INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

Home » Mathematics » Algebra 1 » Graphing Inequalities with Two Variables

Dr. Carleen Eaton

Dr. Carleen Eaton

Graphing Inequalities with Two Variables

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Table of Contents

Section 1: Basic Concepts
Variables and Expressions

11m 22s

Intro
0:00
History of Algebra
0:12
Origin of Word
0:21
Real World Problems
0:35
Definitions
0:58
Variable
1:03
Algebraic Expression
1:37
Operations
2:02
Example 1: Words into Expressions
3:02
Example 2: Words into Expressions
5:20
Example 3: Words into Expressions
6:45
Example 4: Words into Expressions
9:46
Order of Operations

15m 59s

Intro
0:00
Example
0:17
Definition
0:57
Procedure to Evaluate an Arithmetic Expression
1:08
Grouping Symbols (Parentheses, Brackets, Braces)
1:19
Powers
1:42
Multiply/Divide Left to Right
1:57
Add/Subtract Left to Right
2:21
Example: Fraction Bar
2:49
Example 1: Evaluate Arithmetic Expression
3:45
Example 2: Evaluate Arithmetic Expression
7:28
Example 3: Evaluate Arithmetic Expression
10:11
Example 4: Evaluate with Variables
13:12
Distributive Property

9m 50s

Intro
0:00
Distributive Property Statements
0:23
Moving Forward
0:49
Rule for Subtraction
1:14
Reverse Order
1:40
Several Numbers
2:17
Example 1: Evaluate Using Distributive Property
2:56
Example 2: Multiply Using Distributive Property
4:10
Example 3: Simplify Using Distributive Property
4:59
Example 4: Simplify Using Distributive Property
7:03
Real Number System

17m 58s

Intro
0:00
Real Number System
0:31
Natural Numbers
0:39
Whole Numbers
1:11
Integers
1:23
Rational Numbers
1:52
Cannot Divide by Zero
2:18
Decimals
2:27
Example: Terminating or Repeating
2:39
Real Number System, Cont.
3:37
Square Roots
3:42
Examples
3:54
Irrational Numbers
4:36
Examples
5:02
Perfect Square
5:54
Real Number System, Cont.
6:49
Example: Number Line
7:02
Example 1: Which Set of Numbers
7:54
Example 2: Graph on Number Line
10:04
Example 3: Approximate Irrational Number
12:47
Example 4: Order Largest to Smallest
13:57
Functions and Graphs

34m 39s

Intro
0:00
Functions
0:15
Example: Function
0:29
Example: Not Functions (Relations)
1:15
Graphs
4:44
Visual Display
4:53
Example: X and Y
5:03
Coordinate Pairs
5:53
Discrete Function
8:19
Continuous Function
8:55
Vertical Line Test
10:55
Test if Function
11:12
Example: Pass Through Points
11:43
Domain and Range
14:13
Example
14:43
Example 1: Function Given by Table
16:24
Example 2: Cost of Gas
18:46
Example 3: Cost of Gas
23:15
Example 4: Cost of Mail
29:07
Section 2: Solving Linear Equations
From Sentences to Equations

16m 5s

Intro
0:00
Real World Applications
0:18
Strategy
0:26
Using Variables
0:32
Translate Phrases
0:48
Identity Equality Words
1:07
Example 1: Write Equation
1:32
Example 2: Write Equation
4:14
Example 3: Sisters' Ages
8:26
Example 4: Surface Area of Cylinder
12:52
Addition and Subtraction Techniques

15m 24s

Intro
0:00
Techniques
0:21
Addition Principle
0:24
Example
0:37
Subtraction Principle
1:44
Example
1:48
Strategy
2:33
Isolate the Variable
2:41
Example
2:55
Example 1: Solve Equation
3:39
Example 2: Solve Equation
5:38
Example 3: Word Problem
7:38
Example 4: Word Problem
11:14
Multiplication and Division Techniques

15m 41s

Intro
0:00
Isolating the Variable
0:08
Techniques
0:34
Multiplication Principle
0:41
Example
0:57
Division Principle
2:32
Example
2:47
Strategy
3:12
Example
3:30
Opposite Operation
3:53
Example 1: Solve Equation
5:07
Example 2: Solve Equation
6:50
Example 3: Solve Equation
10:05
Example 4: Word Problem
12:07
Techniques for Multistep Equations

