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by Jacqueline Chau
Education 014
* Algebraic Mathematics
Linear Inequality
& System of Linear Inequality
Mon 1/3/2017 8:10 AMJacqueline B. Chau 2
 Review
1. 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
 Lesson
1. Algebraic Compound & Absolute Value of Linear Inequalities
2. 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 Examples
4. Summary of Linear Inequalities & Inequality Systems
 Comprehensive Quiz
Mon 1/3/2017 8:10 AMJacqueline 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
Mathematics
Variable
Term
Equation
Expression
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 1/3/2017 8:10 AMJacqueline B. Chau 4

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Solving linear inequalities
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A linear inequality is similar to a linear equation but uses inequality symbols like < or > instead of =. A solution to a linear inequality is any coordinate pair that makes the inequality true. A linear inequality describes a half-plane region on a coordinate plane where all points in the region satisfy the inequality, with the boundary line given by the related equation. To graph a linear inequality, you solve it for y, graph the boundary line as solid or dotted, and shade the correct half-plane above or below the line.

Quadratic Equation and discriminant
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This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.

Graph
Slope
Cartesian
Coordinates
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 1/3/2017 8:10 AMJacqueline 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)}
Union
Compound
Inequality
Intersection
Continued
Inequality
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 1/3/2017 8:10 AMJacqueline B. Chau 6
System of
Equations
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 .
Consistent
System
Inconsistent
System
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.
Dependent
System
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 1/3/2017 8:10 AMJacqueline B. Chau 7
Independent
System
A system of unequivalent equations, where all
variables are eliminated and the resulting
statement is FALSE, has no common solutions.
BA
BA
1. No Intersection
A ∪ 𝑩 = {∅}
A ∩ 𝑩 = {∅}
2. Some Intersection
A ∪ 𝑩 = {𝑨 − 𝑪, 𝑩 − 𝑪}
A ∩ 𝑩 = 𝑪
3. Complete Intersection
A ∪ 𝑩 = {𝑨, 𝑩} = 𝑫
A ∩ 𝑩 = {𝑨, 𝑩} = 𝑫
C
Mon 1/3/2017 8:10 AMJacqueline B. Chau 8
VENN DIAGRAM
Disjoint
Subset
Overlapping
B
A
D

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1) An inequality is a mathematical statement that uses inequality symbols like <, ≤, >, ≥ to show the relationship between two quantities. 2) When graphing inequalities on a number line, closed circles are used for ≤ and ≥, and open circles are used for < and >. 3) Linear inequalities in two variables can be written as Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C. An ordered pair (x,y) is a solution if it makes the inequality true.

Integer
…,-99,-8, -1, 0, 1, 2, 88, …
Whole
0,1,2,3,4,5,6,…
Counting
1,2,3,4,5,…
Rational
…, -77.33, -2, 0, 5.66, 25/3, …
Real
Mon 1/3/2017 8:10 AMJacqueline B. Chau 9
Counting
Integer
Rational
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 1/3/2017 8:10 AMJacqueline B. Chau 10
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 1/3/2017 8:10 AMJacqueline B. Chau 11
ax + by = c
y = mx + b
y-y = m(x-x)
slope = m= m
slope = m = -1/m
y
x
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 Variables
Standard 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 Form
where slope=m, y-intercept=b
where slope=m, point=(x , y ) 2 1
Slopes of Parallel Lines
2
Slopes of Perpendicular Lines
1 2
21
Standard Form
-x + y = 4
X-Intercept = (c/a) = -4
Y-Intercept = (c/b) = 4
Slope = -(a/b) = 1
Slope-Intercept
y = x + 4
X-Intercept = (-4, 0)
Y-Intercept = b = 4
Slope = m = 1
1 1
(x , y )2 2
(x , y )1 1
slope = m = ────────
[ Rise | Fall ]
Run
Mon 1/3/2017 8:10 AMJacqueline B. Chau 12

