INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Solving Systems of Inequalities By Graphing

Dr. Carleen Eaton

Dr. Carleen Eaton

Solving Systems of Inequalities By Graphing

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Table of Contents

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
Add/Subtract Left to Right
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
Addition Property
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
Addition Property
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
Quadrants
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
Shaded Region
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
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
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
Example 1: Matrix Addition
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
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
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
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
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
Solving Quadratic Equations by Factoring

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
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
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
Add to Both Sides
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
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
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
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
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
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
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
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
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
Leading Coefficient
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
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
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
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
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
Leading Coefficient Equal to One
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
Radicand
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
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
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
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
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
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
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
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
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
Example 2: Add Rational Expressions
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
Example 4: Solve Log Inequality
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
Example: Log Inequality
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
Example 4: Log Inequality
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
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
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
Radioactive Decay
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
Radius
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
Example 2: Center and Radius
11:51
Example 3: Radius
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
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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Lecture Comments (4)

1 answer

Last reply by: Dr Carleen Eaton
Thu Jul 31, 2014 7:09 PM

Post by Philippe Tremblay on July 23, 2014

I draw a short dotted line (like that: ---) next to the equation if it's a strict inequality, and a short solid line when it's not. It takes no time and I find it useful.

1 answer

Last reply by: Dr Carleen Eaton
Tue Oct 9, 2012 11:45 PM

Post by Carroll Fields on October 1, 2012

In the latter part of Example III, 4x+2y> or = to -2. The test point is (1,2) , so the result should be 2 is greater than or equal to -2(1)-1; not 2 is > or = to -2(2)-1. Making the final answer 2> or = to -3, not 2> or = to -5 ( as listed )

Solving Systems of Inequalities By Graphing

  • To solve a system, graph each inequality. The solution of the system is the intersection of the graphs of the inequalities.
  • If the intersection is empty, the system has no solution.

