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Compound Inequalities

Slide Duration:

Section 1: Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
Section 2: Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
1:30
Solving Linear Equations in One Variable Cont.
2:00
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
Section 3: Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
Section 4: Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
Section 5: Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
Section 6: Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
Section 7: Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16

18m 33s

Intro
0:00
Objectives
0:07
0:25
Terms
0:33
Coefficients
0:51
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
Section 8: Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12

23m 38s

Intro
0:00
Objectives
0:08
0:19
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22

29m 27s

Intro
0:00
Objectives
0:12
0:29
Linear Factors
0:38
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
7:28
Discriminants
8:25
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07

16m 47s

Intro
0:00
Objectives
0:08
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14

29m 4s

Intro
0:00
Objectives
0:09
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50

26m 53s

Intro
0:00
Objectives
0:06
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
Section 10: Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43

20m 24s

Intro
0:00
Objectives
0:07
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
0:26
Product Rule to Simplify Square Roots
1:11
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09

17m 22s

Intro
0:00
Objectives
0:07
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
3:55
Example 1
4:47
Example 2
6:00
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10

19m 24s

Intro
0:00
Objectives
0:08
0:25
0:26
1:11
Don’t Distribute Powers
2:54
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25

15m 5s

Intro
0:00
Objectives
0:07
0:17
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
7:04
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33

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 1 answerLast reply by: Professor Eric SmithSat Aug 10, 2019 2:22 PMPost by Eva Wu on August 10 at 01:09:29 PMOn the slide for how the 'And' and 'Or' connections, there was a minor grammar mistake. It should be "Both conditions are met" and "If one of the conditions are met" not "Both conditions are meet" and "If one of the condition are meet" 0 answersPost by Karlin Hilai on February 17, 20174x+1 is greater than or equal to -7 should be x is equal to or greater than -2. 1 answerLast reply by: Professor Eric SmithThu May 29, 2014 10:47 AMPost by Mukesh Jain on May 24, 2014There is a "subtle" typo, as subtle was spelled subtitle :) 2 answersLast reply by: sherman boeyWed Aug 13, 2014 9:26 AMPost by Mankaran Cheema on August 20, 2013In example 3 why did you write -1 when it was supposed to be 1 and the answer would be x is greater than or equal to -2 where as you wrote x is -3 over 2

### Compound Inequalities

• When using the connector AND, both conditions must be satisfied. (Intersection)
• When using the connector OR, at least one of the conditions must be satisfied. (Union)
• A compound inequality is more than one inequality connected using AND or OR.
• Some inequalities can be put together into three parts. For these, whatever you do to one piece, you must do to the other two.

### Compound Inequalities

Solve the compound inequality:
3x − 2 < 74x + 6 > 10
• 3x − 2 < 7
• 3x < 9
• x < 3
• 4x + 6 > 10
• 4x > 4
• x > 1
1 < x < 3
7s − 5 ≤ 16 3x + 10 ≥ 22
• 7s − 5 ≤ 16
• 7s ≤ 21
• s ≤ 3
• 3s + 10 ≥ 22
• 3s ≥ 12
• s ≥ 4
3 ≥ s ≥ 4
6d + 2 > 87d − 5 < 9
• 6d + 2 > 8
• 6d > 6
• d > 1
• 7d − 5 < 9
• 7d < 14
• d < 2
1 < d < 2
3k − 4 ≤ 8
7k − 7 > 35
• 3k − 4 ≤ 8
• 3k ≤ 12
• k ≤ 4
• 7k − 7 > 14
• 7k > 21
• k > 3
3 < k ≤ 4
2n − 3 < 5
4n − 1 ≥ 19
• 2n − 3 < 5
• 2n < 8
• n < 4
• 4n − 1 ≥ 19
• 4n ≥ 20
• n ≥ 5
n < 4 and n ≥ 5
4 < n ≥ 5
8m − 2 ≤ 22
5m − 4 ≥ 51
• 8m − 2 ≤ 22
• 8m ≤ 24
• m ≤ 3
• 5m − 4 ≥ 51
• 5m ≥ 55
• m ≥ 11
m ≤ 3 and m ≥ 11
3 ≥ m ≥ 11
− 9 < 6x − 7 < 11
• − 9 < 6x − 7 and 6x − 7 < 11
• − 9 < 6x − 7
• − 2 < 6x
• − [2/6] < x
• − [1/3] < x
• 6x − 7 < 11
• 6x < 18
• x < 3
− [1/3] < x < 3
− 34 ≥ 6x + 2 > 26
• − 34 ≥ 6x + 26x + 2 > 26
• − 34 ≥ 6x + 2
• − 36 ≥ 6x
• − 6 ≥ x
• 6x + 2 > 26
• 6x > 24
• x > 4
− 6 ≥ x > 4
Solve the compound inequality:
− 2h − 11 > 9 or − 5h + 7 ≤ − 13
• − 2h − 11 > 9
• − 2h > 20
• h <− 10
• − 5h + 7 ≤ − 13
• − 5h ≤ − 20
• h ≥ 4
h <− 10 or h ≥ 4
− 5x − 4 ≥ 11 or − 3x + 12 < 36
• − 5x − 4 ≥ 11
• − 5x ≥ 15
• x ≤ − 3
• − 3x + 12 < 36
• − 3x < 24
• x >− 8
x >− 8 or x ≤ − 3

