INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Solving Compound and Absolute Value Inequalities

Slide Duration:

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
0:18
Same Dimensions
0:25
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities

31m 48s

Intro
0:00
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09

27m 3s

Intro
0:00
0:16
Standard Form
0:18
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
10:12
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56

22m 48s

Intro
0:00
0:21
Standard Form
0:29
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
13:50
16:03
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23

27m 5s

Intro
0:00
0:11
Test Point
0:18
0:29
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
0:13
Like Terms and Like Monomials
0:23
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
9:23
Odd Degrees
12:51
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
8:22
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
12:23
13:29
16:07
18:18

41m 11s

Intro
0:00
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
3:24
3:47
6:33
7:16
Rationalizing Denominators
8:27
9:05
11:47
Conjugates
12:07
13:11
16:12
16:20
16:28
19:04
Distributive Property
19:10
19:20
24:11
28:43
32:00
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22

31m 27s

Intro
0:00
0:11
0:22
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
11:27
Restriction: Index is Even
11:53
12:29
15:41
17:44
20:24
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
16:54
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50

47m 4s

Intro
0:00
0:22
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
12:48
13:09
21:42
29:13
35:02
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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• ## Related Books 1 answer Last reply by: Dr Carleen EatonWed May 2, 2018 10:54 PMPost by Pranav Brahmandam on April 13, 2018Hello Dr. Carleen Eaton, I question on the 3rd example. When u did the 2nd equation which was 2z - 6 >/ -8. When u add the 6 to the other  side, Shouldn't it be a -2 instead of -6.Thank you 1 answer Last reply by: Dr Carleen EatonSun Jun 1, 2014 10:11 PMPost by Leigh Sharpless on May 18, 2014i get that its the same solution in this example regardless of multiplying the 5 first or not, but what if the five were -5, then that would flip the sign when you multiplied it and made it an "and" equation. 1 answer Last reply by: Dr Carleen EatonSun Jun 1, 2014 10:10 PMPost by Leigh Sharpless on May 18, 2014for the 4th example, why dont you multiply the 5 in the denominator first before writing out the two possible equations? 1 answer Last reply by: Dr Carleen EatonSat Nov 16, 2013 3:15 PMPost by Jeffrey Tao on November 4, 2013For the first part, a number can't be both x<3 AND x>=4, so shouldn't it be OR? 1 answerLast reply by: Kavita AgrawalSun May 19, 2013 1:13 PMPost by Kavita Agrawal on May 19, 2013In the "And" Inequality slide, you said {x|2=

### Solving Compound and Absolute Value Inequalities

• A compound inequality combines two inequalities using either “and” or “or”. First solve each inequality separately. If “and” was used, the solution set is the set of all numbers in both solution sets of the two inequalities. If “or” was used, the solution is all numbers in either or both of the solution sets of the two inequalities.
• To solve an inequality involving absolute value, convert the original inequality into a compound inequality that does not involve absolute value, using the definition of absolute value. For example, |2x + 3| > 4 would become: either 2x + 3 > 4 or
• 2x + 3 < −4.
• Describe the solution set of a compound inequality using either a number line or set builder notation.