Solving Systems of Inequalities By Graphing

Solve by graphing:
y > 4x + 3
y ≥ − 2x − 3
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = [4/1] or [(−4)/(−1)]
  • Equation 1: b = 3
  • Equation 2: m = [rise/run] = [(−2)/1] or [2/(−1)]
  • Equation 2: b = -3
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y > 4x + 3
    0 > 4(0) + 3
    0 > 3
    Not True; Shade away from (0,0)
  • Test Equation 2: (0,0)
    y ≥ − 2x − 3
    0 ≥ − 2(0) − 3
    0 ≥ − 3
    True; Shade Towards (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
y > 4x − 3
y >− x + 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = [4/1] or [(−4)/(−1)]
  • Equation 1: b = 3
  • Equation 2: m = [rise/run] = [(−1)/1] or [1/(−1)]
  • Equation 2: b = 2
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y > 4x − 3
    0 > 4(0) − 3
    0 >− 3
    True; Shade Towards (0,0)
  • Test Equation 2: (0,0)
    y >− x + 2
    0 >− (0) + 2
    0 > 2
    Not True; Shade Away From (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
x ≤ 3
y ≥ − [1/3]x − 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = Undefined
  • Equation 1: b = none
  • Equation 1: Solid Vertical Line Through x=3
  • Equation 2: m = [rise/run] = [(−1)/3] or [1/(−3)]
  • Equation 2: b = -2
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    x ≤ 3
    0 ≤ 3
    True; Shade Towards (0,0)
  • Test Equation 2: (0,0)
    y ≥ − [1/3]x − 2
    0 ≥ − [1/3](0) − 2
    0 ≥ − 2
    True; Shade Towards (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
y > 1
y < [3/2]x − 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = 0
  • Equation 1: b = 1
  • Equation 2: m = [rise/run] = [3/2] or [(−3)/(−2)]
  • Equation 2: b = -2
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y > 1
    0 > 1
    Not True; Shade Away From (0,0)
  • Test Equation 2: (0,0)
    y < [3/2]x − 2
    0 < [3/2](0) − 2
    0 <− 2
    Not True; Shade Away From (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
y < [1/2]x − 1
y ≤ 2x + 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = [1/2] or [(−1)/(−2)]
  • Equation 1: b = -1
  • Equation 2: m = [rise/run] = [2/1] or [(−2)/(−1)]
  • Equation 2: b = 2
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y < [1/2]x − 1
    0 < [1/2](0) − 1
    0 <− 1
    Not True; Shade Away From (0,0)
  • Test Equation 2: (0,0)
    y ≤ 2x + 2
    0 ≤ 2(0) + 2
    0 ≤ 2
    True; Shade Towards (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
y > [2/3]x + 3
y < [2/3]x − 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = [2/3] or [(−2)/(−3)]
  • Equation 1: b = 3
  • Equation 2: m = [rise/run] = [2/3] or [(−2)/(−3)]
  • Equation 2: b = -2
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y > [2/3]x + 3
    0 > [1/2](0) + 3
    0 > 3
    Not True; Shade Away From (0,0)
  • Test Equation 2: (0,0)
    y < [2/3]x − 2
    0 < [2/3](0) − 2
    0 <− 2
    Not True; Shade Away From (0,0)
  • Step 3: Notice how this system of equations have no common shading in common.
  • In this case, the solution is the Empty Set, or No Solution.
Solve by graphing:
y < [3/2]x − 1
y < [1/2]x + 1
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to use y - intercept (b) and the slope (m)
  • Equation 1: m = [rise/run] = [3/2] or [(−3)/(−2)]
  • Equation 1: b = -1
  • Equation 2: m = [rise/run] = [1/2] or [(−1)/(−2)]
  • Equation 2: b = 1
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    y < [3/2]x − 1
    0 < [3/2](0) − 1
    0 <− 1
    Not True; Shade Away From (0,0)
  • Test Equation 2: (0,0)
    y < [1/2]x + 1
    0 < [1/2](0) + 1
    0 < 1
    True; Shade Towards (0,0)
  • Step 3:Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
2x + y < 3
3x − y ≥ 2
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to find the x - and y - intercepts for both equations. You may replace inequalities with equal sign.
  • Equation 1: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    2x + y = 3
    2x + 0 = 3
    2x = 3
    x = [3/2]
    ( [3/2],0 )
  • Equation 1: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    2x + y = 3
    0 + y = 3
    y = 3
    (0,3)
    Draw Dashed Line
  • Equation 2: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    3x − y = 2
    3x − 0 = 2
    3x = 2
    x = [2/3]
    ([2/3],0)
  • Equation 2: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    3x − y = 2
    0 − y = 2
    − y = 2
    y = − 2
    (0, - 2)
    Draw Solid Line
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    2x + y < 3
    2(0) + 0 < 3
    0 < 3
    True; Shade Towards (0,0)
  • Test Equation 2: (0,0)
    3x − y ≥ 2
    3(0) − (0) ≥ 2
    0 ≥ 2
    Not True; Shade Away From (0,0)
  • Step 3:Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
x + 2y ≤ − 4
3x + y ≤ 3
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to find the x - and y - intercepts for both equations. You may replace inequalities with equal sign.
  • Equation 1: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    x + 2y = − 4
    x + 0 = − 4
    x = − 4
    ( − 4,0)
  • Equation 1: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    x + 2y = − 4
    0 + 2y = − 4
    2y = − 4
    y = − 2
    (0, - 2)
    Draw a Solid Line
  • Equation 2: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    3x + y = 3
    3x + 0 = 3
    3x = 3
    x = 1
    (1,0)
  • Equation 2: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    3x + y = 3
    0 + y = 3
    y = 3
    (0,3)
    Draw Solid Line
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    x + 2y ≤ − 4
    0 + 2(0) ≤ − 4
    0 ≤ − 4
    Not True; Shade Away From (0,0)
  • Test Equation 2: (0,0)
    3x + y ≤ 3
    3(0) + (0) ≤ 3
    0 ≤ 3
    True; Shade Towards (0,0)
  • Step 3:Add a darker shade to the common shaded areas between the two lines.
Solve by graphing:
2x − y < 3
4x + y < 3
  • Things to remember when graphing inequalities:
  • a) Inequalities with " > " or " < " are dashed lines
  • b) Inequalities with " ≥ " or " ≤ " are solid lines
  • c) Solution is the common shaded area between the two inequalitites, if there is any.
  • If the lines share no shading in common, it is said that the solution is the empty set, and there is no solution.
  • Step 1: Graph both system of equations. Best method is to find the x - and y - intercepts for both equations. You may replace inequalities with equal sign.
  • Equation 1: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    2x − y = 3
    2x + 0 = 3
    2x = 3
    x = [3/2]
    ( [3/2],0)
  • Equation 1: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    2x − y = 3
    0 − y = 3
    − y = 3
    y = − 3
    (0, − 3)
    Draw a Dashed Line
  • Equation 2: x - Intercept
    Set y to zero. Solve for x.
    Point is (x,0)
    4x + y = 3
    4x + 0 = 3
    4x = 3
    x = [3/4]
    ( [3/4],0)
  • Equation 2: y - Intercept
    Set x to zero. Solve for y.
    Point is (0,y)
    4x + y = 3
    0 + y = 3
    y = 3
    (0,3)
    Draw Dashed Line
  • Step 2: Using a test point, check which way to shade for each graph. Always use the easiest point (0,0) if and only if is not on any of the lines
  • Test Equation 1: (0,0)
    2x − y < 3
    2(0) − 0 < 3
    0 < 3
    True; Shade Towards (0,0)
  • Test Equation 2: (0,0)
    4x + y < 3
    4(0) + (0) < 3
    0 < 3
    True; Shade Towards (0,0)
  • Step 3: Add a darker shade to the common shaded areas between the two lines.

