Algebraic Mathematics of Linear Inequality & System of Linear Inequality

by Jacqueline ChauEducation 014

*Algebraic MathematicsLinear Inequality & System of Linear Inequality

LINEAR INEQUALITY – LESSON OUTLINE

Mon 05/01/2023 09:55 PMJacqueline B. Chau 2

Review1. Relevant Math Terminologies & Fundamental Concepts (Fundamental Math Concepts/Terminologies, Set Theory, Union /Intersection)2. Algebraic Linear Equations of Two Variables (Linear Equation using Graph, Elimination, Substitution Methods & its Special Cases)3. Algebraic Systems of Linear Equations of Two Variables (Linear System using Graph, Elimination, Substitution Methods & its Special Cases)4. Summary of Linear Equations & Linear Systems Lesson1. Algebraic Compound & Absolute Value of Linear

Inequalities2. Algebraic Linear Inequalities with Two Variables (Linear Inequality using Graph, Elimination, Substitution Methods & its Special Cases)2. Algebraic Systems of Linear Inequalities of 2 Variables (Linear Inequality System using Graph/Elimination/Substitution Method & Special Cases)3. Algebraic Linear Inequalities & Its Applicable Examples4. Summary of Linear Inequalities & Inequality Systems Comprehensive Quiz

LINEAR INEQUALITY – LESSON OBJECTIVE


Focus on all relevant Algebraic concepts from fundamental vocabulary, to Linear Equality and Systems of Linear Equalities, which are essential to the understanding of today’s topic

Present today’s topic on Linear Inequality encompassing from Linear Inequalities of one variable and two variables, to Compound Linear Inequalities, to Absolute Value of Linear Inequalities, and to Systems of Linear Inequalities

Assess audience’s knowledge of fundamental principles of Mathematics, their ability to communicate and show clear and effective understanding of the content, their utilization of various problem solving techniques, and their proficiency in logical reasoning with a comprehensive quiz

REVIEW – TERMINOLOGY (1/4)Mathematics

VariableTerm

EquationExpression

The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

A symbol, such as a letter of the alphabet, that represents an unknown quantity.

Algebra A branch of math that uses known quantities to find unknown quantities. In algebra, letters are sometimes used in place of numbers.

A series of numbers or variables connected to one another by multiplication or division operations.A mathematical statement that shows the equality of two expressions.

A mathematical statement that shows the equality of two expressions.


REVIEW – TERMINOLOGY (2/4)

Graph

Slope

CartesianCoordinates

A visual representation of data that conveys the relationship between the Input Data and the Output Data.A two-dimensional representation of data invented by Philosopher & Mathematician Rene Descartes (also known as Cartesius), 1596-1650, with X-Axis going left-right, Y-Axis going up-down, and the Origin (0,0) at the center of the intersection of the axes, having four Quadrants (I,II,III,IV) going counter-clockwise begin at the top-right.Known as Gradient, which measures the steepness of a Linear Equation using the ratio of the Rise or Fall (Change of Y) over the Run (Change of X) .X-Intercept Interception of the Linear Equation at the X-Axis.Y-Intercept Interception of the Linear Equation at the Y-Axis. Mon 05/01/2023 09:55 PMJacqueline B.

Chau 5

Function A mathematical rule that conveys the many-to-one relationship between the Domain (Input Data) and the Range (Output Data), f(x)= x² ─> {(x,f(x)),(±1,1)}

REVIEW – TERMINOLOGY (3/4)Union

Compound Inequality

Intersection

ContinuedInequality

Short-hand form a x b for the Logical AND Compound Inequality of form a x AND x b.

A Union B is the set of all elements of that are in either A OR B. A B {a, b}.A Intersect B is the set of all elements of that are contained in both A AND B. A B {ab, ba}.Two or more inequalities connected by the mathematic logical term AND or OR. For example:a x AND x b or a x OR x b.

Absolute Value |x| represents the distance that x is from 0 on the Number Line.


System ofEquations

A system comprises of 2 or more equations that are being considered at the same time with all ordered pairs or coordinates as the common solution set .

