A brief, yet thorough look into the Linear Inequality & System of Linear Inequality and how these Math Concepts would be useful in solving our everyday life problems.
2. Mon 1/3/2017 8:10 AMJacqueline B. Chau 2
οΆ 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
οΆ 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
οΆ Comprehensive Quiz
3. Mon 1/3/2017 8:10 AMJacqueline B. Chau 3
οΆ Focus on all relevant Algebraic concepts from fundamental
vocabulary, to Linear Equality and Systems of Linear
Equalities, which are essential to the understanding of
todayβs topic
οΆ Present todayβs 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
οΆ Assess audienceβs 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)= xΒ² β> {(x,f(x)),(Β±1,1)}
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.
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, β¦ }
Whole Numbers of both Negative Numbers and
Positive Numbers, excluding Fractions.
{ β¦, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
Whole Natural Numbers that are used to count for
quantity of something βwholeβ as in counting a
whole table or a whole chair. { 0,1, 2, 3, 4, 5, β¦ }
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 β (Circumference/Diameter)
is not equal to the Ratio of any two numbers), or
any Square Roots of a Non-Perfect square number.
{ β2, β3, β , 3.141592653β¦, β5, β6, β7, β8, β10, β¦}
Any numbers that can be found on a Number Line,
which includes Rational Numbers and Irrational
Numbers. { β¦, -2, -1/2, 0, 0.33, 0.75, β , 3.3, β¦ }
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, β¦ }
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.
{ β¦, -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 {1Β½, 2ΒΎ}, respectively.
Any numbers that are not Real Numbers and
whose squares are negative Real Numbers, i = β-1
{ β-1, 1i, β-βΒ², βi, β-4, 2i, β-9, 3i, β-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 + βi}
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) β [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 = ββββββββ
[ 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 β y ) = m(x β 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
= ββββ = 3
5 β (-4)
1 β (-2)
Mon 1/3/2017 8:10 AMJacqueline B. Chau 13
Solve by Graphing
14. y
x(0,0)
x=11ο 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 ο 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
19. ax + by β€ c
y β€ mx + b
y-y β€ 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) β [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
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 βbβ 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 β 0 at once
y = mx + b
No
Positive, Negative, 0, Undefined
Audiencesβ Feedback
Steepness = Rise/Run; +, -, 0, β
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
-
Audiencesβ 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