14m 31s

Intro
0:00
What are Multistep Equations
0:06
Addition/Subtraction and Multiplication/Division
0:31
Strategy
0:43
Identify Last Operation
0:47
Example 1: Solve Equation
1:51
Example 2: Solve Equation
5:27
Example 3: Find Numbers
7:39
Example 4: Solve Equation
11:27
When the Variable is on Both Sides of the Equation

20m 17s

Intro
0:00
Solving More Complicated Equations
0:28
Distributive Property
0:41
Review of Distributive Property
0:55
Factoring
1:28
Subtracting
1:50
Applying with Addition/Subtraction
2:08
Possible Outcomes
2:45
Exactly One Solution
2:52
No Solution
3:08
True for All Real Numbers
4:45
Identities
5:01
Example 1: Solve Equation
6:03
Example 2: Solve Equation
9:08
Example 3: Solve Equation
14:06
Example 4: Solve Equation
17:28
Ratios and Proportion

16m 5s

Intro
0:00
Definitions
0:07
Ratio
0:10
Different Representations
0:14
Proportion
0:33
Example
0:40
Cross Product
1:08
Cross Multiplication
1:32
Example
2:13
Rates
3:33
Rates in Real Life
3:46
Example 1: Form a Proportion
4:43
Example 2: Cross Multiply
7:15
Example 3: How Long to Drive
9:00
Example 4: Cross Products
12:13
Applications of Percents

13m 46s

Intro
0:00
Definitions
0:15
Percent of Increase
0:27
Percent of Decrease
0:34
Examples
0:42
Sales Tax
1:48
Discount
2:44
Example 1: Temperature Change
3:12
Example 2: Sales Tax
5:44
Example 3: Clothing Discount
7:04
Example 4: Sales and Discount
9:15
More Than One Variable

20m 38s

Intro
0:00
More Than One Variable
0:21
Real Life Examples
0:30
Strategy
1:08
Possible Techniques
1:17
Typical Application
1:43
Solving for a Different Variable
1:59
Example 1: Solve for Y
5:06
Example 2: Solve for Q
7:38
Example 3: Solve for H
12:56
Example 4: Solve for X
16:04
Section 3: Functions
Relations

16m 58s

Intro
0:00
Definition
0:04
Relation
0:06
Table
0:18
Set of Ordered Pairs
1:01
Graph
1:38
Domain and Range
2:40
Example: Relation
2:51
Example: Broader Cases
3:48
Inverse of a Relation
4:42
Example
4:59
Example 1: Relation as Table/Graph
6:15
Example 2: Domain and Range
8:41
Example 3: Table, Graph, Domain, Range
10:36
Example 4: Inverse of a Relation
13:36
Functions

19m 27s

Intro
0:00
Definition
0:14
Review of Relations
0:27
Violation of Function
1:43
Example: Function
2:00
Vertical Line Test
3:18
Example
3:41
Function Notation
6:15
Using f(x)
6:26
Example: Value Assigned
7:12
Example 1: Relation a Function
8:10
Example 2: Relation a Function
9:39
Example 3: Using f(x) Notation
12:20
Example 4: g(x) Notation
15:01
Linear Functions

20m 15s

Intro
0:00
Definition
0:07
Standard Form
0:18
Example
0:52
Graph and Intercepts
2:39
Example: Graph
2:48
X-Intercept
2:56
Y-Intercept
3:35
Graphing Linear Equations
4:29
Example
4:47
Linear Functions
7:51
Example
8:15
Example 1: Linear
10:16
Example 2: Linear Equation
12:58
Example 3: Intercepts
14:23
Example 4: Equation from Intercepts
16:47
Section 4: Linear Functions and Their Graphs
Slope and Rate of Change

19m 46s

Intro
0:00
Rate of Change
0:06
Other Words
0:14
Example
0:24
Slope
2:12
Two Points
2:39
Steepness of a Line
2:57
Possible Slopes
4:29
Positive Slope
5:02
Negative Slope
5:29
Zero Slope (Horizontal Line)
6:23
Undefined Slope (Vertical Line)
7:08
Example 1: Rate of Change of Table
8:19
Example 2: Slope Through Points
10:52
Example 3: Increasing/Decreasing
13:06
Example 4: Slope Through Points
16:02
Direct Variation

13m 54s

Intro
0:00
Definitions
0:10
Constant of Variation k
0:21
Example: Gas and Miles Driven
0:59
Graph
1:50
k is Slope
2:04
Examples
2:27
Applications
2:47
Write, Graph, Solve
2:58
Example 1: Constant of Variation
3:11
Example 2: Graph Direct Variation
4:59
Example 3: Direct Variation
6:50
Example 4: Distance Car Travels
9:18
Slope Intercept Form of an Equation

