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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
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