*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

Solving Systems of Inequalities By Graphing

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
  • Solving by Graphing 0:08
    • Graph Each Inequality
    • Overlap
    • Corresponding Linear Equations
    • Test Point
    • Example: System of Inequalities
  • No Solution 7:06
    • Empty Set
    • Example: No Solution
  • 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

Transcription: Solving Systems of Inequalities By Graphing

Welcome to Educator.com.0000

Today, we will be covering solving systems of inequalities by graphing.0002

In order to solve a system of inequalities by graphing, you need to graph the solution set of each inequality.0009

So, previously we discussed techniques; and you can review that lecture about how to find the solution set of an inequality by graphing.0015

Here, all you are going to do is graph each inequality.0024

And the solution of the system of the inequalities put together is the intersection of the solution sets of the inequalities.0027

So, what this is, really, is the overlap--the area of overlap--the points that are in common between each of the solution sets.0035

For example, if you are given a system of inequalities: y ≤ x - 3, and y > -2x + 1,0044

recall the techniques for graphing an inequality to determine the solution.0056

First, you want to graph the corresponding linear equation to find the boundary line of the solution set.0063

Then, use a test point to determine the half-plane containing the solution set.0082

Let's go ahead and do that, and then talk about how to find, then, the solution set.0099

This is just graphing each one--finding the solution set for each one; and then we have to talk about how to find the solution set for the system.0105

So, I am going to start with this one; and that is y ≤ x - 3; and the corresponding linear equation would be y = x - 3.0111

So, this is in slope-intercept form, y = mx + b, so I can easily graph this, because I know that the y-intercept is -3 and the slope is 1.0126

Since the slope is 1, increase y by 1; increase x by 1; increase y by 1; increase x by 1; and on.0139

Now, looking back here, this says y is less than or equal to -3; and what that tells me is that I am going0148

to use a solid line here for the boundary line, because the line is included in the solution set; so use a solid line to graph this.0158

The next thing is to determine which half-plane--the upper or lower half-plane--this solution set is in.0174

And I want to use an easy test point, and that is the origin (0,0), as my test point.0179

And it is well away from the boundary line, so I can use that.0185

If it were near or on the boundary, I would want to pick a different test point.0189

I am going to take my test point, (0,0), and substitute those values in to the inequality.0194

So, 0 is less than or equal to 0 minus 3; 0 is less than or equal to -3.0200

Well, that is not true; since that is not true, that means that the solution set is not in this upper half-plane.0208

(0,0) is not part of the solution set; therefore, the solution set is in this lower half-plane.0216

OK, that is the solution set for this inequality; now let's work on the other one.0223

Here, y is greater than -2x + 1; the corresponding linear equation is y = -2x + 1.0229

So, here I have the y-intercept at 1 and a slope of -2.0240

So, decrease y by 2...1, 2; increase x by 1; decrease y by 2; increase x by 1; and so on.0246