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Standard Form
ax + by = c
-3x + 1y = 2
Note: 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 Form
y = mx + b
y = 3x + 2
Point-Slope Form
(y – y ) = m(x – x )
(y + 4) = 3(x + 2 )
y = 3x + 2
Slope = m = 3
X-Intercept = -b/m = -2/3
Y-Intercept = b = 2
1 1
y
x
x y
-1
0
1
2
3
-1
2
5
8
11
(0,0)
(x , y ) = (-2, -4)1 1
Linear Equation with 2 Variables
Slope = -(a/b) = 3
X-Intercept = c/a = -2/3
Y-Intercept = c/b = 2
Slope = m = 3
X-Intercept = -b/m = -2/3
Y-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
= ──── = 3
5 – (-4)
1 – (-2)
Mon 1/3/2017 8:10 AMJacqueline B. Chau 13
Solve by Graphing
y
x(0,0)
x=11UndefinedSlope
x
y
X-Intercept=(11,0)
Y-Intercept=none
Slope=m=undefined
X-Intercept = none
Y-Intercept = (0, -10)
Slope = m = 0
y = -10  Zero Slope
Through the Origin
y = mx
Slope = m
X-Intercep = (0, 0)
Y-Intercep = (0, 0)
Special Cases of Linear Equations with Two Variables
Vertical Line
x = a
Slope = undefined
X-Intercep = (a, 0)
Y-Intercep = none
Horizontal Line
y = b
Slope = 0
X-Intercep = none
Y-Intercep = (0, b)
Positive & Negative Slopes Zero & Undefined Slopes
Mon 1/3/2017 8:10 AMJacqueline B. Chau 14
x
y
One Solution
A ∩ 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 Slope
3x - 2y = 6
y = (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
Solve by Graphing
Solve by Elimination
2x + 4y = 20
2(3x – 2y = 6)
8x = 32
x = 4
2(4) + 4y = 20
4y = 12
y = 3
 (x, y ) = (4, 3)
∴ A ∩ B є {(4,3)}
Solve by Substitution
2x + 4y = 20
3x – 2y = 6
2x + 2(3x - 6) = 20
8x – 12 = 20
8x = 32
x = 4
2(4) + 4y = 20
4y = 12
y = 3
 (x, y ) = (4, 3)
Negative-Slope & Positive-Slope Equations
(4, 3)
Mon 1/3/2017 8:10 AMJacqueline B. Chau 15
Negative Slope
2x + 4y = 20
y = (-1/2)x + 5
Slope = -(1/2)
X-Intercep = (10, 0)
Y-Intercep = (0, 5)
Special Cases of System of Parallel Linear-Equations with Two Variables
x
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 Equation2
3x - 2y = 6
y = (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
Parallel Equation1
3x - 2y = -2
y = (3/2)x + 1
Slope = (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 ∅.
No Intersection
No Solution
A ∩ B є {∅}
Parallel Linear Equations
where a / a = b / b = 1
and c / c = {Integers}1
1 1
2
2 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 16

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Special Cases of System of Equivalent Linear-Equations with Two Variables
x
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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 Equation2
3x - 2y = 6
y = (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
Dependent Equation1
6x - 4y = 12
y = (3/2)x -3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
Solve by Graphing
Solve by Elimination
6x - 4y = 12
2(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 Coincide
Infinite Solution
A ∩ B є {(2,0),(0,-3), …}
11 1 22 2
Dependent Equations
Mon 1/3/2017 8:10 AMJacqueline B. Chau 17
Linear Equations with 2 Variables
The 3 Forms of Linear Equations:
1. Standard Form
ax + by = c
2. Slope-Intercept Form
y = mx + b
3. Point-Slope Form
y-y = m(x-x )
The 4 Special Cases of Linear Equations:
1. Positive Slope m = n  y = mx + b
2. Negative Slope m = -n  y = -mx + b
3. Zero Slope m = 0  y = n
4. 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 Graphing
2. Solve by Elimination
3. Solve by Substitution
The 3 Special Cases of Linear System:
1. Linear System of One Solution
A∩ 𝐁={(x,y)}
2. Linear System of No Solution
A∩ 𝐁={∅}
3. Linear System of Infinite Solution
A∩ 𝐁={∞}
Linear Systems with 2 Variables
11
Cautions: The coefficient b in the Standard Form is
not the same b in the Slope-Intercept Form.
11 1
22 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 18
ax + by ≤ c
y ≤ mx + b
y-y ≤ m(x-x)
slope = m= m
slope = m = -1/m
1
11
Linear Inequalities
Standard Form
Slope-Intercept Form
Point-Slope Form
where slope=m, y-intercept=b
where slope=m, point=(x , y )
Slopes of Parallel Lines
2
Slopes of Perpendicular Lines
1 2
where {a,b,c}=Real; ([a|b]=0) ≠ [b|a]
y
x(0,0)
x y
-1
0
1
2
3
-1
2
5
8
11
Solve by Graphing
1 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 19
Case 1: Linear Inequality of Finite Solution
𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩
−𝟑 ≤ 𝒙 ≤ 𝟒
𝒙 ≥ −𝟑 𝑨𝑵𝑫 𝒙 ≤ 𝟒
Case 2: Linear Inequality of No Solution
𝐀 ∪ 𝑩 = {𝑨 𝑶𝑹 𝑩}
−𝟑 ≥ 𝒙 ≥ 𝟒
𝒙 ≤ −𝟑 𝑶𝑹 𝒙 ≥ 𝟒
y
x
(0,0)
−𝟑 ≤ 𝒙 ≤ 𝟒
Solve by Graphing
𝒙 ≤ 𝟒
𝒙 ≥ −𝟑
𝒙 ≤ −𝟑
𝒙 ≥ 𝟒
−𝟑 ≥ 𝒙 ≥ 𝟒 x
Linear Inequalities of One Variable
Mon 1/3/2017 8:10 AMJacqueline B. Chau 20
Special Cases of Compound Linear-Inequalities with One Variables

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This document provides steps for solving radical equations: 1) Isolate the radical on one side of the equation by performing inverse operations 2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical 3) Solve the remaining polynomial equation It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.