12m 6s

Intro
0:00
Slope Intercept Form
0:12
m (Slope) and b (Y Intercept)
0:31
Example
1:12
Example 1: Slope Intercept Form Equation
2:39
Example 2: Graph the Equation
5:11
Example 3: Slope Intercept Form Equation
6:51
Example 4: Slope Intercept Form Equation
8:50
Point Slope Form of an Equation

9m 7s

Intro
0:00
Point Slope Form
0:07
Manipulating to Other Forms
0:35
m (Slope), x1 y1 (Point)
0:47
Example 1: Point Slope Form Equation
1:03
Example 2: Point Slope Form Equation
2:50
Example 3: Point Slope Form Equation
4:18
Example 4: Point Slope Form Equation
6:50
Parallel Lines and Perpendicular Lines

18m 2s

Intro
0:00
Parallel Lines
0:08
Example
0:15
Vertical Lines
0:40
Perpendicular Lines
1:19
Negative Reciprocal
1:31
Example
2:05
Example 1: Slope Intercept Form
3:25
Example 2: Parallel or Perpendicular
6:15
Example 3: Slope Intercept Form
9:27
Example 4: Slope Intercept Form
12:35
Section 5: Systems of Equations
Graphing Systems of Equations

22m 45s

Intro
0:00
Systems of Equations
0:10
Definition
0:15
Example
0:31
Solution
0:47
Solving by Graphing
1:23
Points of Intersection
1:36
Example
1:56
Number of Solutions
3:09
Independent
3:20
Dependent
3:50
Inconsistent
4:46
Example 1: Solve by Graphing
5:45
Example 2: Solve by Graphing
9:50
Example 3: Solve by Graphing
14:17
Example 4: Solve by Graphing
18:03
Solving by Substituting

22m 41s

Intro
0:00
Substitution
0:09
Example
0:45
Number of Solutions
2:47
Infinite Solutions
3:11
No Solutions
4:28
Example 1: Solve by Substitution
5:44
Example 2: Solve by Substitution
10:01
Example 3: Solve by Substitution
15:17
Example 4: Solve by Substitution
19:41
Solving by Addition and Subtraction

16m 13s

Intro
0:00
Fundamental Principle
0:10
Example
0:23
Example 1: Solve the System
1:52
Example 2: Solve the System
5:53
Example 3: Solve the System
10:15
Example 4: Solve the System
14:08
Section 6: Inequalities
Addition & Subtraction Techniques

11m 34s

Intro
0:00
Fundamental Principle
0:09
Example
0:36
Solutions of Inequalities
1:51
Inequality
1:59
Set Builder Notation
2:02
Graph on a Number Line
2:08
Examples
2:18
Example 1: Solve the Inequality
4:59
Example 2: Solve the Inequality
7:00
Example 3: Solve the Inequality
8:10
Example 4: Solve the Inequality
9:47
Multiplication & Division Techniques

10m 49s

Intro
0:00
Fundamental Principle
0:10
Only Positive Numbers
0:23
Example
0:51
Fundamental Principle, Cont.
2:01
Negative Numbers
2:12
Reverse Inequality Sign
2:28
Example
2:48
Example 1: Solve the Inequality
4:26
Example 2: Solve the Inequality
5:45
Example 3: Solve the Inequality
6:50
Example 4: Solve the Inequality
8:28
Techniques for Multistep Inequalities

16m 56s

Intro
0:00
Similarity to Multistep Equations
0:16
Negative Numbers
0:32
Example
0:49
Inequalities Containing Grouping Symbols
1:24
Example
1:35
Special Cases
2:45
Example: All Real Numbers
3:04
Example: Empty Set
4:10
Example 1: Solve the Inequality
6:05
Example 2: Solve the Inequality
7:39
Example 3: Solve the Inequality
9:57
Example 4: Solve the Inequality
13:56
Compound Inequalities

21m 32s

Intro
0:00
What is a Compound Inequality
0:07
Joined by 'And' or 'Or'
0:16
Inequalities Combined by 'And'
0:36
Intersection/Overlap
0:53
Example
1:08
Inequalities Combined by 'Or'
4:23
Union
4:41
Example
5:27
Example 1: Solve the Inequality
6:39
Example 2: Solve the Inequality
11:30
Example 3: Solve the Inequality
13:43
Example 4: Solve the Inequality
18:19
Equations with Absolute Value