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Compound Inequalities

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:07
• Compound Inequalities 0:37
• 'And' vs. 'Or'
• 'And'
• 'Or'
• 'And' Symbol, or Intersection
• 'Or' Symbol, or Union
• Inequalities
• Example 1 6:22
• Example 2 9:30
• Example 3 11:27
• Example 4 13:49

### Transcription: Compound Inequalities

Welcome back to www.educator.com.0000

In this lesson, we are going to take care of compound inequalities.0002

In compound inequalities we will look at connecting inequalities using some very special words and, and or.0010

We will have to first recognize some subtleties between using both of these words.0017

Blabbering about some nice new vocabulary, we will learn about union and intersection.0023

Once we know more about how to connect them then we can finally get to our compound inequalities and how we can solve these.0029

In some situations, you have more than one condition and they could be connected using the word and or they could be connected using or.0040

To see the subtle difference between using one of these things, let us just try them out on a list of numbers.0050

I have the numbers 1 all the way up to 10 and we will see what number should be included in the various different situations below.0056

Let us see if we can first take care of all the numbers greater than 3 and less than 7.0066

Notice in this one I’m using that word, and.0071

Numbers that are greater than 3, we will have 4 or 5 and 6 and all of these are also less than 7.0075

The things I will include in my list 4, 5, and 6.0083

Let us think on how we will approach this problem a little bit different for the next one.0090

Maybe list out all numbers that are greater than 4 or they are less than 2.0095

The numbers greater than 4, that would we 5, 6, 7, 8, 9, and 10 and the numbers that are less than 2, 1.0101

Notice how in that situation I was looking for 2 things.0112

I was looking for all numbers that were greater than 4 and I have highlighted those first.0115

And then I went back and I looked for ones that were less than 2.0119

Let us give it a few little marks here and try the third situation.0127

All numbers greater than 6 and they are less than 3.0132

That is a tough one to do because I can go ahead and highlight the numbers greater than 6, that will be 7, 8, 9, 10 but they are not less than 3.0137

I can not include them.0148

Notice how in that situation since I'm dealing with and, it is like I have to satisfy both of these conditions must be greater than 6 must be less than 3.0150

If it does not satisfy both of them then I cannot include them.0164

Let us try another one and see how it works out.0167

All numbers greater than 4 so I have 5, 6, 7, 8, 9, 10 or they are less than 7.0170

What numbers less than 7 would be 6, 5, 4, 3, 2,1.0178

All the numbers fall into one category or they have fallen to the other one.0183

In fact, using or seems like it is a little bit more relaxed as long as it satisfies one of my conditions, then I will go ahead and include it in my list.0188

In fact, that highlights the difference between using and using the word or to connect these 2 conditions.0197

Let us make it even more clear.0204

When you use and to connect two conditions, then you must have both conditions met in order to include it.0207

However, when you are using or, then as long as you have one of the conditions met then you can go ahead and include that.0214

The way you will see and and or used is when we combine our intervals in our solutions.0222

If an object meets both conditions using our and connection, then it is said to be in the intersection of the conditions0230

and we can use the symbol to connect those.0239

Some people might think of this as and but it stands for the intersection.0242

On the flip side, if an object meets at least one of the conditions using the word or,0253

then it is said to be in the union of the conditions.0257

We will use the symbols, think of or and union.0261

You will see these symbols for sure and watch how we connect our solutions using either and or or.0267

We know a little bit more about the connections, let us get into how we can actually solve these things.0277

We will connect these things using and or or and the way you go about solving is you can actually solved each of the inequalities separately.0285

Just take care of one at a time and the part where these and or or come into play is when we want to connect together our solutions.0294

There are some situations where we can actually connect the inequalities at the very beginning.0305

One of those situations is when you are dealing with and and 2 of the parts are exactly the same.0310

Just like this example that I have highlighted below.0316

I have 5x < or = 3 + 11x and I also have a -3 + 11x on the other side over here.0321

We are connected using and.0329

I'm going to put these together into what is known as a compound inequality.0332

You will notice that everything on this side of the inequality comes from the left side.0336

Everything on the other side of inequality over here comes from this inequality.0344

It encapsulates both of them at the same time.0351

If you do connect one like this, the way you end up solving it is just remember that whatever you do to one part, do it to all 3 parts.0354

If you subtract 3, subtract 3 from all parts.0362

This also includes if you have to flip a sign.0366

If you multiply by a negative number then flip both of those inequality signs that are present then you should be okay.0368

Just remember if you want to solve each one separately, that works to.0377

Let us see some of our examples, see the solving process in action.0383

We want to solve the following inequality and of course write our answer using a number line and using interval notation.0388