Case 1: Linear Inequality of One Solution
|𝒙| ≤ 𝟎
𝒙 ≥ 𝟎 𝑨𝑵𝑫 𝒙 ≤ 𝟎
∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {0}
𝒙 ≥ 𝟎
𝒙 ≤ 𝟎 𝑶𝑹 𝒙 ≥ 𝟎
∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {0}
Case 2: Linear Inequality of Finite Solution
𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩
𝒙 ≤ 𝟒
𝒙 ≥ −𝟒 𝑨𝑵𝑫 𝒙 ≤ 𝟒
∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩
= {-3,-2,-1,0,1,2,3,4}
Case 3: Linear Inequality of No Solution
𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩
|𝒙| ≥ 𝟒
𝒙 ≤ −𝟒 𝑶𝑹 𝒙 ≥ 𝟒
∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {∅}
y
x
(0,0)
−𝟒 ≤ 𝒙 ≤ 𝟒
Solve by Graphing
𝒙 ≤ 𝟒
𝒙 ≥ −𝟒
𝒙 ≤ −𝟒
𝒙 ≥ 𝟒
−𝟒 ≥ 𝒙 ≥ 𝟒 x
Linear Inequalities with One Variable
Mon 1/3/2017 8:10 AMJacqueline B. Chau 21
x𝒙 ≤ 𝟎 𝒙 ≥ 𝟎0 ≤ 𝒙 ≤ 𝟎
y
x
(0,0)
x y
-1
0
1
2
3
-5
-2
1
4
7
Solve by GraphingLinear Inequalities with Two Variables
Case 1: Linear Inequality of Infinite Solution
−𝟑𝒙 + 𝒚 ≤ 𝟐
 ± (−𝟑𝒙 + 𝒚) ≤ 𝟐 (A) AND (B)
 − (−𝟑𝒙 + 𝒚) ≤ 𝟐 (A)
𝟑𝒙 − 𝒚 ≤ 𝟐
𝒚 ≥ 𝟑𝒙 − 𝟐
 + (−𝟑𝒙 + 𝒚) ≤ 𝟐 (B)
−𝟑𝒙 + 𝒚 ≤ 𝟐
𝒚 ≤ 𝟑𝒙 + 𝟐
 𝒚 ≥ 𝟑𝒙 − 𝟐 (A)
𝑨𝑵𝑫 𝒚 ≤ 𝟑𝒙 + 𝟐 (B)
 ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 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),… , ∞}
Mon 1/3/2017 8:10 AMJacqueline B. Chau 22
x y
-1
0
1
2
3
-1
2
5
8
11
Linear Inequalities with Two Variables
Case 2: Linear Inequality of No Solution
−𝟑𝒙 + 𝒚 ≥ 𝟐
 ± (−𝟑𝒙 + 𝒚) ≥ 𝟐 (A) AND (B)
 − (−𝟑𝒙 + 𝒚) ≥ 𝟐 (A)
𝟑𝒙 − 𝒚 ≥ 𝟐
𝒚 ≤ 𝟑𝒙 − 𝟐
 + (−𝟑𝒙 + 𝒚) ≥ 𝟐 (B)
−𝟑𝒙 + 𝒚 ≥ 𝟐
𝒚 ≥ 𝟑𝒙 + 𝟐
 𝒚 ≤ 𝟑𝒙 − 𝟐 (A)
𝑨𝑵𝑫 𝒚 ≥ 𝟑𝒙 + 𝟐 (B)
 ∴ 𝐀 ∩ 𝑩 = {𝑨 𝑨𝑵𝑫 𝑩}
= {∅}
Mon 1/3/2017 8:10 AMJacqueline B. Chau 23
x
ySolve by Graphing
x y
-1
0
1
2
3
-5
-2
1
4
7
x y
-1
0
1
2
3
-1
2
5
8
11
Linear Inequalities with Two Variables
Case 3: Linear Inequality of Infinite Solution
𝒙 + 𝒚 ≥ 𝟎
 ± (𝒙 + 𝒚) ≥ 𝟎 (A) AND (B)
 − (𝒙 + 𝒚) ≥ 𝟎 (A)
− 𝒙 − 𝒚 ≥ 𝟎
𝒚 ≤ −𝒙
 + (𝒙 + 𝒚) ≥ 𝟎 (B)
𝒙 + 𝒚 ≥ 𝟎
𝒚 ≥ −𝒙
 𝒚 ≤ −𝒙 (A)
𝑨𝑵𝑫 𝒚 ≥ −𝒙 (B)
 ∴ 𝐀 ∩ 𝑩 = {𝑨 𝑨𝑵𝑫 𝑩}
= {(0,0),(1,-1),(2,-2),(3,-3),
(1,1),(2,2),(3,3),(4,4),(5,5),
(10,1),(10,-1),(3,-1),,… , ∞}
Mon 1/3/2017 8:10 AMJacqueline B. Chau 24
x
ySolve by Graphing
x y
-1
0
1
2
3
1
0
-1
-2
-3
x y
-1
0
1
2
3
-1
0
1
2
3

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DONE BY, SLAYERIX 11 B K.V.PATTOM TRIVANDRUM

