## Similar to Lesson 1.2 NT (Equation and Inequalities).pdf

Expresiones algebraicas
Expresiones algebraicas
simaraalexandrasanch

DaniloFrondaJr

Ecuaciones lineales
Ecuaciones lineales
Fabio Pinto Oviedo

María Antonio y Norneris Meléndez
María Antonio y Norneris Meléndez
Mariantonio

MIT Math Syllabus 10-3 Lesson 6: Equations
MIT Math Syllabus 10-3 Lesson 6: Equations
Lawrence De Vera

2.6 Other Types of Equations
2.6 Other Types of Equations
smiller5

05. s3 ecuaciones polinómicas
05. s3 ecuaciones polinómicas
Carlos Sánchez Chuchón

Cipriano De Leon

IGCSEFM-FactorTheorem.pptx
IGCSEFM-FactorTheorem.pptx
AngieMichailidou

edwinllantoy2

October. 27, 2014
October. 27, 2014
khyps13

Math 8 - Systems of Linear Inequalities in Two Variables
Math 8 - Systems of Linear Inequalities in Two Variables
Carlo Luna

MCA_UNIT-1_Computer Oriented Numerical Statistical Methods
MCA_UNIT-1_Computer Oriented Numerical Statistical Methods
Rai University

Solution of linear equation & inequality
Solution of linear equation & inequality
Dalubhasaan ng Lungsod ng Lucena

cesaramaro8

Jeancarlos freitez
Jeancarlos freitez
JeancarlosFreitez

Lesson 5: Polynomials
Lesson 5: Polynomials
Kevin Johnson

Lesson 1: The Real Number System
Lesson 1: The Real Number System
Kevin Johnson

JOANNAMARIECAOILE

Expresiones algebraicas
Expresiones algebraicas
CarlosRamosAzuaje

### Similar to Lesson 1.2 NT (Equation and Inequalities).pdf(20)

Expresiones algebraicas
Expresiones algebraicas

Ecuaciones lineales
Ecuaciones lineales

María Antonio y Norneris Meléndez
María Antonio y Norneris Meléndez

MIT Math Syllabus 10-3 Lesson 6: Equations
MIT Math Syllabus 10-3 Lesson 6: Equations

2.6 Other Types of Equations
2.6 Other Types of Equations

05. s3 ecuaciones polinómicas
05. s3 ecuaciones polinómicas

IGCSEFM-FactorTheorem.pptx
IGCSEFM-FactorTheorem.pptx

October. 27, 2014
October. 27, 2014

Math 8 - Systems of Linear Inequalities in Two Variables
Math 8 - Systems of Linear Inequalities in Two Variables

MCA_UNIT-1_Computer Oriented Numerical Statistical Methods
MCA_UNIT-1_Computer Oriented Numerical Statistical Methods

Solution of linear equation & inequality
Solution of linear equation & inequality

Jeancarlos freitez
Jeancarlos freitez

Lesson 5: Polynomials
Lesson 5: Polynomials

Lesson 1: The Real Number System
Lesson 1: The Real Number System

Expresiones algebraicas
Expresiones algebraicas

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
Nguyen Thanh Tu Collection

How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17
Celine George

2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference
KlettWorldLanguages

How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS
Celine George

Imagination in Computer Science Research
Imagination in Computer Science Research
Abhik Roychoudhury

What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
Celine George

Genetics Teaching Plan: Dr.Kshirsagar R.V.
Genetics Teaching Plan: Dr.Kshirsagar R.V.
DrRavindrakshirsagar1

How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17
Celine George

C# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdf
Scholarhat

New Features in Odoo 17 Sign - Odoo 17 Slides
New Features in Odoo 17 Sign - Odoo 17 Slides
Celine George

Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx
monicaaringo1

Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx
ibtesaam huma

AZ-900 Microsoft Azure Fundamentals Summary.pdf
AZ-900 Microsoft Azure Fundamentals Summary.pdf
OlivierLumeau1

1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
AnneMarieJacildo

Jerry Chew

SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
Murugan Solaiyappan

artenzmartenkai

RDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEW
Murugan Solaiyappan

How to Add a Filter in the Odoo 17 - Odoo 17 Slides
How to Add a Filter in the Odoo 17 - Odoo 17 Slides
Celine George

Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
MysoreMuleSoftMeetup

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...