Now, since this is a strict inequality, I am going to use a dashed line, because the boundary line here is not part of the solution set.0256

OK, so here is my dashed line; next, I need to find a test point.0267

And again, I am going to use (0,0); I am going to use that test point.0281

Actually, let's use a different one, because that is a bit close; so let's go ahead and pick one0293

that is farther away, so we don't have any chance of causing confusion.0297

Let's use something up here...(2,1); (2,1) would be good for the test point.0301

So, my test point is going to be (2,1): I am going to insert (2,1) into that inequality.0308

1 is greater than -2; x is 2; plus 1; this is my test point; and 1 is greater than -4 plus 1; 1 is greater than -3.0316

And this is true; since this is true, that means that the test point is in the correct half-plane where the solution set lies.0331

So, for this line, this is the upper half-plane, and this is part of the solution set; so I am going to go ahead and shade that in.0340

OK, so far, I had just been going along, doing what I usually do when I would find the solution set for an inequality.0356

But remember, we are looking for the solution set for a system of inequalities.0364

So, this lower half-plane is the solution set for this inequality; this half-plane is the solution set for this inequality.0369

Now, I am looking for the intersection between the two solution sets, and that is right here, in this quadrant right here.0376

Darken that in even more; I am going to go ahead and use another color to emphasize that.0382

OK, so the area bounded by this line and including that line--that is the boundary line for the system of equations.0396

And then, the boundary line over here is the dashed line; so this line is not part of the solution set.0406

The technique: graph each of the inequalities; find their solution sets; and then find the area that is the intersection of the two solution sets.0414

You may come across a situation where there is no solution to a system of inequalities.0427

And this is because the two inequalities may not have any points in common.0433

Remember: the common points in each of the solution sets comprise the solution set for the system of inequalities.0438

If the two inequalities have no points in common, then the solution set of the system is the empty set.0446

For example, let's say I am given y < x - 4 and y ≥ x - 2.0454

So, I am going to go ahead and graph those, starting with y < x - 4.0462

I need to find the boundary line, so the corresponding linear equation is y = x - 4.0467

So, - 4 is the y-intercept, and the slope is 1.0474

So, increase x by 1, and increase y by 1, with each step.0480

Since this is a strict inequality, I am going to use a dashed line.0487

I have my boundary line; now I need to use a test point, and I am going to use this (the origin) as the test point.0495

And I am going to substitute (0,0) into the inequality and see what happens.0503

This tells me that 0 is less than -4; and that is not true.0510

Since that is not true, this is not part of the solution set; the solution set is actually below this line--the lower half-plane.0516

OK, that was my first inequality; now, my second inequality is y ≥ x - 2.0525

The corresponding linear equation is y = x - 2.0532

Here, I have a y-intercept of -2 and a slope of 1; so increase x by 1 and y by 1 each time.0538

Now, here I am going to use a solid line; this line is part of the solution set.0551

And a test point: I can use (0,0) again--that is well away from this boundary line.0561

Substituting these values into this inequality gives me 0 ≥ -2; and that is true: 0 is greater than -2.0567

So, this upper half-plane describes the solution set.0578

Now, you can see what happened here; and you may have already noticed that these two lines have the same slope.0585

Since they have the slope, and they are parallel lines, that means they are never going to intersect.0592

Since this solution set is above this line, and this one is below the line, there are not going to be any points in common.0598

These two lines will go along forever, and the points above and below them will never intersect.0605

So, this is a situation where there is no solution; we just say it is the empty set.0611

So here, we saw a situation where there is no solution, because there are no points in common.0622

OK, the first example is very straightforward: x ≥ 2; y > 3.0628

Starting with the x ≥ 2: the corresponding linear equation would be x = 2, and that just tells me that, if x equals 2,0635

no matter what the y-value is, it is just saying x is 2; so that is going to be a vertical line.0650

And since this is greater than or equal to, I am going to make this a solid line.0657

Now, you could certainly use a test point; or you could just look at this and say,0674

"Well, it is saying that x is greater than or equal to 2, which means that it is going to be values to the right."0678

You could certainly say, "OK, I want to do a test point at (0,0); 0 is greater than or equal to 2--not true, so this is not part of the solution set."0685