A ∩ B є
{(0,0),(4,3),(0,5),(2,0),(0,-3), …}
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 ≤ 20
2(3x – 2y ≤ 6)
8x ≤ 32
x ≤ 4
2(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 ≤ 6
2x + 2(3x - 6) ≤ 20
8x – 12 ≤ 20
8x ≤ 32
x ≤ 4
2(4) + 4y ≤ 20
4y ≤ 12
y ≤ 3
 (x, y ) ≤ (4, 3)
Negative-Slope & Positive-Slope Equations
x
Positive Slope
3x - 2y = 6
y = (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
Negative Slope
2x + 4y = 20
y = (-1/2)x + 5
Slope = -(1/2)
X-Intercep = (10, 0)
Y-Intercep = (0, 5)
(4, 3)
Special Cases of System of Linear Inequalities with Two Variables
Mon 1/3/2017 8:10 AMJacqueline B. Chau 25
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!
∴ A∩B ∈ {(0,0),(1,1),(2,2),(3,3),(4,4),…}
Parallel Linear Equations
x
Special Cases of System of Linear Inequalities with Two Variables
Parallel Equation 1
3x - 2y ≥ -2
y ≤ (3/2)x + 1
Slope = (a₁/b₁) = (3/2)
X-Intercep = (-2/3, 0)
Y-Intercep = (0, 1)
Parallel Equation 2
3x - 2y ≤ 6
y ≥ (3/2)x - 3
Slope = (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 1
2
2 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 26
1 2
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 є {⏀}
Parallel Linear Equations
x
Special Cases of System of Linear Inequalities with Two Variables
Parallel Equation 1
3x - 2y ≤ -2
y ≥ (3/2)x + 1
Slope = (3/2)
X-Intercep = (-2/3, 0)
Y-Intercep = (0, 1)
Parallel Equation 2
3x - 2y ≥ 6
y ≤ (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
and a / a = b / b = 1
and c / c = {Integers}1
1 1
2
2 2
Mon 1/3/2017 8:10 AMJacqueline 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 ≤ 12
2(3x – 2y ≤ 6)
0 ≤ 0
True!
∴ A∩B є {(2,0),(0,-3),(6,6),(-4,-9),…}
Coincide Linear Equations
x
Dependent Equation 1
6x - 4y ≤ 12
y ≥ (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)
and a / a = b / b = c / c = 111 1 22 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 28
Special Cases of System of Linear Inequalities with Two Variables
Dependent Equation 2
3x - 2y ≥ 6
y ≤ (3/2)x - 3
Slope = (3/2)
X-Intercep = (2, 0)
Y-Intercep = (0, -3)

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The document discusses linear equations and inequalities in one variable. It defines linear equations and inequalities, and describes methods for solving them including: guess-and-check, cover-up, and working backwards. It also covers properties of equality and inequality, and provides examples of solving linear equations and inequalities using these properties and graphical representations of solution sets on number lines.

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Linear Inequalities with 2 Variables
The 3 Forms of Linear Inequalities:
1. Standard Form of Infinite Solution
ax + by ≤ c
2. Absolute-Value Form of Infinite Solution
|ax + by| ≤ c
3. Absolute-Value Form of No Solution
|ax + by| ≥c
The 4 Special Cases of Linear Inequalities:
1. Linear Inequality of Infinite Solution with
2 intersected lines (Note: when a or b is 0, A)
2. 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)
3. 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 Graphing
2. Solve by Elimination
3. Solve by Substitution
The 4 Special Cases of Inequality Systems:
1. Inequality System of One Solution with 2
intersected lines
A∩ 𝐁={(x,y)}
2. Inequality System of Infinite Solution with
2 parallel-intersected lines OR with 2
coincide lines
A∩ 𝐁 ={(x,y), …, ∞}
3. Inequality System of No Solution with 2
parallel lines of no intersections
A∩ 𝐁={∅}
Linear-Inequality Systems with 2 Variables
11 1
22 2
Mon 1/3/2017 8:10 AMJacqueline B. Chau 29
Mon 1/3/2017 8:10 AMJacqueline B. Chau 30
Absolute-Value Inequalities of 1 Variable
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|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:
1. Linear Inequality of One Solution with 2
intersecting lines
A∩ 𝐁={(x,y)}
2. Linear Inequality of Finite Solution with
2 overlapping lines
A∩ 𝐁={(x,y), …,(x,y)}
3. Linear Inequality of Infinite Solution with
2 parallel-intersecting lines
A∩ 𝐁={(x,y), …, ∞}
4. Linear Inequality of No Solution with 2
parallel non-intersecting lines
A∩ 𝐁={∅}
The 2 Absolute-Value Forms of Inequalities:
1. Absolute-Value Form of Infinite Solution
|ax + by| ≤ c
2. Absolute-Value Form of No Solution
|ax + by| ≥ c
3. Solve by
The 4 Special Cases of Inequality Systems:
1. Inequality System of Infinite Solution with
2 intersected lines
A∩ 𝐁={(x,y), …, ∞}
2. Inequality System of Infinite Solution with
2 parallel-intersecting lines
A∩ 𝐁 ={(x,y), …, ∞}
3. Inequality System of No Solution with 2
parallel non-intersecting lines
A∩ 𝐁={∅}
Absolute-Value Inequalities of 2 Variables
*
*
*
*
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Circumference & Radius, Volume & Pressure,
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Mon 1/3/2017 8:10 AMJacqueline B. Chau 31
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Mon 1/3/2017 8:10 AMJacqueline B. Chau 33
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 line
A set or subset of numbers
Cartesian Coordinates
Philosopher & Mathematician Rene Descartes
ax + by = c where a, b ≠ 0 at once
y = mx + b
No
Positive, Negative, 0, Undefined
Audiences’ Feedback
Steepness = 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 1/3/2017 8:10 AMJacqueline B. Chau 34
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?
False
Graph, Elimination, Substitution
One Ordered-Pair, No Solution, Infinite
+ & - Slope Equations
All variables are eliminated & statement is True
Infinite Set of common Sol
True or False
-
Audiences’ Feedback
Reals
Irrationals
-
-
-
-
Mon 1/3/2017 8:10 AMJacqueline B. Chau 35
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?
-
-
Mon 1/3/2017 8:10 AMJacqueline B. Chau 36