REVIEW – TERMINOLOGY (4/4)Consistent

System

InconsistentSystem

A system of Independent Equations that shares no common solutions or no common coordinates, meaning these linear equations result with 2 non-intersecting parallel lines, and when solving by elimination, all variables are eliminated and the resulting statement is FALSE .

A system of equations that has either only one solution where the 2 lines intersect at a coordinate (x,y), or infinite number of solutions where the 2 parallel lines share a common set of coordinates, or infinite number of solutions where the 2 parallel lines are coincide.

DependentSystem

A system of equations that shares an infinite number of solutions since these equations are equivalent, therefore, resulting with 2 coincide lines. When solving by elimination, all variables are eliminated and the resulting statement is TRUE.


IndependentSystem

A system of unequivalent equations, where all variables are eliminated and the resulting statement is FALSE, has no common solutions.

REVIEW – SET THEORY

BA

BA

1.No Intersection A A2.Some Intersection A A3.Complete

Intersection A A

C


VENN DIAGRAM

Disjoint

Subset

Overlapping

BA

D

REVIEW – NUMBER THEORY (1/3)

Integer…,-99,-8, -1, 0, 1, 2, 88,

…Whole

0,1,2,3,4,5,6,…Countin

g1,2,3,4,5,…

Rational…, -77.33, -2, 0, 5.66,

25/3, …

Real


REVIEW – NUMBER THEORY (2/3)Counting

IntegerRational

Real

Irrational

Natural Numbers or Whole Numbers that are used to count for quantity; but without zero, since one cannot count with zero. { 1, 2, 3, 4, 5, … }

Whole Numbers of both Negative Numbers and Positive Numbers, excluding Fractions. { …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

Whole Natural Numbers that are used to count for quantity of something “whole” as in counting a whole table or a whole chair. { 0,1, 2, 3, 4, 5, … }

Quotients of two Integer Numbers where Divisor cannot be zero; which means they include Integer Numbers and Fractions. { -3.5, -2.9,1.6, 3.33,...}Any numbers that are not Rational Numbers (that means they are non-repeating and non-terminating Decimals like ∏ (Circumference/Diameter) is not equal to the Ratio of any two numbers), or any Square Roots of a Non-Perfect square number. { √2, √3, ∏ , 3.141592653…, √5, √6, √7, √8, √10, …}Any numbers that can be found on a Number Line, which includes Rational Numbers and Irrational Numbers. { …, -2, -1/2, 0, 0.33, 0.75, ∏ , 3.3, … } Mon 05/01/2023 09:55 PMJacqueline B.

Chau 10

REVIEW – NUMBER THEORY (3/3)Ratio

Percentage

Mixed

Complex

Imaginary

Quotients of two Rational Numbers that convey the relationship between two or more sets of things. { 1:2, 3/4, 5:1, 9/2, 23:18, … }

Part of a whole expressed in hundredth or a result of multiply a number by a percent. Percent is a part in a hundred. { 5%, 3/100, 0.003, 40%, 8/20 }

Fraction Quotients of two Integer Numbers that convey the partial of a whole. { …, -3.5, -2.9, -1.5, -0.5, 0, 0.75, 3.33, 4.2, ...}

Improper Fraction {3/2, 11/4} must be simplified into Mixed Number format {1½, 2¾}, respectively.Any numbers that are not Real Numbers and whose squares are negative Real Numbers, i = √-1{ √-1, 1i, √-∏², ∏i, √-4, 2i, √-9, 3i, √-16, 4i, ...}Any numbers that consist of a Real Number and an Imaginary Number; Complex is the only Number that cannot be ordered. A Complex Number can become a Real Number or an Imaginary Number when one of its part is zero. { 0 + 1i, 4 + 0i, 4 + ∏i} Mon 05/01/2023 09:55 PMJacqueline B.