How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17

2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference

How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS

Imagination in Computer Science Research
Imagination in Computer Science Research

What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides

Genetics Teaching Plan: Dr.Kshirsagar R.V.
Genetics Teaching Plan: Dr.Kshirsagar R.V.

How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17

C# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdf

New Features in Odoo 17 Sign - Odoo 17 Slides
New Features in Odoo 17 Sign - Odoo 17 Slides

Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx

Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx

AZ-900 Microsoft Azure Fundamentals Summary.pdf
AZ-900 Microsoft Azure Fundamentals Summary.pdf

1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx

SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence

RDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEW

How to Add a Filter in the Odoo 17 - Odoo 17 Slides
How to Add a Filter in the Odoo 17 - Odoo 17 Slides

Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...

### Lesson 1.2 NT (Equation and Inequalities).pdf

• 1. PHILIPPINE NORMAL UNIVERSITY VISAYAS The National Center for Teacher Education The Environment and Green Technology Education Hub Cadiz City, Negros Occidental
• 2. PHILIPPINE NORMAL UNIVERSITY VISAYAS The National Center for Teacher Education The Environment and Green Technology Education Hub Cadiz City, Negros Occidental
• 3. After this lesson, one is expected to solve equations and inequalities.
• 4. An equation is a statement that two expressions are equal. Example 1.4 The following are examples of equation. 2𝑥 − 6 = 0 𝑥2 − 5𝑥 + 6 = 0 To solve an equation means to find all the values of the variables that will make the statement true. The values of the variables that make the statement true are called solutions or roots of the equation. Example 1.5 The equation 2𝑥 − 6 = 0 is a true statement for 𝑥 = 3 but it is false for any other number. The root or solution of this equation is 3.
• 5. Properties of Equations Let 𝑎, 𝑏, 𝑐 ∈ ℝ. 1. Reflexive Property 𝑎 = 𝑎 Example 2 = 2 2. Symmetric Property (or Replacement Property) If 𝑎 = 𝑏, then 𝑏 = 𝑎. If 𝑎 = 𝑏, then 𝑎 can replace 𝑏 in any instance. Example If 5 = 𝑥, then 𝑥 = 5.
• 6. 3. Transitive Property If 𝑎 = 𝑏, and 𝑏 = 𝑐, then 𝑎 = 𝑐. Example If 𝑥 = 𝑦 and 𝑦 = 5, then 𝑥 = 5. 4. Addition Property If 𝑎 = 𝑏, then 𝑎 + 𝑐 = 𝑏 + 𝑐. Example If 𝑥 = 𝑦 then 𝑥 + 3 = 𝑦 + 3. 5. Multiplication Property If 𝑎 = 𝑏, then 𝑎𝑐 = 𝑏𝑐. Example If 𝑥 = 𝑦, then 5𝑥 = 5𝑦.
• 7. Example 1.6 Use the properties of equations and properties of real numbers to solve the equation 2 3 𝑥 − 1 = 𝑥 − 3. Solution. 3 2 3 𝑥 − 1 = 𝑥 − 3 3 Multiplication Property 2 𝑥 − 1 = 𝑥 − 3 3 2𝑥 − 2 = 3𝑥 − 9 Distributive Property 2𝑥 − 2 + 2 − 3𝑥 = 3𝑥 − 9 + (2 − 3𝑥) Addition Property (2𝑥 − 3𝑥) + −2 + 2 = (3𝑥 − 3𝑥) + (−9 + 2) Associative Property −𝑥 + 0 = 0 − 7 Additive Inverse −𝑥 = −7 Additive Identity Thus, 𝑥 = 7 Multiplication Property
• 8. Solve the equation 2 3 𝑥 − 1 = 𝑥 − 3. Solution: 𝑥 = 7 To check, using replacement property, 2 3 𝑥 − 1 = 𝑥 − 3 2 3 7 − 1 = 7 − 3; 4 = 4 which is true by reflexive property of equality.