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Algebraic Mathematics of Linear Inequality & System of Linear Inequality

  • 1. by Jacqueline Chau Education 014 * Algebraic Mathematics Linear Inequality & System of Linear Inequality
  • 2. Mon 1/3/2017 8:10 AMJacqueline B. Chau 2  Review 1. 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  Lesson 1. Algebraic Compound & Absolute Value of Linear Inequalities 2. 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 Examples 4. Summary of Linear Inequalities & Inequality Systems  Comprehensive Quiz
  • 3. Mon 1/3/2017 8:10 AMJacqueline 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
  • 4. Mathematics Variable Term Equation Expression 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 1/3/2017 8:10 AMJacqueline B. Chau 4
  • 5. Graph Slope Cartesian Coordinates 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 1/3/2017 8:10 AMJacqueline 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)}
  • 6. Union Compound Inequality Intersection Continued Inequality 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 1/3/2017 8:10 AMJacqueline B. Chau 6 System of Equations 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 .
  • 7. Consistent System Inconsistent System 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. Dependent System 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 1/3/2017 8:10 AMJacqueline B. Chau 7 Independent System A system of unequivalent equations, where all variables are eliminated and the resulting statement is FALSE, has no common solutions.
  • 8. BA BA 1. No Intersection A ∪ 𝑩 = {∅} A ∩ 𝑩 = {∅} 2. Some Intersection A ∪ 𝑩 = {𝑨 − 𝑪, 𝑩 − 𝑪} A ∩ 𝑩 = 𝑪 3. Complete Intersection A ∪ 𝑩 = {𝑨, 𝑩} = 𝑫 A ∩ 𝑩 = {𝑨, 𝑩} = 𝑫 C Mon 1/3/2017 8:10 AMJacqueline B. Chau 8 VENN DIAGRAM Disjoint Subset Overlapping B A D
  • 9. Integer …,-99,-8, -1, 0, 1, 2, 88, … Whole 0,1,2,3,4,5,6,… Counting 1,2,3,4,5,… Rational …, -77.33, -2, 0, 5.66, 25/3, … Real Mon 1/3/2017 8:10 AMJacqueline B. Chau 9
  • 10. Counting Integer Rational 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 1/3/2017 8:10 AMJacqueline B. Chau 10
  • 11. 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 1/3/2017 8:10 AMJacqueline B. Chau 11
  • 12. ax + by = c y = mx + b y-y = m(x-x) slope = m= m slope = m = -1/m y x 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 Variables Standard 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 Form where slope=m, y-intercept=b where slope=m, point=(x , y ) 2 1 Slopes of Parallel Lines 2 Slopes of Perpendicular Lines 1 2 21 Standard Form -x + y = 4 X-Intercept = (c/a) = -4 Y-Intercept = (c/b) = 4 Slope = -(a/b) = 1 Slope-Intercept y = x + 4 X-Intercept = (-4, 0) Y-Intercept = b = 4 Slope = m = 1 1 1 (x , y )2 2 (x , y )1 1 slope = m = ──────── [ Rise | Fall ] Run Mon 1/3/2017 8:10 AMJacqueline B. Chau 12
  • 13. Standard Form ax + by = c -3x + 1y = 2 Note: 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 Form y = mx + b y = 3x + 2 Point-Slope Form (y – y ) = m(x – x ) (y + 4) = 3(x + 2 ) y = 3x + 2 Slope = m = 3 X-Intercept = -b/m = -2/3 Y-Intercept = b = 2 1 1 y x x y -1 0 1 2 3 -1 2 5 8 11 (0,0) (x , y ) = (-2, -4)1 1 Linear Equation with 2 Variables Slope = -(a/b) = 3 X-Intercept = c/a = -2/3 Y-Intercept = c/b = 2 Slope = m = 3 X-Intercept = -b/m = -2/3 Y-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 = ──── = 3 5 – (-4) 1 – (-2) Mon 1/3/2017 8:10 AMJacqueline B. Chau 13 Solve by Graphing