Chau 11

ax + by = c

y = mx + by-y = m(x-x)

REVIEW – LINEAR EQUATION (1/3)

slope = m = m slope = m = -1/m

y

x

-x + y =

4 y

= x +

4

1

(0,0)

Quadrant I(positive, positive)

Quadrant II(negative, positive)

Quadrant III(negative, negative)

Quadrant IV(positive, negative)

Run = X - X

Rise

= Y

-

Y

11

Linear Equation with 2 VariablesStandard Form

Slope-Intercept Form

where {a,b,c}=Real; ([a|b]=0) ≠ [b|a]x-intercept=(c/a), y-intercept=(c/b)and slope=-(a/b)

Point-Slope Formwhere slope=m, y-intercept=b

where slope=m, point=(x , y ) 2 1

Slopes of Parallel Lines2

Slopes of Perpendicular Lines1 2

21

Standard Form-x + y = 4X-Intercept = (c/a) = -4Y-Intercept = (c/b) = 4 Slope = -(a/b) = 1

Slope-Intercept y = x + 4X-Intercept = (-4, 0) Y-Intercept = b = 4Slope = m = 1

1 1

(x , y ) 2 2

(x , y ) 1 1

slope = m = ────────

[ Rise | Fall ]Run


Standard Formax + by = c-3x + 1y = 2Note: b in the Standard Form is NOT the same b in the Slope-Intercept Form. b in the Standard Form is a coefficient/constant. b in the Slope-Intercept Form is the Y-Intercept.

Slope-Intercept Formy = mx + by = 3x + 2

Point-Slope Form(y – y ) = m(x – x )(y + 4) = 3(x + 2 ) y = 3x + 2Slope = m = 3X-Intercept = -b/m = -2/3Y-Intercept = b = 2

1 1

REVIEW – LINEAR EQUATION (2/3)y

x

-3x

+ y

= 2

y

= 3

x

+ 2

x y-10123

-125811

(0,0)

(x , y ) = (-2, -4)

1 1

Linear Equation with 2 VariablesSlope = -(a/b) = 3X-Intercept = c/a = -2/3Y-Intercept = c/b = 2

Slope = m = 3X-Intercept = -b/m = -2/3Y-Intercept = b = 2

Cautions: Study all 3 forms to understand that the notations & formulas only applied to its own format not others. Then pick one format of the equation (most common is Slope-Intercept Form) to memorize.

(x , y ) = (1, 5) 2 2

slope = m = ──── = 35 – (-4)

1 – (-2)


Solve by Graphing

REVIEW – LINEAR EQUATION (3/3)

y

x(0,0)

x =

11

Und

efine

d Sl

opex

y = x

Pos

itive

Slope

y = -x Negative

Slope

y

X-Intercept = (0, 0)

Y-Intercept = (0, 0)

Slope = m = -1

X-Inte

rcept

= (0, 0

)

Y-Inter

cept =

(0, 0

)

Slope

= m = 1

X-In

terc

ept =

(11,

0)

Y-

Inte

rcep

t = n

one

Slop

e =

m =

un

defin

ed

X-Intercept = noneY-Intercept = (0, -10)Slope = m = 0y = -10 Zero Slope

Through the Originy = mxSlope = mX-Intercep = (0, 0)Y-Intercep = (0, 0)

Special Cases of Linear Equations with Two Variables

Vertical Linex = aSlope = undefinedX-Intercep = (a, 0)Y-Intercep = none

Horizontal Liney = bSlope = 0X-Intercep = noneY-Intercep = (0, b)

Positive & Negative Slopes Zero & Undefined Slopes


REVIEW – SYSTEM OF LINEAR EQUATION (1/3)

x

3x –

2y =

6

2x + 4y = 20

yOne SolutionA B є {(4,3)}

Special Cases of System of Linear Equations with Two Variables

a x + b y = c (A)

a x + b y = c (B)

11

Case 1: Linear System of One Solution

where {a,b,c} = Real; ([a|b]=0) ≠ [b|a]

1

22 2

Positive Slope3x - 2y = 6y = (3/2)x - 3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

