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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
Mon 05/01/2023 09:55 PMJacqueline B. Chau 3
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.
Mon 05/01/2023 09:55 PMJacqueline B. Chau 4
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.
Mon 05/01/2023 09:55 PMJacqueline B. Chau 6
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.
Mon 05/01/2023 09:55 PMJacqueline B. Chau 7
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
Mon 05/01/2023 09:55 PMJacqueline B. Chau 8
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
Mon 05/01/2023 09:55 PMJacqueline B. Chau 9
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
Mon 05/01/2023 09:55 PMJacqueline B. Chau 12
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)
Mon 05/01/2023 09:55 PMJacqueline B. Chau 13
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
Mon 05/01/2023 09:55 PMJacqueline B. Chau 14
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)
Mon 05/01/2023 09:55 PMJacqueline B. Chau 15
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 16
REVIEW – SYSTEM OF LINEAR EQUATION (2/3)
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 17
REVIEW – SYSTEM OF LINEAR EQUATION (3/3)
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 18
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 19
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 20
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 21
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 22
ABSOLUTE VALUE OF LINEAR INEQUALITY (2/4)
x y-10123
-125811
Linear Inequalities with Two VariablesCase 2: Linear Inequality of No Solution
( (A) AND (B) ( (A)
( (B)
(A) (B) ∴ {}
Mon 05/01/2023 09:56 PMJacqueline B. Chau 23
x
ySolve by Graphing
y ≤
3x -
2
x y-10123
-5-2147y
3x
+ 2
x y-10123
-125811
ABSOLUTE VALUE OF LINEAR INEQUALITY (3/4)
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
ABSOLUTE VALUE OF LINEAR INEQUALITY (4/4)
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
where {a,b,c} = Real; ([a|b]=0) ≠ [b|a]
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 25
3x –
2y ≤
6
y
a x + b y ≥ c (A)
a x + b y ≤ c (B)
11
Case 2: Inequality System of Infinite Solution
where {a,b,c} = Real; ([a|b]=0) ≠ [b|a]
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),…}
Parallel Linear Equations
SYSTEM OF LINEAR INEQUALITY (2/4)
x
3x –
2y ≥
-2
Special Cases of System of Linear Inequalities with Two Variables
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 26
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
where {a,b,c} = Real; ([a|b]=0) ≠ [b|a]
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 {}
Parallel Linear Equations
SYSTEM OF LINEAR INEQUALITY (3/4)
x
3x –
2y ≤
-2
Special Cases of System of Linear Inequalities with Two Variables
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)
X-Intercep = (2, 0)Y-Intercep = (0, -3)
and a / a = b / b = 1 and c / c = {Integers}1
1 122 2
Mon 05/01/2023 09:56 PMJacqueline B. Chau 27
y
a x + b y ≤ c (A)
a x + b y ≤ c (B)
11
Case 4: Inequality System of Infinite Solution
where {a,b,c} = Real; ([a|b]=0) ≠ [b|a]
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
SYSTEM OF LINEAR INEQUALITY (4/4)
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 28
Special Cases of System of Linear Inequalities with Two Variables
3x –
2y ≤
6
6x –
4y ≤
12
Dependent Equation 23x - 2y ≥ 6
y ≤ (3/2)x - 3Slope = (3/2)
X-Intercep = (2, 0)Y-Intercep = (0, -3)
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 29
Mon 05/01/2023 09:56 PMJacqueline B. Chau 30
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*
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* 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, …
Mon 05/01/2023 09:56 PMJacqueline B. Chau 31
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
Mon 05/01/2023 09:56 PMJacqueline B. Chau 34
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----
Mon 05/01/2023 09:56 PMJacqueline B. Chau 35
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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Mon 05/01/2023 09:56 PMJacqueline B. Chau 36