### Solving Compound and Absolute Value Inequalities

Solve the compound inequality: − 3 <− 5 − 2m < 1
• 1)Add 5 to both sides
• 2 <− 2m < 6
• 2) Divide both sides by − 2
• − 1 > m >− 3
• Change the order for clarity
− 3 < m <− 1
Solve the compound inequality: 19 < 3m − 5 < 22
• 1)Add 5 to both sides
• 24 < 3m < 27
• 2) Divide both sides by 3
8 < m < 9
Solve the compound inequality: − 83 <− 8p − 3 ≤ − 27
• 1)Add 3 to both sides
• − 80 <− 8p ≤ − 24
• 2) Divide both sides by − 8
• 10 > p3
• Arrange for clarity
3 ≤ p < 10
Solve the compound inequality: − 5k ≥ 34 or 2k + 1 ≥ 3
• Isolate the variable for both inequalities
• − 5k ≥ 35
• 2k ≥ − 4
• Divide so that the variable is alone
• − 5k ≥ 35
k ≤ − 7
• 2k ≥ −4
k ≥ −2
k ≤ − 7 or k − 2
Solve the compound inequality: − 5k + 9 ≤ − 36 or 3k + 6 ≤ − 21
• Isolate the variable for both inequalities and solve
• 5k + 9 ≤ − 36
− 5k ≤ − 45
k ≥ 9
• 3k + 6 ≤ − 21
3k ≤ − 27
k ≤ −9
k ≥ 9 or k ≤ − 9
Solve the compound inequality: 9 − 6k ≥ 21 or 7k + 5 ≥ 12
• Isolate the variable for both inequalities and solve
• 9 − 6k ≥ 21
− 6k ≥ 12
k ≤ −2
• 7k + 5 ≥ 12
7k ≥ 7
k ≥ 1
k ≥ 1 or k ≤ − 2
Solve the inequality: 4 − 2|9 + 4m|<− 26
• Subtract 4 from both sides
• − 2|9 + 4m|<− 30
• Divide both sides by − 2
• |9 + 4m|> 15
• Break inequality into two to eliminate the absolute value
• 9 + 4m > 15 and 9 + 4m <− 15
• Solve the inequalities
• 9 + 4m > 15
4m > 6
m > [6/4]
m > [3/2]
• 9 + 4m <− 15
4m <−24
m <−6
m > [3/2] or m <− 6
Solve the inequality: 6|4 − 8x| − 6 > 114
• Add 6 to both sides
• 6|4 − 8x|> 120
• Divide both sides by 6
• |4 − 8x|> 20
• Break inequality into two to eliminate the absolute vale
• 4 − 8x > 20 and 4 − 8x <− 20
• Solve the inequalities
• 4 − 8x > 20
−8x > 16
x <−2
• 4 − 8x <− 20
−8x <−24
x > 3
x > 3 or x <− 2
Solve the inequality: − 4|5m + 9| + 9 ≤ 5
• Subtract 9 from both sides
• − 4|5m + 9| ≤ − 4
• Divide both sides by − 4
• |5m + 9|1
• Break inequality into two to eliminate the absolute vale
• 5m + 91 and 5m + 9 ≤ − 1
• Solve the inequalities
• 5m + 91
5m ≥ −8
m ≥ −[8/5]
• 5m + 9 ≤ − 1
5m ≤ − 10
m ≤ − 2
m − [8/5] or m ≤ − 2
Solve the inequality: 6|7 − 8x| − 8 − 2
• Add 8 to both sides
• 6|7 − 8x|6
• Divide both sides by 6
• |7 − 8x|1
• Break inequality into two to eliminate the absolute vale
• 7 − 8x ≥ 1 and 7 − 8x ≤ − 1
• Solve the inequalities
• 7 − 8x ≥ 1
− 8x − 6
x ≤ [6/8]
x ≤ [3/4]
• 7 − 8x ≤ − 1
−8x ≤ −8
x ≥ 1
x ≥ 1 or x ≤ [3/4]

*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.

### Solving Compound and Absolute Value 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
• Compound Inequalities 0:08
• And and Or
• Example: And
• Example: Or
• And Inequality 1:41
• Intersection
• Example: Numbers
• Example: Inequality
• Or Inequality 4:35
• Example: Union
• Example: Inequality
• Absolute Value Inequalities 7:19
• Definition of Absolute Value
• Examples: Compound Inequalities
• Example: Complex Inequality
• Example 1: Solve the Inequality 12:54
• Example 2: Solve the Inequality 17:21
• Example 3: Solve the Inequality 18:54
• Example 4: Solve the Inequality 22:15

### Transcription: Solving Compound and Absolute Value Inequalities

Welcome to Educator.com.0000

In today's lesson, we are going to be solving compound and absolute value inequalities.0002

A compound inequality consists of two or more inequalities combined by either "and" or "or."0009

To solve a compound inequality, solve each part.0016

For example, if you are given 4x - 2 < 10, and x - 1 ≥ 4, it is connected by the word and, so it is a compound inequality.0022

So, I am going to solve each part of that.0040

4x...add 2 to both sides; that would be less than 12; divide both sides by 4, so x is less than 3.0044

And if I add 1 to both sides, I am going to get that x is greater than or equal to 4.0054

We will talk, in a second, about how the solution set works for this; but the first step is to just solve both parts of the inequality.0063

The same idea applies if an inequality is a compound inequality connected by the word or.0074