  • 14. y x(0,0) x=11UndefinedSlope x y X-Intercept=(11,0) Y-Intercept=none Slope=m=undefined X-Intercept = none Y-Intercept = (0, -10) Slope = m = 0 y = -10  Zero Slope Through the Origin y = mx Slope = m X-Intercep = (0, 0) Y-Intercep = (0, 0) Special Cases of Linear Equations with Two Variables Vertical Line x = a Slope = undefined X-Intercep = (a, 0) Y-Intercep = none Horizontal Line y = b Slope = 0 X-Intercep = none Y-Intercep = (0, b) Positive & Negative Slopes Zero & Undefined Slopes Mon 1/3/2017 8:10 AMJacqueline B. Chau 14
  • 15. x y One Solution A ∩ 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 Slope 3x - 2y = 6 y = (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) Solve by Graphing Solve by Elimination 2x + 4y = 20 2(3x – 2y = 6) 8x = 32 x = 4 2(4) + 4y = 20 4y = 12 y = 3  (x, y ) = (4, 3) ∴ A ∩ B є {(4,3)} Solve by Substitution 2x + 4y = 20 3x – 2y = 6 2x + 2(3x - 6) = 20 8x – 12 = 20 8x = 32 x = 4 2(4) + 4y = 20 4y = 12 y = 3  (x, y ) = (4, 3) Negative-Slope & Positive-Slope Equations (4, 3) Mon 1/3/2017 8:10 AMJacqueline B. Chau 15 Negative Slope 2x + 4y = 20 y = (-1/2)x + 5 Slope = -(1/2) X-Intercep = (10, 0) Y-Intercep = (0, 5)
  • 16. Special Cases of System of Parallel Linear-Equations with Two Variables x 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 Equation2 3x - 2y = 6 y = (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) Parallel Equation1 3x - 2y = -2 y = (3/2)x + 1 Slope = (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 ∅. No Intersection No Solution A ∩ B є {∅} Parallel Linear Equations where a / a = b / b = 1 and c / c = {Integers}1 1 1 2 2 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 16
  • 17. Special Cases of System of Equivalent Linear-Equations with Two Variables x 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 Equation2 3x - 2y = 6 y = (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) Dependent Equation1 6x - 4y = 12 y = (3/2)x -3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) Solve by Graphing Solve by Elimination 6x - 4y = 12 2(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 Coincide Infinite Solution A ∩ B є {(2,0),(0,-3), …} 11 1 22 2 Dependent Equations Mon 1/3/2017 8:10 AMJacqueline B. Chau 17
  • 18. Linear Equations with 2 Variables The 3 Forms of Linear Equations: 1. Standard Form ax + by = c 2. Slope-Intercept Form y = mx + b 3. Point-Slope Form y-y = m(x-x ) The 4 Special Cases of Linear Equations: 1. Positive Slope m = n  y = mx + b 2. Negative Slope m = -n  y = -mx + b 3. Zero Slope m = 0  y = n 4. 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 Graphing 2. Solve by Elimination 3. Solve by Substitution The 3 Special Cases of Linear System: 1. Linear System of One Solution A∩ 𝐁={(x,y)} 2. Linear System of No Solution A∩ 𝐁={∅} 3. Linear System of Infinite Solution A∩ 𝐁={∞} Linear Systems with 2 Variables 11 Cautions: The coefficient b in the Standard Form is not the same b in the Slope-Intercept Form. 11 1 22 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 18
  • 19. ax + by ≤ c y ≤ mx + b y-y ≤ m(x-x) slope = m= m slope = m = -1/m 1 11 Linear Inequalities Standard Form Slope-Intercept Form Point-Slope Form where slope=m, y-intercept=b where slope=m, point=(x , y ) Slopes of Parallel Lines 2 Slopes of Perpendicular Lines 1 2 where {a,b,c}=Real; ([a|b]=0) ≠ [b|a] y x(0,0) x y -1 0 1 2 3 -1 2 5 8 11 Solve by Graphing 1 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 19
  • 20. Case 1: Linear Inequality of Finite Solution 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 −𝟑 ≤ 𝒙 ≤ 𝟒 𝒙 ≥ −𝟑 𝑨𝑵𝑫 𝒙 ≤ 𝟒 Case 2: Linear Inequality of No Solution 𝐀 ∪ 𝑩 = {𝑨 𝑶𝑹 𝑩} −𝟑 ≥ 𝒙 ≥ 𝟒 𝒙 ≤ −𝟑 𝑶𝑹 𝒙 ≥ 𝟒 y x (0,0) −𝟑 ≤ 𝒙 ≤ 𝟒 Solve by Graphing 𝒙 ≤ 𝟒 𝒙 ≥ −𝟑 𝒙 ≤ −𝟑 𝒙 ≥ 𝟒 −𝟑 ≥ 𝒙 ≥ 𝟒 x Linear Inequalities of One Variable Mon 1/3/2017 8:10 AMJacqueline B. Chau 20 Special Cases of Compound Linear-Inequalities with One Variables