24m 16s

Intro
0:00
Absolute Value
0:06
Number Line
0:22
Example
0:41
Absolute Value is N
1:52
Absolute Value Function
3:17
Example
3:40
g(x) and f(x)
4:31
Solving Absolute Value Equations
6:23
Absolute Value in Words
6:39
Split Into Two Parts
7:58
Solve Both Equations
8:22
Example 1: Solve the Absolute Value
10:34
Example 2: Solve the Absolute Value
13:09
Example 3: Solve the Absolute Value
14:52
Example 4: Solve the Absolute Value
20:23
Inequalities with Absolute Values

17m 37s

Intro
0:00
Inequalities of the Form |x|< n
0:07
Values that Satisfy Both Inequalities
0:46
Example
1:27
Inequalities of the Form |x|> n
3:58
Values that Satisfy Either Inequalities
4:19
Example
4:47
Example 1: Solve the Inequality
6:38
Example 2: Solve the Inequality
9:54
Example 3: Solve the Inequality
12:05
Example 4: Solve the Inequality
14:50
Graphing Inequalities with Two Variables

24m 33s

Intro
0:00
Graph
0:08
Half Plane and Boundary
0:51
Technique for Graphing
1:57
Graph Equation
2:01
Solid Line or Dashed Line
2:07
Example
2:32
Choosing a Test Point
5:10
Example
5:26
Example 1: Solve the Inequality
7:49
Example 2: Solve the Inequality
11:37
Example 3: Solve the Inequality
15:44
Example 4: Solve the Inequality
19:10
Graphing Systems of Inequalities

24m 4s

Intro
0:00
System of Inequalities
0:05
Example
0:22
Solving a System of Inequalities
0:38
Solution Set
0:46
Graph Each Inequality
0:57
Area of Overlap
1:45
Example 1: Solve the System of Inequalities
2:44
Example 2: Solve the System of Inequalities
6:33
Example 3: Solve the System of Inequalities
11:40
Example 4: Solve the System of Inequalities
17:36
Section 7: Polynomials
Multiplying Monomials

22m 19s

Intro
0:00
What is a Monomial
0:09
Examples
0:17
Power
0:55
Base and Exponent
1:52
Properties of Exponents
2:16
Add Exponents
2:25
Multiply Exponents
4:00
Product Exponent
4:39
Simplified Form
7:26
Examples
7:47
Example 1: Simplify the Monomial
8:26
Example 2: Simplify the Monomial
10:32
Example 3: Simplify the Monomial
12:48
Example 4: Simplify the Monomial
17:33
Dividing Monomials

24m 2s

Intro
0:00
Properties of Exponents
0:05
Dividing with Same Base
0:15
Example
0:53
Quotient Raised to Power
2:22
Example
2:53
Raising to 0 Power
4:00
Example
4:21
Negative Exponents
5:45
Example
6:05
Example 1: Simplify the Monomial
7:33
Example 2: Simplify the Monomial
14:56
Example 3: Simplify the Monomial
13:30
Example 4: Simplify the Monomial
17:35
Polynomials

8m 56s

Intro
0:00
What is a Polynomial
0:07
Monomial
0:40
Binomial
1:15
Trinomial
1:25
Degree of a Polynomial
1:56
Example: Degree of Monomial
2:13
Example: Degree of Polynomial
2:47
Ordering Polynomials
3:32
Example
3:47
Example 1: Trinomial or Binomial
4:44
Example 2: Find the Degree
5:27
Example 3: Increasing Powers
6:11
Example 4: Decreasing Powers
7:27
Adding and Subtracting Polynomials

15m 51s

Intro
0:00
Adding Polynomials
0:07
Like Terms
0:18
Example
1:02
Subtracting Polynomials
2:44
Example
2:58
Example 1: Add Polynomials
5:11
Example 2: Subtract Polynomials
7:30
Example 3: Add and Subtract
9:35
Example 4: Add and Subtract
12:09
Multiplying Polynomials by Monomials

18m 17s

Intro
0:00
Distributive Property
0:07
Example
0:54
Solving Equations
1:36
Isolate Variable and Solve
1:46
Example 1: Multiply
1:59
Example 2: Simplify
3:33
Example 3: Simplify
7:20
Example 4: Solve
13:37
Multiplying Polynomials

