation:
____________
Complement: _____________
Interval Notation: _____________
2)
Inequality: _______________
Interval Notation:
____________
Complement: _____________
Interval Notation: _____________
3)
Inequality: _______________
Interval Notation:
____________
Complement: _____________
27
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? _________________________________________
28
Homework
1) Set A defined as A = {2, 3, 5, 7, 11, 13}
Set B defined as B = {2, 5, 7}.
Determine the complement of set B within set 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.
7) Write an inequality for each graph.
Graph the complement of the set shown in the
universe of real numbers.
Write an inequality for each complement.
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}
Set B defined as B = {A, R, T}.
Determine the complement of set B within set A.
a. [-6, 10)
b. [-
5) Consider the set of integers greater than -4 and
less than 8. A subset of this set is the positive
factors of 5. What is the complement of this
subset?
9) Can a set be its own complement? If so,
give an example. If not, explain why not.
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?
29
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.
(b)
30
(c)
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 =
2) A = {G, O, A, T}
A B =
B = {P, O, P, E}
A B =
3) A = {apples, bananas}
A B =
B = {orange, plums}
A B =
4) A = {–3, –6, –9}
A B =
B = {0, –2, –4, –6}
A B =
5) A = {H, A, I, R}
A B =
A B =
B = {B, R, U, S, H}
A B =
31
6)
(hint: remember
is a symbol for the empty set)
7)
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: _______________
Inequality: ______________
Inequality: ______________
32
Inequality: _______________
Inequality: ______________
Inequality: ______________
Exercise
Describe each set using an inequality. Find and graph
and
.
Inequality: _______________
Inequality: ______________
Inequality: ______________
Inequality: _______________
Inequality: ______________
Inequality: ______________
33
Summary
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!
34
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 =
3. A = {10, 12, 14}
A B =
B = {–10, –12, –14}
A B =
A B =
Complete the following table.
35
The graphs of two sets, A and B, are shown below. Graph
Then write an inequality that describes each graph.
36
Regents Review
37
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.
1)
3)
2)
4) -3x + 1 ≥ 10
38
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.
6)
5)
Graph:
Graph:
Interval Notation:
Interval Notation:
Complete the following chart:
Inequality
Graph
Interval Notation
7)
8)
9)
[-1, 7]
10)
39
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
G
H=
G
H=
S
T=
S
T=
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}
S = {x │0< x ≤ -1}
T = {x │- 1 ≤ x < 5}
40
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)
16)
17)
(the complement of A within the universe U)
18)
19)
20)
41
Chapter 3:
Inequalities and
Set Theory
Mrs. Steptoe
42