Solve by Graphing

Solve by Elimination 2x + 4y = 202(3x – 2y = 6) 8x = 32 x = 42(4) + 4y = 20 4y = 12 y = 3 (x, y ) = (4, 3)

A B є {(4,3)}

Solve by Substitution 2x + 4y = 20 3x – 2y = 62x + 2(3x - 6) = 20 8x – 12 = 20 8x = 32 x = 42(4) + 4y = 20 4y = 12 y = 3 (x, y ) = (4, 3)

Negative-Slope & Positive-Slope Equations

(4, 3)


Negative Slope2x + 4y = 20y = (-1/2)x + 5Slope = -(1/2)X-Intercep = (10, 0)Y-Intercep = (0, 5)

Special Cases of System of Parallel Linear-Equations with Two Variables

x

3x –

2y =

6

y

a x + b y = c (A)

a x + b y = c (B)

11

Case 2: Linear System of No Solution

1

22 2

Parallel Equation23x - 2y = 6y = (3/2)x - 3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

Parallel Equation13x - 2y = -2y = (3/2)x + 1Slope = (3/2)X-Intercep = (-2/3, 0)Y-Intercep = (0, 1)

Solve by Graphing

Solve by Elimination 3x - 2y = -2 3x – 2y = 6 0 = 4 False Statement! No Solution in common A B є {} System is Inconsistent That means the solution set is

empty

3x –

2y =

-2

No IntersectionNo SolutionA B є {

Parallel Linear Equations

where a / a = b / b = 1 and c / c = {Integers}1

1 122 2



Special Cases of System of Equivalent Linear-Equations with Two Variables

x

3x –

2y =

6

y

a x + b y = c (A)a x + b y = c (B)

11

Case 3:Linear System of Infinite Solution

where a / a = b / b = c / c = 1

1

22 2

Dependent Equation23x - 2y = 6y = (3/2)x - 3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

Dependent Equation16x - 4y = 12y = (3/2)x -3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

Solve by Graphing

Solve by Elimination 6x - 4y = 122(3x – 2y = 6) 0 = 0 True! Infinite Set of All Solutions in common A B є {(2,0),(0,-3) ,…} Whenever you have a system of

Dependent Equations, both variables are eliminated and the resulting statement is True; solution set contains all ordered pairs that satisfy both equations.

Lines CoincideInfinite SolutionA B є {(2,0),(0,-3), …}

11 1 22 2

Dependent Equations

6x –

4y =

12



SUMMARY – LINEAR EQUATION & SYSTEMLinear Equations with 2 Variables

The 3 Forms of Linear Equations:1. Standard Form ax + by = c2. Slope-Intercept Form y = mx + b3. Point-Slope Form y-y = m(x-x )

The 4 Special Cases of Linear Equations:

4. Positive Slope m = n y = mx + b

5. Negative Slope m = -n y = -mx + b

6. Zero Slope m = 0 y = n

7. Undefined Slope m = ∞ x = n

The 3 Ways to solve Linear Systems:a x + b y = c (A)a x + b y = c (B)1. Solve by Graphing2. Solve by Elimination3. Solve by Substitution

The 3 Special Cases of Linear System:

4. Linear System of One Solution A={(x,y)}

2. Linear System of No Solution A={}

3. Linear System of Infinite Solution A={}

Linear Systems with 2 Variables

11Cautions: The coefficient b in the Standard Form is not the same b in the Slope-Intercept Form.

11 122 2


LINEAR INEQUALITY WITH TWO VARIABLES

ax + by ≤ cy ≤ mx + by-y ≤ m(x-x) slope = m = m

slope = m = -1/m 1

11

Linear InequalitiesStandard Form

Slope-Intercept Form

Point-Slope Formwhere slope=m, y-intercept=b

where slope=m, point=(x , y )Slopes of Parallel Lines

2Slopes of Perpendicular Lines

1 2

where {a,b,c}=Real; ([a|b]=0) ≠ [b|a]

y

x

y ≤

3x +

2

(0,0)

x y-10123

-125811

Solve by Graphing

1 2


COMPOUND OF LINEAR INEQUALITY

Case 1: Linear Inequality of Finite Solution

Case 2: Linear Inequality of No Solution

y

x

(0,0)