Or, like I said, you could just look back here and say, "OK, it is telling me that the values of x are greater than or equal to 2."0697

So, this half-plane contains the solution set.0704

The second inequality, y > 3, has a corresponding linear equation of y = 3.0711

So, if y equals 3, that is going to be a horizontal line right here.0720

And it is a strict inequality, so I am going to make that a dashed line, making it a different color so it shows up.0725

So, this blue line is that y = 3; OK.0735

Again, I could either just go back and say, "All right, this is y > 3, so those points would be up here";0743

or I can always use my test point, (0,0), and say 0 is greater than 3--that is not true, so I know this is not part of the solution set.0750

So, I need to go up here; OK.0761

So again, the solution set for the system of inequalities is the area of intersection of the two solution sets.0774

For this first inequality, the solution set is over here.0782

For the second inequality, the solution set is up here; and I can see the area of intersection is right up here.0786

And it is to the right of this solid vertical line, and it includes the points on the line.0793

And it is above the dashed blue line, and it does not include the points on the line.0799

We are just graphing each inequality and finding this area of intersection.0805

Here it is slightly more complicated, but the same technique.0811

Again, this is a system of inequalities, starting with the first one: y > x + 1.0815

The corresponding linear equation is y = x + 1.0822

The y-intercept is 1, and the slope is 1.0828

So, increase y by 1; increase x by 1; increase y by 1; increase x by 1.0835

And this is also going to be a dashed line, since it is a strict inequality; this boundary line is not part of the solution set.0842

Take a test point, (0,0); substitute back into that inequality to give 0 is greater than 0 + 1; so that says 0 is greater than 1.0858

Well, that is, of course, not true--which means that this half-plane where the test point is, is not the solution set.0874

Instead, it is the upper half-plane; so I am going to shade that in to indicate that this is the solution set for the first inequality.0884

The second inequality is y = -x + 2; well, the inequality is y > -x + 2; the corresponding linear equation is y = -x + 2.0893

So, that gives me a y-intercept of 2 and a slope of -1.0907

Decrease y by 1; increase x by 1; decrease y by 1; increase x by 1; and on down.0914

Again, it is a strict inequality, so we have another dashed line; I am making this a different color so it stands out as a separate line.0923

This blue line is the boundary line for the solution set of the second inequality.0941

OK, again, I am using (0,0) as my test point.0949

I am substituting into that inequality: 0 is greater than -0 + 2, so 0 is greater than 2.0956

And again, this is not true; so looking at this line, this point is not part of the solution set.0966

So, this lower half-plane is not the solution set.0976

I am going to go to the upper half-plane and shade that in.0979

OK, therefore, the solution set for the system of inequalities is right up here in this corner.0996

It is the points above the black line (but not including this line) and above the blue line (but not including that boundary line).1004

The technique, again: graph the first inequality (which was right here); we found the solution set in the upper half-plane.1014

We graphed the second inequality and found its solution set in the upper half-plane.1022

And then, we noted that this area up here is the overlap between the two inequalities; and that contains the solution set for the system.1027

OK, Example 3: 2x - y > 4 is my first inequality.1041

And one thing that I see right away is that this is not in the standard form we usually use.1050

So, I am going to work with this to put it in a standard form, where y is isolated on the left; and that is going to give me -y > -2x + 4.1055

Now, I have to divide by -1; and recall that when you divide an inequality by a negative number,1067

then you have to reverse the direction of the inequality symbol.1073

So, this is going to become y < 2x - 4; and it is absolutely crucial that, if you are multiplying1079

or dividing by a negative number, you immediately reverse the inequality symbol; or you won't end up with the correct solution set.1089

OK, so that is my first inequality; I am going to go ahead and do that same thing with the second inequality before I work with either one of these.1096

So, here I have 4x + 2y ≥ -2, so I am going to subtract 4x from both sides.1107

And then, I am going to divide both sides by 2; and since 2 is a positive number, I can just keep that inequality symbol as it was.1118

This is -4 divided by 2, so that gives me -2x - 1.1125

OK, looking at this first inequality: the corresponding linear equation, now in standard slope-intercept form, is y = 2x -4.1138