I can see I have 2x -5 < or = - 7 or 2x – 5 > 1.0395

I’m going to solve these just separately, just take care of one at a time.0403

Looking at the left, I will add 5 to both sides, 2x > or = -2.0406

Now I can divide both sides by 2 and get that x < or = -1.0418

There is one of my solutions, let us focus on the other one.0429

You will add 5 to both sides of this inequality, 2x > 6, now divide by 2 and get that x > 3.0434

I have 2 intervals and I will be connecting things, let us drop this down using or.0450

I want to think of all the numbers that satisfy one of these, or satisfy the other one.0458

As long as they satisfy one of these conditions, I will go ahead and include it in my overall solution.0463

Let us take this a bit at a time.0469

I will do a little number line here, a number line here and I work on combining them into one number line.0472

First I can look at all the numbers that are less than or equal to 1.0481

Here is -1, I'm using a solid circle because it is or equals to and we are less than that so we will shade in that direction.0489

For the other inequality, I'm looking at numbers like 3 but not included or greater so I’m using an open circle there.0501

What will I put on my final number line is all places where I shaded at least once.0509

Let us see how that looks, I have my -1 and everything less and I have 3 or everything greater.0523

This number line which has both of them shows one condition or the other, and I can include both.0535

Let us go ahead and represent this using our interval notation.0543

We have everything less than - 1 and we include that -1 and we have everything from 3 up to infinity.0546

Since we are dealing with or, let us go ahead and use our union symbol to connect the two.0558

I have the many different ways that you can represent the solution for this inequality.0563

Let us take a look at another one, solve the following inequality and write your answer using a number line and interval notation.0571

This is a special inequality, this is one of our compound inequalities because it have 1, 2, 3 different parts to it.0578

As long as you remember that whatever we do to one part, we should do it to all three,I think it will turn out okay.0585

Let us work on getting that x all by itself in its particular part.0591

We will go ahead and subtract 5 from all three parts and get – 8 < 2x is < or = 2.0597

Let us divide everything by 2.0612

-8 ÷ 2 = -4, x < or = 1.0620

In this one I am looking for all values between -4 and 1.0628

I think we can make a number line for that.0633

Okay, I need to shade in everything in between -4 and all the way up to 1.0649

It looks like the -4 is not included, I will use a nice open circle, but the 1 is included, so we will shade that in.0657

Now that we have our number line, we can describe this using our interval notation.0667

We are starting way down at -4 not included and going all the way up to , that is included.0671

In some of these inequalities, you have to be careful on which conditions it satisfies.0689

If you end up with no numbers that simply do not work, watch how that turns out.0695

We want to solve the following inequality and write our answer using a number line and interval notation.0701

I have 4x + 1 > or = -7 or -2x + 3 > or = 5.0706

Let us begin by solving each of these separately.0716

With 1 on the left, I will start by adding 1 to both sides, 4x < or = -6 and now divide both sides by 4.0721

We are reducing that, I get that x > or = -3/2.0737

We will set that off to this side and solve the other one.0744

Let us subtract 3 from both sides and we will go ahead and divide both sides by -2.0749

Since we are dividing by negative, I’m going to flip my inequalities symbol.0758

This one is very interesting.0766

Notice how I'm looking for numbers that are less than or equal to 1, but I’m also looking for numbers that are greater than or equal to -3/2.0768

If I look at where those two are located, here is -3/2 and here is 1.0779

You will see that we get actually all numbers because I'm looking for things that are greater than or equal to -3/2,0785

that will shade everything on the right side of that - 3/2.0792

Everything less than or equal to 1 would shade everything in the other direction.0796

It will end up shading the entire number line.0801

We will say we would not include all numbers from negative infinity up to infinity.0805

I might make a note, all real numbers.0813

Watch for a very similar situation to happen when we deal with and.0824

Solve the following inequality and write our answer using a number line and interval notation.0830

I have two inequalities and a bunch of different x's, let us work on getting them together first.0837

I'm going to subtract an x from both sides on the left side here, 1x < -5.0842

I can add up 8 on both sides and we will get that x < 3.0851

Let us go over to the right and see what we can do there.0861

I will subtract 15 from both sides, that will give me –x < -10.0864

Now divide both sides by -1, since we are dividing by a negative number, let us flip out the inequality sign.0871

I have x < 3 and x > 10.0882

This one actually proves that when we are dealing with and, it must satisfy both of these conditions in order to be included.0889

If you start thinking what numbers are less than 3 and also greater than 10, you will that you do not get any number.0896

You can not be both things at the same time.0902

They are both in completely different spots on a number line.0906

Let us make another one.0913

We can not shade in numbers that are both less than 3 and greater than 10.0917

What is that mean to our solution then?0942

I mean what can we put down?0943

This is where we say there is no solution.0947

Be careful on using those connectors and how it affects your solution.0953

Remember that when using the word or, it must satisfy one of the conditions.0957

However, when using the word and, it must satisfy both of those conditions before you can include it in your solution.0964

Thanks for watching www.educator.com.0971

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