Solve by Graphing

𝒙≤𝟒

𝒙≤−𝟑

x

Linear Inequalities of One Variable


Special Cases of Compound Linear-Inequalities with One Variables

ABSOLUTE VALUE OF LINEAR INEQUALITY (1/4)

Case 1: Linear Inequality of One Solution

∴ {0}

∴ {}Case 2: Linear Inequality of Finite Solution

∴ {-3,-2,-1,0,1,2,3,4}

Case 3: Linear Inequality of No Solution

∴ {}

y

x

(0,0)

Solve by Graphing

𝒙≤𝟒

𝒙≤−𝟒

x

Linear Inequalities with One Variable


x𝒙≤𝟎

y

x

y 3

x - 2

(0,0)

x y-10123

-5-2147

Solve by GraphingLinear Inequalities with Two VariablesCase 1: Linear Inequality of Infinite Solution ( (A) AND (B) ( (A)

( (B) (A) (B) ∴ = {(0,0),(-2,-4),(-1,-1), (0,2),(1,5),(2,8),(3,11), (-1,-5),(0,-2),(1,1),(2,4), (3,7),}

y ≤

3x +

2



x y-10123

-125811

Linear Inequalities with Two VariablesCase 2: Linear Inequality of No Solution

( (A) AND (B) ( (A)

( (B)

(A) (B) ∴ {}


x

ySolve by Graphing

y ≤

3x -

2

x y-10123

-5-2147y

3x

+ 2

x y-10123

-125811


Linear Inequalities with Two VariablesCase 3: Linear Inequality of Infinite Solution

( (A) AND (B) ( (A)

( (B)

(A) (B) ∴ {,(1,-1),(2,-2),(3,-3), (1,1),(2,2),(3,3),(4,4),(5,5), (10,1),(10,-1),(3,-1),,} Mon 05/01/2023 09:56 PMJacqueline B.

Chau 24


x

ySolve by Graphing

y ≤ x

x y-10123

10-1-2-3

x y-10123

-10123

y x

A B є{(0,0),(4,3),(0,5),(2,0),(0,-3), …}

3x –

2y ≤

6

y

a x + b y ≤ c (A)

a x + b y ≤ c (B)

11

Case 1: Inequality System of One Solution


1

22 2

Solve by Graphing

Solve by Elimination 2x + 4y ≤ 202(3x – 2y ≤ 6) 8x ≤ 32 x ≤ 42(4) + 4y ≤ 20 4y ≤ 12 y ≤ 3 (x, y ) ≤ (4, 3) A B є

{(0,0),(4, 3), …}

Solve by Substitution 2x + 4y ≤ 20 3x – 2y ≤ 62x + 2(3x - 6) ≤ 20 8x – 12 ≤ 20 8x ≤ 32 x ≤ 42(4) + 4y ≤ 20 4y ≤ 12 y ≤ 3 (x, y ) ≤ (4, 3)

Negative-Slope & Positive-Slope Equations

SYSTEM OF LINEAR INEQUALITY (1/4)

x

2x + 4y ≤ 20

Positive Slope3x - 2y = 6y = (3/2)x - 3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

Negative Slope2x + 4y = 20y = (-1/2)x + 5Slope = -(1/2)X-Intercep = (10, 0)Y-Intercep = (0, 5)

(4, 3)

Special Cases of System of Linear Inequalities with Two Variables


3x –

2y ≤

6

y

a x + b y ≥ c (A)

a x + b y ≤ c (B)

11

Case 2: Inequality System of Infinite Solution


1

22 2

Solve by Graphing

Solve by Elimination 3x - 2y ≥ -2 3x – 2y ≤ 6 0 ≤ 4 True!AB {(0,0),(1,1),(2,2),(3,3),(4,4),…}



x

3x –

2y ≥

-2


Parallel Equation 13x - 2y ≥ -2y ≤ (3/2)x + 1Slope = (a₁/b₁) = (3/2)X-Intercep = (-2/3, 0)Y-Intercep = (0, 1)