So, by putting it in that form, we made it much easier to graph.1153

The y-intercept is at -4, and the slope is 2.1158

So, increase y by 2; increase x by 1 for the slope.1162

Looking back, this is a strict inequality, so I need to use a dashed line, indicating that the boundary is not part of the solution set.1169

And then, a test point--I need to determine where the solution set lies--in the upper or lower half-plane.1189

I am using the origin as my test point, substituting (0,0) in for x and y.1198

This gives me 0 - 0 > 4, or 0 is greater than 4; and that is not true.1208

Since that is not valid, that is not part of the solution set; so this is not the correct half-plane.1216

I am going to shade the lower half-plane instead.1221

OK, over here, the corresponding linear equation is y = -2x - 1.1228

So, the y-intercept is -1; the slope is -2x.1237

So, when I decrease y by 2, I am going to increase x by 1; decrease y by 2; increase x by 1.1243

Again, I have to check here, and I see that I am going to use a solid line for this boundary line,1258

because the line is part of the solution set.1264

And then, I need a test point here; and I want to pick a test point that is not quite so close to this boundary line,1278

just in case my graphing wasn't perfect, so I am going to select (1,2) right up here as my test point, just to be safe.1285

Substitute those values into this inequality right here, or back up here--either way.1296

I am going to go ahead and use this one, because it is simpler: 2 ≥ -2(2) - 1.1308

2 is greater than or equal to -4 minus 1; so, 2 is greater than or equal to -5; and that is true.1319

So, my test point is in the half-plane containing the solution set.1333

I am going to shade this upper half-plane.1340

Once I have done that, I can see that the solution set for the set of inequalities, the system of inequalities, is right here.1348

It is the area bounded by this line and including the line, and then the area bounded by the dashed line,1356

right here, but not including it; so, this lower right section of the graph.1363

For this one, we had to take an extra step, just to make it easier--put this in standard form.1372

And then, we graphed the corresponding linear equation to find the boundary line.1376

We used the test point to find that the lower half-plane was the solution set for this first inequality.1381

The same technique for the other inequality: and the upper half-plane turned out to contain the solution set.1388

So, my area of intersection is right here; that is the solution set for the system.1395

OK, Example 4: y < 2x + 1; y ≥ 2x + 3.1403

We are going ahead and starting out with the first inequality, y < 2x + 1.1412

I am finding my boundary line with the corresponding linear equation, y = 2x + 1.1424

Here, the y-intercept is 1; the slope is 2; so increasing y by 2 means increasing x by 1, and continuing on.1429

I am checking back and seeing that I need a dashed line, because this is a strict inequality.1446

So, I am graphing out this line with a dashed line; so this line will not be part of the solution set--the points on the line.1452

The second inequality is y ≥ 2x + 3; the linear equation is y = 2x + 3.1463

I am going to go ahead and graph this out; and this tells me that the y-intercept here is 3.1475

And the slope is 2: increase y by 2, increase x by 1; or decrease y by 2, decrease x by 1; and so on.1484

So, this is going to give me another line, right next to this one.1499

But this time, I am actually going to use a solid line.1504

So, this is a solid line, because this is greater than or equal to.1509

Now, let's look at some test points for each.1517

For this first one, let's use a test point of (0,0) right here for this line and substitute in.1520

0 is less than 2 times 0 plus 1; that gives me 0 < 0 + 1, or 0 is less than 1.1528

And that is true; so I have this as part of my solution set right here, so it is the lower half-plane.1537

I found the solution set for this inequality.1550

For this one, I am also going to use (0,0); that is well away from that line--that is a good test point.1552

Substitute in: 0 is greater than or equal to 2 times 0, plus 3; 0 ≥ 0 + 3; 0 is greater than or equal to 3.1560

That is not true; so, for this line, (0,0) is not part of the solution set; so it is the upper half-plane.1572

Now, it is a little tough to draw: but if you look here, one thing that you will see is that these lines have the same slope.1585

Because of that, I know that these are parallel lines, so I know that these two lines are parallel and that they will never intersect.1593

Since the points are all below this parallel line in that half-plane, and above this line (they are parallel to it), these solution sets are never going to intersect.1602

In this case, the solution is the empty set; there are no points in common, since these are parallel lines.1614

That concludes this session on solving systems of inequalities by graphing at Educator.com; see you again!1625

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