For example, you might be given 3x > 6, or 4x < 4.0080

Again, solve each part: x > 2, or x < 1.0090

Breaking this down and talking about the solution sets for each: if "and" is used, the solution set is the intersection of the solution sets of the individual inequalities.0103

So, I am going to review this here--the idea of intersection--and this is also covered in detail in the Algebra I series.0113

Intersection means the common members of both solution sets.0121

Just looking at numbers: if I have 2, 4, 6, 8, and 10, and that is my first set; and then I have a second set0127

that is 6, 8, 10, 12, and 14; the intersection would be the elements common to both.0136

So, I see that I have 6 in both; 8; and 10; so the intersection would be 6, 8, and 10.0149

Apply that to inequalities connected by the word and: for example, x - 4 < 10, and 2x ≥ 4.0158

OK, solve each part as discussed; so add 4 to both sides...this means that x is less than 14;0173

and (both conditions must hold) if I divide both sides by 2, I get x ≥ 2.0179

To visualize this, look at the number line: if I have a 0 right here, and then 2, 4, 6, 8, 10, 12, 14,0187

this is telling me that x is less than 14; and that would just go on and on and on.0200

This is telling me x is greater than or equal to 2, which is going to start here and go on and on and on and on.0207

But I am looking for the intersection, or the common elements; and the common elements would be greater than or equal to 2, and less than 14.0215

So, the intersection would come out as 0, 2, 4, 6, 8, 10, 12, 14--greater than or equal to 2, and less than 14.0225

And you could write that out as an inequality, or using set notation, that x is greater than or equal to 2, and less than 14.0240

So, this is actually just a more efficient way of writing, instead of two separate sections here with the word and;0255

you can just write it out more efficiently like this; x is greater than or equal to 2 and less than 14; or showing it on the number line like this.0262

So, that is for inequalities joined by the word and.0271

If or is used in a compound inequality, the solution set is the union of the solution sets of the individual inequalities.0276

Now, reviewing what a union is, just using numbers: if you have a set, such as 4, 5, 6, 7, 8; and then you have another set,0285

-2, -1, 2, 3, 4, 5; and you are asked to find the union; well, the union is any elements that are in either one of these, or both.0298

So, if it is in one; if it is in the other; or if it is in both; that would be the union.0315

Now, I am looking, and I have 4, and that is in both; I don't need to write it twice; 5; 6 is just in this one, but it is included;0321

7, just in that one; 8; and then, I already covered 4 and 5, but I also have to include -2, -1, 2, and 3.0332

And I could rewrite this in ascending order, but it is all included here.0344

The union means that it includes the elements that are in either one of these, or both.0348

Applying this concept to inequalities: 3x + 2 > 8, or 4x - 3 < 1 is a compound inequality joined by the word or.0353

First, solve each inequality: 3x >...subtract 2 from both sides...6; divide both sides by 3 to get x > 2.0366

Or: solve this one also--if I add 3 to both sides, I am going to get 4x < 4; I am going to divide both sides by 4 to get x < 1.0380

Now, looking at this on the number line, what this is saying is that...0, 1, 2, 3, 4...x is greater than 2.0391

This is saying x is less than 1; and there is no overlap here, but that is OK, because the union0403

means that, if something is in either one of these or both, it is included.0409

This is saying that the solution set is that x is greater than 2, or x is less than 1.0414

And this can be written in set notation; so, I would write it as x is greater than 2, or x is less than 1.0420

So, when you use "and," it is the intersection of the two solution sets.0429

When "or" joins a compound inequality, the solution set is the union of the two solution sets.0432

Now, we have worked with absolute value equations; and this time, today, we are working with absolute value inequalities.0440

To solve an absolute value inequality, you need to use the definition of absolute value.0448

And recall that the definition of absolute value is the distance that a value is from 0 on the number line.0453

So, the absolute value of 3 is 3, because the distance between this and 0 on the number line is 3.0461

The absolute value of -3 is also 3, because the distance between -3 and 0 on the number line is 3.0481

So, the absolute value of 3 is 3, because it is 1, 2, 3 away from 0 on the number line.0489

The absolute value of -3 is 3, because -3 is also 3 units from 0 on the number line.0494

For more complicated problems, you may need to rewrite the inequality as a compound inequality.0505