18m 2s

Intro
0:00
Distributive Property
0:08
Example
0:54
FOIL Method
2:44
First, Outer, Inner, Last
3:20
Example 1: Multiply
5:32
Example 2: Multiply
7:27
Example 3: Multiply
9:41
Example 4: Multiply
13:56
Special Products

17m

Intro
0:00
Square of a Sum
0:06
Example
1:09
Square of a Difference
2:46
Example
3:22
Difference of Two Squares
4:50
Example
5:31
Example 1: Multiply
6:24
Example 2: Multiply
8:34
Example 3: Multiply
11:03
Example 4: Multiply
12:54
Section 8: Factoring
Special Product

17m 51s

Intro
0:00
Prime and Composite Numbers
0:09
Prime Number
0:12
Composite Number
0:42
Factored Forms
1:39
Prime Factored Form
1:40
Factored Form
2:21
Greatest Common Factor
3:55
Example: GCF for Number
4:19
Example: GCF for Monomial
6:00
Example 1: Prime Factored Form
7:51
Example 2: Factored Form
9:34
Example 3: GCF
11:12
Example 4: GCF
13:28
Factoring Using Greatest Common Factor

25m 21s

Intro
0:00
Distributive Property
0:05
Example: Binomial
0:49
Example: Trinomial
2:18
Factoring by Grouping
4:17
Example: Four Terms
4:40
Zero Product Property
8:21
Example
9:01
Example 1: Factor the Polynomial
10:38
Example 2: Factor the Polynomial
13:43
Example 3: Factor the Polynomial
19:59
Example 4: Solve the Polynomial
22:58
Factoring Trinomials with Leading Coefficient of 1

27m 11s

Intro
0:00
Factoring Trinomials
0:07
Leading Coefficient
0:11
Example
1:20
Rules for Signs
2:42
P and Q Both Positive
2:55
P and Q Both Negative
3:39
P and Q Opposite Signs
4:30
Solving Equations
5:18
Example
6:44
Example 1: Factor the Polynomial
7:41
Example 2: Factor the Polynomial
12:33
Example 3: Factor the Polynomial
16:39
Example 4: Solve the Polynomial
21:35
Factoring General Trinomials

46m 9s

Intro
0:00
Factoring Trinomials
0:15
Example
2:42
Grouping
7:20
Example
7:35
Rules for Signs
10:51
Same as Leading Coefficient is 1
11:05
Greatest Common Factor
12:29
Use Whenever Possible
12:41
Example
12:59
Prime Polynomials
13:58
Example
14:33
Solving Equations
16:55
Example
17:25
Example 1: Factor the Polynomial
18:46
Example 2: Factor the Polynomial
25:23
Example 3: Factor the Polynomial
32:37
Example 4: Solve the Polynomial
36:18
Factoring the Difference of Two Squares

24m 3s

Intro
0:00
Difference of Two Squares
0:08
Example
0:36
Factoring Using Several Techniques
2:23
Factoring the GCF
2:30
Example
3:22
Solving Equations
5:24
Example
5:50
Example 1: Factor the Polynomial
7:34
Example 2: Factor the Polynomial
9:11
Example 3: Factor the Polynomial
12:00
Example 4: Solve the Polynomial
18:31
Factoring Perfect Squares

18m 10s

Intro
0:00
Perfect Squares
0:07
Example: Perfect Square Trinomials
1:12
Solving Equations
2:57
Square Root Property
3:09
Example
3:28
Example 1: Factor the Polynomial
5:09
Example 2: Factor the Polynomial
6:13
Example 3: Solve the Polynomial
8:43
Example 4: Solve the Polynomial
13:35
Section 9: Quadratic Functions
Graphing Quadratic Functions

35m 45s

Intro
0:00
Parabolas
0:14
Standard Form of Quadratic Function
0:28
Examples
1:05
Absolute Value of 'a'
2:19
Parabolas That Open Upward
3:14
Minimum
3:48
Example
3:57
Parabolas That Open Downward
6:57
Example
7:17
Maximum
9:23
Vertex
9:53
Example
10:40
Axis of Symmetry
14:16
Example
15:03
Example 1: Graph the Quadratic
19:54
Example 2: Graph the Quadratic
24:12
Example 3: Vertex Maximum or Minimum
28:32
Example 4: Axis of Symmetry
31:13
Solving Equations by Graphing