Parallel Equation 23x - 2y ≤ 6

y ≥ (3/2)x - 3Slope = (a₂/b₂) = (3/2)

X-Intercep = (2, 0)Y-Intercep = (0, -3)

and m = m a / a = b / b = 1 and c / c = {Integers}1

1 12

2 2


1 2

3x –

2y ≤

6

y

a x + b y ≤ c (A)

a x + b y ≥ c (B)

11

Case 3: Inequality System of No Solution


1

22 2

Solve by Graphing

Solve by Elimination 3x - 2y ≤ -2 3x – 2y ≥ 6 0 ≥ 4 False! System is Inconsistent! Solution Set is Empty.A B {}



x

3x –

2y ≤

-2


Parallel Equation 13x - 2y ≤ -2y ≥ (3/2)x + 1Slope = (3/2)X-Intercep = (-2/3, 0)Y-Intercep = (0, 1)

Parallel Equation 23x - 2y ≥ 6

y ≤ (3/2)x - 3Slope = (3/2)


and a / a = b / b = 1 and c / c = {Integers}1

1 122 2


y

a x + b y ≤ c (A)

a x + b y ≤ c (B)

11

Case 4: Inequality System of Infinite Solution


1

22 2

Solve by Graphing

Solve by Elimination 6x - 4y ≤ 122(3x – 2y ≤ 6) 0 ≤ 0 True!AB {(2,0),(0,-3),(6,6),(-4,-9),…}

Coincide Linear Equations


x

Dependent Equation 16x - 4y ≤ 12y ≥ (3/2)x - 3Slope = (3/2)X-Intercep = (2, 0)Y-Intercep = (0, -3)

and a / a = b / b = c / c = 111 1 22 2



3x –

2y ≤

6

6x –

4y ≤

12

Dependent Equation 23x - 2y ≥ 6

y ≤ (3/2)x - 3Slope = (3/2)


SUMMARY– LINEAR INEQUALITY & SYSTEMLinear Inequalities with 2 Variables

The 3 Forms of Linear Inequalities:1. Standard Form of Infinite Solution ax + by ≤ c2. Absolute-Value Form of Infinite Solution |ax + by| ≤ c3. Absolute-Value Form of No Solution |ax + by| ≥cThe 4 Special Cases of Linear Inequalities:

4. Linear Inequality of Infinite Solution with

2 intersected lines (Note: when a or b is 0, A)

5. Linear Inequality of Infinite Solution (Note:

when a or b is 0, Absolute-Value Form becomes a 1-

variable Linear Inequality with a Finite Solution)

6. Linear Inequality of No Solution

The 3 Ways to solve Inequality Systems:a x + b y ≤ c (A)a x + b y ≤ c (B)1. Solve by Graphing2. Solve by Elimination3. Solve by Substitution

The 4 Special Cases of Inequality Systems:4. Inequality System of One Solution with 2

intersected lines A={(x,y)}2. Inequality System of Infinite Solution with 2

parallel-intersected lines OR with 2 coincide lines

A ={(x,y), …, }3. Inequality System of No Solution with 2 parallel lines of no intersections A={}

Linear-Inequality Systems with 2 Variables

11 122 2



SUMMARY– ABSOLUTE VALUE OF LINEAR INEQUALITYAbsolute-Value Inequalities of 1 Variable

The 2 Forms of Linear Inequalities:1. Absolute-Value Form of Infinite Solution |ax + by| ≤ c , where [a|b] = 0