  • 21. Case 1: Linear Inequality of One Solution |𝒙| ≤ 𝟎 𝒙 ≥ 𝟎 𝑨𝑵𝑫 𝒙 ≤ 𝟎 ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {0} 𝒙 ≥ 𝟎 𝒙 ≤ 𝟎 𝑶𝑹 𝒙 ≥ 𝟎 ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {0} Case 2: Linear Inequality of Finite Solution 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 𝒙 ≤ 𝟒 𝒙 ≥ −𝟒 𝑨𝑵𝑫 𝒙 ≤ 𝟒 ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {-3,-2,-1,0,1,2,3,4} Case 3: Linear Inequality of No Solution 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 |𝒙| ≥ 𝟒 𝒙 ≤ −𝟒 𝑶𝑹 𝒙 ≥ 𝟒 ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 𝑩 = {∅} y x (0,0) −𝟒 ≤ 𝒙 ≤ 𝟒 Solve by Graphing 𝒙 ≤ 𝟒 𝒙 ≥ −𝟒 𝒙 ≤ −𝟒 𝒙 ≥ 𝟒 −𝟒 ≥ 𝒙 ≥ 𝟒 x Linear Inequalities with One Variable Mon 1/3/2017 8:10 AMJacqueline B. Chau 21 x𝒙 ≤ 𝟎 𝒙 ≥ 𝟎0 ≤ 𝒙 ≤ 𝟎
  • 22. y x (0,0) x y -1 0 1 2 3 -5 -2 1 4 7 Solve by GraphingLinear Inequalities with Two Variables Case 1: Linear Inequality of Infinite Solution −𝟑𝒙 + 𝒚 ≤ 𝟐  ± (−𝟑𝒙 + 𝒚) ≤ 𝟐 (A) AND (B)  − (−𝟑𝒙 + 𝒚) ≤ 𝟐 (A) 𝟑𝒙 − 𝒚 ≤ 𝟐 𝒚 ≥ 𝟑𝒙 − 𝟐  + (−𝟑𝒙 + 𝒚) ≤ 𝟐 (B) −𝟑𝒙 + 𝒚 ≤ 𝟐 𝒚 ≤ 𝟑𝒙 + 𝟐  𝒚 ≥ 𝟑𝒙 − 𝟐 (A) 𝑨𝑵𝑫 𝒚 ≤ 𝟑𝒙 + 𝟐 (B)  ∴ 𝐀 ∩ 𝑩 = 𝑨 𝑨𝑵𝑫 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),… , ∞} Mon 1/3/2017 8:10 AMJacqueline B. Chau 22 x y -1 0 1 2 3 -1 2 5 8 11
  • 23. Linear Inequalities with Two Variables Case 2: Linear Inequality of No Solution −𝟑𝒙 + 𝒚 ≥ 𝟐  ± (−𝟑𝒙 + 𝒚) ≥ 𝟐 (A) AND (B)  − (−𝟑𝒙 + 𝒚) ≥ 𝟐 (A) 𝟑𝒙 − 𝒚 ≥ 𝟐 𝒚 ≤ 𝟑𝒙 − 𝟐  + (−𝟑𝒙 + 𝒚) ≥ 𝟐 (B) −𝟑𝒙 + 𝒚 ≥ 𝟐 𝒚 ≥ 𝟑𝒙 + 𝟐  𝒚 ≤ 𝟑𝒙 − 𝟐 (A) 𝑨𝑵𝑫 𝒚 ≥ 𝟑𝒙 + 𝟐 (B)  ∴ 𝐀 ∩ 𝑩 = {𝑨 𝑨𝑵𝑫 𝑩} = {∅} Mon 1/3/2017 8:10 AMJacqueline B. Chau 23 x ySolve by Graphing x y -1 0 1 2 3 -5 -2 1 4 7 x y -1 0 1 2 3 -1 2 5 8 11
  • 24. Linear Inequalities with Two Variables Case 3: Linear Inequality of Infinite Solution 𝒙 + 𝒚 ≥ 𝟎  ± (𝒙 + 𝒚) ≥ 𝟎 (A) AND (B)  − (𝒙 + 𝒚) ≥ 𝟎 (A) − 𝒙 − 𝒚 ≥ 𝟎 𝒚 ≤ −𝒙  + (𝒙 + 𝒚) ≥ 𝟎 (B) 𝒙 + 𝒚 ≥ 𝟎 𝒚 ≥ −𝒙  𝒚 ≤ −𝒙 (A) 𝑨𝑵𝑫 𝒚 ≥ −𝒙 (B)  ∴ 𝐀 ∩ 𝑩 = {𝑨 𝑨𝑵𝑫 𝑩} = {(0,0),(1,-1),(2,-2),(3,-3), (1,1),(2,2),(3,3),(4,4),(5,5), (10,1),(10,-1),(3,-1),,… , ∞} Mon 1/3/2017 8:10 AMJacqueline B. Chau 24 x ySolve by Graphing x y -1 0 1 2 3 1 0 -1 -2 -3 x y -1 0 1 2 3 -1 0 1 2 3
  • 25. A ∩ B є {(0,0),(4,3),(0,5),(2,0),(0,-3), …} 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 ≤ 20 2(3x – 2y ≤ 6) 8x ≤ 32 x ≤ 4 2(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 ≤ 6 2x + 2(3x - 6) ≤ 20 8x – 12 ≤ 20 8x ≤ 32 x ≤ 4 2(4) + 4y ≤ 20 4y ≤ 12 y ≤ 3  (x, y ) ≤ (4, 3) Negative-Slope & Positive-Slope Equations x Positive Slope 3x - 2y = 6 y = (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) Negative Slope 2x + 4y = 20 y = (-1/2)x + 5 Slope = -(1/2) X-Intercep = (10, 0) Y-Intercep = (0, 5) (4, 3) Special Cases of System of Linear Inequalities with Two Variables Mon 1/3/2017 8:10 AMJacqueline B. Chau 25
  • 26. 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! ∴ A∩B ∈ {(0,0),(1,1),(2,2),(3,3),(4,4),…} Parallel Linear Equations x Special Cases of System of Linear Inequalities with Two Variables Parallel Equation 1 3x - 2y ≥ -2 y ≤ (3/2)x + 1 Slope = (a₁/b₁) = (3/2) X-Intercep = (-2/3, 0) Y-Intercep = (0, 1) Parallel Equation 2 3x - 2y ≤ 6 y ≥ (3/2)x - 3 Slope = (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 1 2 2 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 26 1 2
  • 27. 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 є {⏀} Parallel Linear Equations x Special Cases of System of Linear Inequalities with Two Variables Parallel Equation 1 3x - 2y ≤ -2 y ≥ (3/2)x + 1 Slope = (3/2) X-Intercep = (-2/3, 0) Y-Intercep = (0, 1) Parallel Equation 2 3x - 2y ≥ 6 y ≤ (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) and a / a = b / b = 1 and c / c = {Integers}1 1 1 2 2 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 27