You may see inequalities in the form |x| < n.0516

If you see that, rewrite the inequality as a compound inequality, as follows.0523

Looking at this, I have an example: the absolute value of x is less than 4.0532

Think about what this is saying; it is saying that the absolute value of x is less than 4 units from 0.0541

OK, so anything that is less than 4...it could be 3, 2, 1, but it is less than 4 away from 0.0552

That could be satisfied by anything between 0 and 4; but it can also be satisfied by anything between 0 and -4,0562

because the distance between -4 and 0 on the number line is less than 4.0580

-3: the distance between that and 0 is less than 4.0585

So, anything in this range is going to satisfy it; so what this is really saying is that x is less than 4, and x is greater than -4.0589

So, in order for the answers to fall in this range, they can't just be less than 4 and go all the way down;0605

because then you will get numbers way over here, that have an absolute value that is much bigger than 4.0610

So, it is in this range, where x is greater than -4, but less than 4.0614

So, in general, when you see an absolute value in this form, where it is less than something, you rewrite it0621

as two related inequalities: x < 4, and x > -4.0630

We are just generalizing out: |x| < n can be rewritten as x < n, and x > -n.0637

OK, the other possibility is that you have inequalities that are in the form |x| > n.0647

For example, |x| > 3; well, what that is saying is that the absolute value of the number is more than 3 away from 0 on the number line.0660

So, anything bigger than 3 is going to have an absolute value of greater than 3.0673

4 has an absolute value greater than 3; and on up.0680

In addition, however, if I look over here on the left, any number smaller than -3 is also going to have an absolute value that is greater than 3.0685

If I took -4, the absolute value of that is 4, which is greater than 3.0695

So, this would translate to x > 3, or x < -3.0700

You need to memorize these two forms and be familiar with them.0709

And generalizing this out, this would say that |x| > n could be rewritten as x > n, or x < -n.0712

So, you need to keep these in mind: when you see |x| < n, you rewrite it as x < n, and x > -n.0722

When you see |x| > n, then x > n, or x < -n.0731

And you can apply these to more complex inequalities, such as |4x + 1| ≥ 12.0741

I would recognize that it is in this form, and I would rewrite it as 4x + 1 ≥ 12, or 4x + 1 ≤ -12.0752

So, just follow this pattern; and we will work on this in the examples.0770

OK, starting with Example 1: this is a shorthand way of a compound inequality that is joined by the word "and."0775

Instead of writing out 7 < 14x - 42, and 14x - 42 ≤ 35, they just combine the same term, put it in the middle, and left these on the outside.0785

But this is really a compound inequality that is joined by the word "and."0805

And remember that, in order to solve these, you solve each inequality; and then you find the intersection of their solution sets.0810

So, for absolute inequalities joined by the word "and," we are going to need to find the intersection of the solution sets.0818

All right, let's solve each of these; that is the first step.0824

I am going to rewrite this with the x on the left; 14x - 42 ≥ 7--more standard.0828

It is saying the same thing: 14x - 42 ≥ 7; I just reversed the sides of the inequality.0845

Next, add 42 to both sides; and that is going to give me 49; 14x ≥ 49.0854

Now, divide both sides by 14; and since that is a positive number, I don't need to reverse the inequality symbol.0869

I can simplify this, because they have a common factor of 7; so to simplify this, remove the common factor; that is going to give me 7/2.0878

I just pulled out the 7 from both the numerator and the denominator--factored it out.0890

OK, solving this inequality--my second inequality: adding 42 to both sides is going to give me 14x ≤ 77.0895

Again, dividing both sides by 14 is going to give me 77/14; again, I have a common factor of 7, so this is going to give me 11/2.0909

Now, the intersection of these, graphing this out: well, this is (2, 4, 6)...about 3.5, 3 and 1/2; and this is equal to 5 and 1/2.0922

OK, to help me graph it out, I am going to write it in decimal form: 1, 2, 3, 4, 5...a little more room....6.0945

OK, so what this is saying is x ≥ 3.5, which is right here; and it includes that point 3.5, so I am going to put a closed circle.0961

Here, the other restriction is that x ≤ 5.5; so this is greater than--it is going up this way;0975

but I have to stop when I get to here, because the intersection of the solution set has to meet both conditions.0983