40m 42s

Intro
0:00
Solving a Quadratic Equation
0:08
Example
0:56
Two Distinct Solutions/Roots
8:10
Roots
8:23
Example: Graphs
8:40
One Double Root
9:19
Example: One X-Intercept
9:54
No Real Roots
14:03
Example
14:53
Estimating Solutions
18:41
Example: Not Integers
19:18
Example 1: Solve by Graphing
20:18
Example 2: Solve by Graphing
26:36
Example 3: Solve by Graphing
30:18
Example 4: Estimate by Graphing
34:59
Solving Equations by Completing the Square

28m 13s

Intro
0:00
Perfect Square Trinomials
0:15
Example
0:36
Completing the Square
4:55
Example
6:20
Completing the Square to Solve Equations
9:19
Example
9:40
When the Leading Coefficient is Not 1
13:17
Example
14:01
Example 1: Solve the Equation
15:05
Example 2: Complete the Square
20:16
Example 3: Solve by Completing the Square
22:31
Example 4: Solve by Completing the Square
25:02
Solving Equations Using the Quadratic Formula

17m 17s

Intro
0:00
Quadratic Formula
0:17
Standard Form
0:24
Example
1:00
Discriminant
3:14
Two Solutions and Both Real
3:40
One Real Solution
4:07
No Real Solutions
4:28
Example 1: Solve the Equation
6:25
Example 2: Solve the Equation
8:42
Example 3: Solve the Equation
12:02
Example 4: Number of Real Roots
15:23
Section 10: Radical Expressions and Equations
Simplifying Radical Expressions

41m 30s

Intro
0:00
Radical Expression
0:12
Example: Radicand Simplest Form
0:29
Example: Not Simplest Form
1:16
Principal Square Root (Positive)
2:43
Product Property
3:40
Examples
4:05
Square Roots of Variables with Even Powers
7:01
Eliminate Radical Sign
7:42
Divide Exponent by 2
7:57
Absolute Value of Result
8:29
Examples
8:52
Quotient Rule
14:12
Example
14:31
Rationalizing Denominators
16:08
Example
16:43
Conjugates
18:33
Example
19:53
Simplest Radical Form
20:58
Three Criteria
21:10
Example 1: Simplify Expression
21:57
Example 2: Simplify Expression
25:12
Example 3: Simplify Expression
31:37
Example 4: Simplify Expression
35:29
Operations with Radical Expressions

21m 52s

Intro
0:00
Adding and Subtracting Radical Expressions
0:13
Like Radicals
0:28
Distributive Property
1:10
Multiplying Radical Expressions
4:24
Example: Use FOIL
4:44
Example 1: Simplify Expression
7:07
Example 2: Simplify Expression
8:51
Example 3: Simplify Expression
12:14
Example 4: Simplify Expression
16:06
Solving Radical Equations

27m

Intro
0:00
Radical Equations
0:15
Examples
0:30
Solving a Radical Equation
1:13
Isolate Radical
1:18
Square Both Sides
1:38
Example
1:44
Extraneous Solutions
2:57
Example: Check Solutions
3:30
Example 1: Solve Equation
6:29
Example 2: Solve Equation
9:52
Example 3: Solve Equation
14:29
Example 4: Solve Equation
20:53
Pythagorean Theorem

17m 24s

Intro
0:00
Right Triangles
0:06
Vertex
0:32
Hypotenuse
0:56
Legs
1:11
Pythagorean Theorem
1:21
Graphical Representation
1:37
Example
2:39
Pythagorean Triples
3:40
Example
3:56
Converse of the Pythagorean Theorem
4:36
Example
6:23
Example 1: Length of Hypotenuse
7:24
Example 2: Length of Legs
9:02
Example 3: Area of Triangle
12:00
Example 4: Length of Side
14:59
Distance Formula

26m 50s

Intro
0:00
Distance Formula
0:09
Similarity to Pythagorean Theorem
0:21
Missing Coordinates
5:50
Example
6:22
Example 1: Distance Between Points
11:43
Example 2: Distance Between Points
14:05
Example 3: Distance Between Points
18:18
Example 4: Missing Coordinate
21:57
Section 11: Rational Expressions and Equations
Inverse Variation

24m 13s

Intro
0:00
Direct Variation
0:12
Inverse Variation
0:24
Constant of Variation k
0:50
Y Varies Inversely as X
0:59
Graphing Inverse Variation
3:09
Real World Applications
3:24
Example
3:59
Product Rule
10:19
Alternate Form
11:10
Finding Missing 4th Point
11:24
Example 1: Graph Inverse Variation
11:36
Example 2: Graph Inverse Variation
14:47
Example 3: Find Missing Point
19:39
Example 4: Find Missing Point
21:53
Rational Expressions