2. Absolute-Value Form of No Solution |ax + by| ≥ c , where [a|b] = 0

The 4 Special Cases of Linear Inequalities:3. Linear Inequality of One Solution with 2

intersecting lines A={(x,y)}2. Linear Inequality of Finite Solution with 2

overlapping lines A={(x,y), …,(x,y)}3. Linear Inequality of Infinite Solution with 2

parallel-intersecting lines A={(x,y), …, ∞}4. Linear Inequality of No Solution with 2

parallel non-intersecting lines A={}

The 2 Absolute-Value Forms of Inequalities:1. Absolute-Value Form of Infinite Solution

|ax + by| ≤ c2. Absolute-Value Form of No Solution

|ax + by| ≥ c3. Solve by

The 4 Special Cases of Inequality Systems:4. Inequality System of Infinite Solution with 2

intersected lines A={(x,y), …, }2. Inequality System of Infinite Solution with 2

parallel-intersecting lines A ={(x,y), …, }3. Inequality System of No Solution with 2 parallel non-intersecting lines A={}

Absolute-Value Inequalities of 2 Variables

APPLICABLE EXAMPLE*

*

*

*

* Age, Office Number, Cell Phone Number, …

Travel Distance, Rate, Arrival/Departure Time, Gas Mileage, Length of Spring,…

Hourly Wage, Phone Bill, Ticket Price, Profit, Revenue, Cost, …

Circumference & Radius, Volume & Pressure, Volume & Temperature, Weight & Surface Area

Temperature, Gravity, Wave Length, Surface Area of Cylinder, Mixture of Two Elements, …


APPLICABLE EXAMPLE


QUIZ (1/2)1. What is the visual representation of a Linear Equation?2. What is the visual representation of a Linear Inequality?3. What is the 2-dimensional coordinates that represents data visually?4. Who was the creator of this system?5. What is the Standard Form of a Linear Equation?6. What is the Slope-Intercept Form of a Linear Equation?7. Is the Coefficient “b” in these 2 equation formats the same? 8. List all 4 Special Cases in Linear Equations?9. Which form of the Linear Equations is your favorite?10.What is its slope? List the 4 different types of slopes.11.What is X-Intercept? What is Y-Intercept? 12.How do you represent these interceptions in ordered pairs?13.What is the slope of the line perpendicular to another line?14.What is an Equivalent Equation?15.What is a System of Linear Equations?

A straight lineA set or subset of numbers

Cartesian CoordinatesPhilosopher & Mathematician Rene Descartes

ax + by = c where a, b ≠ 0 at oncey = mx + b

NoPositive, Negative, 0, Undefined

Audiences’ FeedbackSteepness = Rise/Run; +, -, 0, ∞

The crossing at the x-axis or y-axis(x, 0) or (0, y)

Negative Reciprocal-

2+ equations considered @ same time with solution set satisfy all equations


QUIZ (2/3)16.True or False - Absolute Value |x|=-5? 17.List 3 different approaches to solve Linear Systems? 18.List all 3 Special Cases of Linear Systems.19.Which Linear System is the result of the 1 Ordered-Pair Solution?20.What is a Dependent Equation?21.What solution you get for a System of Dependent Equations? 22.Graphs of Linear-Dependent System are Coincide Lines.23.What is Linear Inequality?24.How difference do you find Linear Equation versus Inequalities?25.Which number set(s) would most likely be the solution of Linear Inequality?26.Which number set is so negatively impossible? (Hint: this is a trick question.)27.So now, what is the definition of Irrational Numbers?28.Graphs of Linear-Dependent Inequalities are Coincide Lines.29.Solutions for Special Cases of both Linear Equality and Inequality are Union

Sets.30.Is Linear Inequality more useful than Linear Equality in solving problems?

FalseGraph, Elimination, Substitution

One Ordered-Pair, No Solution, Infinite+ & - Slope Equations

All variables are eliminated & statement is TrueInfinite Set of common Sol

True or False-

Audiences’ FeedbackReals

Irrationals----


QUIZ (3/3)31.After this presentation, what do you think of Linear Inequality as a tool to solve

everyday problem?32.Having asked the above question, would you think learning Algebra is

beneficial due to its practicality in real life as this lesson of Linear Inequality has just proven itself?

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Algebraic Mathematics of Linear Inequality & System of Linear Inequality

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Transcript of Algebraic Mathematics of Linear Inequality & System of Linear Inequality