It has to be greater than or equal to 3.5, and less than or equal to 5.5.0993

So, I can write it like this as an inequality; I can graph it here, or do set notation: where x is greater than or equal to 7/2 and less than or equal to 11/2.0998

OK, again, recognize that this is really a compound inequality; and it is joined by the word and, but it is just a shorthand way of writing it.1025

I wrote this out as two related inequalities, solved each, and found the intersection of their solution sets.1033

The second example is a compound inequality joined by the word or.1042

I am going to go ahead and solve both of these and find the union of the solution sets.1045

Adding 7 to both sides gives me 3y ≤ 15; divide both sides by 3: y ≤ 5.1058

On this side, I am subtracting 10 from both sides to get 2y > 16; dividing both sides by 2 gives me y > 8.1067

OK, since this is or, it is the union of the solution sets; so any element of either set,1082

or that is in both sets, would be included in the solution set for this compound inequality.1089

So here, I have y ≤ 5, and here I have y > 8.1104

So, all of this is included, and all of this is part of the solution set.1115

It could also be written as y ≤ 5, or y > 8.1122

OK, the third example involves an absolute value inequality.1132

When I look at this, I just want to think about which form this is in; and this is in the general form |x| < n.1138

And recall, for those, that you could rewrite this and get rid of the absolute value bars by saying x < n, and x > -n.1145

That is the general form; so I am going to rewrite this as 2z - 6 ≤ 8 (that is this form), and 2z - 6 ≥ -8 (which is this form).1158

It is really important to memorize these or understand them well enough that you can apply them to more complex situations.1174

Now, I am going to solve each.1181

2z - 6 ≤ 8: I am adding 6 to both sides to get 14, then dividing both sides by 2 to get z ≤ 7.1184

And I need to solve this other one.1200

I am going to add 6 to both sides to get 2z ≥ -6, z ≥ -3.1205

And since I am dividing by a positive number, I don't have to worry about reversing the inequality symbol.1218

OK, I end up with...again, I had an absolute value inequality in this form; I solved both inequalities.1226

Actually, I made a little mistake here; let me go ahead and correct that.1246

2z - 6 ≥ -8, so 2z ≥...this is actually going to be -2; so adding 6 to both sides is going to give me -2.1250

Now, if I divide both sides by 2, that is going to give me z ≥ -1.1263

OK, you can either leave this as it is, or you can go ahead and graph it out.1271

And since this is "and," the solution set needs to meet both of these conditions.1277

So, -1, 0, 1, 2, 3, 4, 5, 6, 7...a little more...let's go to 8; OK, z is less than or equal to 7, so this is going to go on and continue on;1284

but it also has to meet the condition that z is greater than or equal to -1.1301

So, anything in this range is going to be the intersection of the solution set for these inequalities.1305

Also, writing it out using set notation: what we have is that z is greater than or equal to -1, and less than or equal to 7.1313

These are three different ways of writing out the solution set.1331

OK, the next example is also an absolute value inequality.1335

And this one is in the form |x| > n, in this general form, which can be rewritten as x > n or x < -n.1340

So, rewriting this and removing the absolute value bars as two different inequalities: 3w - 8, divided by 5, is greater than 4;1354

so that is this first form; or 3w - 8, divided by 5, is less than -4.1364

OK, the first thing to do is get rid of the fraction: multiply both sides of this inequality by 5.1374

Next, add 8 to both sides; and finally, divide both sides by 3.1385

For this inequality, again, get rid of the fraction; multiply both sides by 5; add 8 to both sides; and divide by 3.1398

This is joined by the word "or"; and 28/3...I am going to rewrite that as 9 and 1/3.1415

So, w is greater than 9 and 1/3, or w is less than -4.1425

To graph this, -5, -4, -3, -2, -1, 0, 1...all the way to 9 and 1/3.1430

OK, so this is saying that w is greater than 9.3, or it is less than -4.1445

Open circle, because it is a strict inequality...1456

Set notation would be w is greater than 28/3, or w is less than -4.1460

Recognizing that this is in the general form |x| > n, I rewrote this as 3w - 8, divided by 5, is greater than 4, or the same expression is less than -4.1472

Solve each one: and the solution set is the intersection of the solution sets of those two inequalities.1487

That concludes this lesson of Educator.com; and I will see you soon!1496

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