34m 22s

Intro
0:00
Rational Expressions
0:10
Examples
0:28
Excluded Values
1:03
Dividing by 0
1:29
Example
2:49
Simplifying Rational Expressions
7:12
Eliminating the GCF
7:17
Example: Regular Fraction
7:30
Example: Rational Expression
8:12
Simplifying and Excluded Values
10:15
Original Rational Expression
10:24
Example
10:47
Example 1: Find Excluded Values
13:47
Example 2: Simplify and Find Excluded Values
16:10
Example 3: Simplify and Find Excluded Values
22:04
Example 4: Simplify and Find Excluded Values
26:29
Multiplying Rational Expressions

22m 58s

Intro
0:00
Procedure
0:08
Examples
0:29
Cancel Before Multiplication
1:53
Example
2:04
Rational Expressions Containing Polynomials
3:18
Example
3:46
Example 1: Multiply Rational Expressions
6:04
Example 2: Multiply Rational Expressions
9:11
Example 3: Multiply Rational Expressions
11:19
Example 4: Multiply Rational Expressions
17:36
Dividing Rational Expressions

21m 49s

Intro
0:00
Procedure
0:10
Reciprocal of Expression
0:22
Example: Regular Fractions
0:44
Example: Rational Expressions
1:46
Cancel Before Multiplying
3:23
Why Cancel
3:45
Example
4:15
Rational Expressions Containing Polynomials
6:46
Example
7:06
Example 1: Divide Rational Expressions
9:15
Example 2: Divide Rational Expressions
13:11
Example 3: Divide Rational Expressions
15:39
Dividing Polynomials

35m 57s

Intro
0:00
Dividing a Polynomial by a Monomial
0:11
Example: Regular Fractions
0:36
Example: Polynomials
1:24
Dividing a Polynomial by a Binomial
2:56
Example: Dividend and Divisor
3:30
Long Division
5:28
Example: Regular Numbers
5:49
Example: Polynomials
7:17
Missing Terms
12:20
Definition
12:40
Example
12:55
Example 1: Divide the Polynomials
18:42
Example 2: Divide the Polynomials
20:54
Example 3: Divide the Polynomials
23:28
Example 4: Divide the Polynomials
28:52
Adding and Subtracting Rational Expressions with Like Denominators

17m 38s

Intro
0:00
Adding with Like Denominators
0:09
Example: Regular Numbers
0:19
Example: Rational Expressions
1:05
Subtracting with Like Denominators
2:35
Example: Regular Fractions
2:52
Example: Rational Expressions
3:05
Denominators That Are Additive Inverses
4:08
What Are Additive Inverses
4:35
Example
5:53
Example 1: Add Rational Expressions
7:54
Example 2: Subtract Rational Expressions
8:43
Example 3: Add Rational Expressions
10:39
Example 4: Subtract Rational Expressions
11:48
Adding and Subtracting Rational Expressions with Unlike Denominators

37m 16s

Intro
0:00
Least Common Multiple of Polynomials
0:21
Example: Regular Fractions
0:42
Example: Rational Expressions
5:18
Equivalent Rational Expressions Using LCM
7:23
Example
8:09
Adding and Subtracting
14:24
Summary of Techniques
14:32
Example 1: Find the LCM
15:09
Example 2: Add Rational Expressions
17:53
Example 3: Subtract Rational Expressions
22:19
Example 4: Add Rational Expressions
30:44
Complex Fractions

25m 38s

Intro
0:00
Mixed Expressions
0:10
Analogy to Mixed Fractions
0:23
Polynomial and Rational Expression
0:59
Example: Combining
1:55
Converting to Rational Expression
2:29
Complex Fraction
5:16
Examples
5:30
Simplifying Complex Fractions
6:08
Example
6:27
Example 1: Write as Rational Expression
9:43
Example 2: Simplify Complex Fractions
12:44
Example 3: Simplify Complex Fractions
15:03
Example 4: Simplify Complex Fractions
19:55
Rational Equations

38m 9s

Intro
0:00
Definition
0:11
Example: Cross Multiplication
0:39
Example: Rational Expressions
1:13
Solving Rational Equations
3:12
Multiply by LCM of Denominators
3:33
Example
4:02
Work Problems
7:19
Example: Complete a Project
8:17
Extraneous Solutions
12:41
Check All Solutions
13:18
Example
13:54
Example 1: Solve Rational Equation
17:28
Example 2: Solve Rational Equation
19:45
Example 3: Work Problem
27:15
Example 4: Solve Rational Equation
31:10
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Lecture Comments (4)

1 answer

Last reply by: Catherine MOLAKAL
Wed Jul 27, 2016 6:30 PM

Post by Catherine MOLAKAL on July 27, 2016

Dr Eathon,
how did you get -1 1/3 for -intercept, wont it be -4/3.