  • 28. 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 ≤ 12 2(3x – 2y ≤ 6) 0 ≤ 0 True! ∴ A∩B є {(2,0),(0,-3),(6,6),(-4,-9),…} Coincide Linear Equations x Dependent Equation 1 6x - 4y ≤ 12 y ≥ (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3) and a / a = b / b = c / c = 111 1 22 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 28 Special Cases of System of Linear Inequalities with Two Variables Dependent Equation 2 3x - 2y ≥ 6 y ≤ (3/2)x - 3 Slope = (3/2) X-Intercep = (2, 0) Y-Intercep = (0, -3)
  • 29. Linear Inequalities with 2 Variables The 3 Forms of Linear Inequalities: 1. Standard Form of Infinite Solution ax + by ≤ c 2. Absolute-Value Form of Infinite Solution |ax + by| ≤ c 3. Absolute-Value Form of No Solution |ax + by| ≥c The 4 Special Cases of Linear Inequalities: 1. Linear Inequality of Infinite Solution with 2 intersected lines (Note: when a or b is 0, A) 2. 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) 3. 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 Graphing 2. Solve by Elimination 3. Solve by Substitution The 4 Special Cases of Inequality Systems: 1. Inequality System of One Solution with 2 intersected lines A∩ 𝐁={(x,y)} 2. Inequality System of Infinite Solution with 2 parallel-intersected lines OR with 2 coincide lines A∩ 𝐁 ={(x,y), …, ∞} 3. Inequality System of No Solution with 2 parallel lines of no intersections A∩ 𝐁={∅} Linear-Inequality Systems with 2 Variables 11 1 22 2 Mon 1/3/2017 8:10 AMJacqueline B. Chau 29
  • 30. Mon 1/3/2017 8:10 AMJacqueline B. Chau 30 Absolute-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: 1. Linear Inequality of One Solution with 2 intersecting lines A∩ 𝐁={(x,y)} 2. Linear Inequality of Finite Solution with 2 overlapping lines A∩ 𝐁={(x,y), …,(x,y)} 3. Linear Inequality of Infinite Solution with 2 parallel-intersecting lines A∩ 𝐁={(x,y), …, ∞} 4. Linear Inequality of No Solution with 2 parallel non-intersecting lines A∩ 𝐁={∅} The 2 Absolute-Value Forms of Inequalities: 1. Absolute-Value Form of Infinite Solution |ax + by| ≤ c 2. Absolute-Value Form of No Solution |ax + by| ≥ c 3. Solve by The 4 Special Cases of Inequality Systems: 1. Inequality System of Infinite Solution with 2 intersected lines A∩ 𝐁={(x,y), …, ∞} 2. Inequality System of Infinite Solution with 2 parallel-intersecting lines A∩ 𝐁 ={(x,y), …, ∞} 3. Inequality System of No Solution with 2 parallel non-intersecting lines A∩ 𝐁={∅} Absolute-Value Inequalities of 2 Variables
  • 31. * * * * * 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 1/3/2017 8:10 AMJacqueline B. Chau 31
  • 32. Mon 1/3/2017 8:10 AMJacqueline B. Chau 32
  • 33. Mon 1/3/2017 8:10 AMJacqueline B. Chau 33
  • 34. 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 line A set or subset of numbers Cartesian Coordinates Philosopher & Mathematician Rene Descartes ax + by = c where a, b ≠ 0 at once y = mx + b No Positive, Negative, 0, Undefined Audiences’ Feedback Steepness = 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 1/3/2017 8:10 AMJacqueline B. Chau 34
  • 35. 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? False Graph, Elimination, Substitution One Ordered-Pair, No Solution, Infinite + & - Slope Equations All variables are eliminated & statement is True Infinite Set of common Sol True or False - Audiences’ Feedback Reals Irrationals - - - - Mon 1/3/2017 8:10 AMJacqueline B. Chau 35
  • 36. 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? - - Mon 1/3/2017 8:10 AMJacqueline B. Chau 36