1 answer

Last reply by: Dr Carleen Eaton
Thu Jun 20, 2013 8:59 PM

Post by Taylor Wright on June 20, 2013

would it be easier to configure the equation in slope intercept form in order to determine the correct half plane?

Graphing Inequalities with Two Variables

  • The graph of an inequality in two variables is a half plane bounded by a straight line.
  • If the inequality is strict (< or >), then draw the boundary as a dashed line, otherwise draw it as a solid line.
  • Use a test point, normally the origin, to determine which half plane is the solution of the inequality. Shade that region.

Graphing Inequalities with Two Variables

Graph y > x - 2
  • Find slope and intercept
    b = - 2
    m = 1
  • Determine line type
  • Dashed since >
  • Test point (0,0)
    0 > 0 - 2
    0 > - 2
    Which is false
Graph utilizing properties
Graph y < 2x
  • Find slope and intercept
    b = 0
    m = 2
  • Determine line type
    Dashed since <
  • Test point (1,0)
    0 < 2(1)
    0 < 2
    Which is true
Graph utilizing properties
Graph y ≥ 5x + 1
  • Find slope and intercept
    b = 5m = 1
  • Determine line type
  • Solid since ≥
  • Test point (0,0)
    0 ≥ 5(0) + 1
    0 ≥ 1
    Which is false
Graph utilizing properties
Graph y ≥ [x/3] + 2
  • Find slope and intercept
    b = 2
    m = [1/3]
  • Determine line type
    Solid since
  • Test point (0,0)
    0 ≥ [0/3] + 2
    0 ≥ 2
    Which is false
Graph utilizing properties
Graph y ≤ x + 7
  • Find slope and intercept
    b = 7
    m = 1
  • Determine line type
  • Solid since £
  • Test point (0,0)
    0 ≤ 0 + 7
    0 ≤ 7
    Which is true
Graph utilizing properties
Graph y − x > 5
  • Find slope and intercept
    b = 5
    m = 1
  • Determine line type
  • Dashed since >
  • Test point (0,0)
    0 - 0 > 5
    0 > 5
    Which is false
Graph utilizing properties
Graph 2y + x <− 1
  • Find slope and intercept
    b = − 1
    m = − [1/2]
  • Determine line type
    Dashed since <
  • Test point (0,0)
    2(0) + 0 < - 1
    0 < - 1
    Which is false
Graph utilizing properties
Graph y + 2x > 3
  • Find 2 points using intercepts
    x = 0 → (0,3)y = 0 → (3,0)
  • Determine line type
    Dashed since >
  • Test point (0,0)
    0 + 2(0) > 3
    0 > 3
    Which is false
Graph utilizing properties
Graph 3y - x > - 5
  • Find 2 points using intercepts
    x = 0 → (0, − [5/3])y = 0 → (5,0)
  • Determine line type
    Dashed since >
  • Test point (0,0)
    3(0) - 0 > - 5
    0 > - 5
    Which is true
Graph utilizing properties
Graph [(4y + 5x)/2] ≤ 2
  • Find 2 points using intercepts
    x = 0 → (0,1)y = 0 → ([4/5],0)
  • Determine line type
    Solid since ≤
  • Test point (0,0)
    [(4(0) + 5(0))/2] ≤ 2
    [0/2] ≤ 2
    0 ≤ 2
    Which is true
Graph utilizing properties

*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.

Answer

Graphing Inequalities with Two Variables

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
  • Graph 0:08
    • Half Plane and Boundary
  • Technique for Graphing 1:57
    • Graph Equation
    • Solid Line or Dashed Line
    • Example
    • Choosing a Test Point
    • Example
  • Example 1: Solve the Inequality 7:49
  • Example 2: Solve the Inequality 11:37
  • Example 3: Solve the Inequality 15:44
  • Example 4: Solve the Inequality 19:10

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