• 9. An equation that is true for all values of the variable is called an identity. An equation that is true for some values of the variables but not true for other values is called a conditional equation. An equation that has no solutions is called a contradiction. Example 1.7 Classify each equation as an identity, a conditional equation, or a contradiction. a) 𝑥2 + 2𝑥 = 𝑥(𝑥 + 2) b) 5𝑥 − 2 = 𝑥 − 10 c) 3𝑥 + 2 = 1 + 3𝑥
• 10. Classify each equation as an identity, a conditional equation, or a contradiction. a) 𝑥2 + 2𝑥 = 𝑥(𝑥 + 2) Solution Getting the product of the right side of 𝑥2 + 2𝑥 = 𝑥(𝑥 + 2), we have 𝑥2 + 2𝑥 = 𝑥2 + 2𝑥 Since the left and right expressions are the same, the equation will hold true for whatever value of 𝑥, thus we have an identity.
• 11. Classify each equation as an identity, a conditional equation, or a contradiction. 𝑏. 5𝑥 − 2 = 𝑥 − 10 Solution Adding both sides of 5𝑥 − 2 = 𝑥 − 10 by (2 − 𝑥), we have 5𝑥 − 2 + 2 − 𝑥 = 𝑥 − 10 + (2 − 𝑥) 4𝑥 = −8. Multiplying both sides of 4𝑥 = −8 by 1 4 , we get 𝑥 = −2 The original equation is satisfied by 𝑥 = −2 but not by other values. Thus, the equation is a conditional equation.
• 12. Classify each equation as an identity, a conditional equation, or a contradiction. 𝑐. 3𝑥 + 2 = 1 + 3𝑥 Solution Subtracting 3𝑥 from both sides of the equation 3𝑥 + 2 − 3𝑥 = 1 + 3𝑥 − 3𝑥 we get 2 = 1 which is a false statement. The equation is a false statement no matter what the value of 𝑥 is and thus, it has no solution. Therefore, it is a contradiction.
• 13. Real numbers are shown on the number line with larger numbers written to the right. For any two real numbers, the one to the left is less than the one to the right. The symbol < means “is less than” and the symbol > means “is greater than”. A statement that one quantity is greater than or less than another quantity is called an inequality.
• 14. ▪ If 𝑎 and 𝑏 are real numbers, 𝑎 is greater than 𝑏 or 𝑎 > 𝑏 means 𝑎 is to the right of 𝑏 on the number line, and 𝑎 − 𝑏 is positive. ▪ If 𝑎 is less than 𝑏, or 𝑎 < 𝑏, this means 𝑎 is to the left of 𝑏 on the number line, and 𝑎 − 𝑏 is negative
• 15. Note the following: 𝑎 > 𝑏 and 𝑏 < 𝑎 have the same meaning. 𝑎 > 0 means that 𝑎 is positive, 𝑎 < 0 means that 𝑎 is negative.
• 17. The following statements are also inequalities: 𝑎 ≤ 𝑏 means “𝑎 is less than or equal to 𝑏” 𝑎 ≥ 𝑏 means “𝑎 is greater than or equal to 𝑏 If 𝑎 ≤ 𝑏 and 𝑎 ≥ 𝑏 hold simultaneously, then 𝑎 = 𝑏. For any real number 𝑎, 𝑎 ≤ 0 means 𝑎 is not positive, and 𝑎 ≥ 0 means 𝑎 is not negative.
• 18. Example 1.9 The following are examples of inequalities: a) 2𝑥 + 3 < 12 b) 𝑥2 − 2𝑥 − 8 ≤ 0 c) 𝑥 + 2𝑦 > 1 + 𝑦
• 19. Properties of Inequalities Let 𝑎, 𝑏, 𝑐 ∈ ℝ. 1. Trichotomy Property. If 𝑎, 𝑏 ∈ ℝ, then one and only one of the following relations holds: 𝑎 < 𝑏, 𝑎 = 𝑏, 𝑜𝑟 𝑎 > 𝑏. 2. Transitive Property. If 𝑎, 𝑏, 𝑐 ∈ ℝ, such that 𝑎 > 𝑏 and 𝑏 > 𝑐, then 𝑎 > 𝑐. 3. Addition Property. If 𝑎, 𝑏, 𝑐 ∈ ℝ, such that 𝑎 > 𝑏 , then 𝑎 + 𝑐 > 𝑏 + 𝑐. 4. Multiplication Property. Let 𝑎, 𝑏, 𝑐 ∈ ℝ. 𝑖. If 𝑎 > 𝑏 and 𝑐 > 0, then 𝑎𝑐 > 𝑏𝑐. 𝑖𝑖. If 𝑎 > 𝑏 and 𝑐 < 0, then 𝑎𝑐 < 𝑏𝑐.
• 20. Example 1.10 Solve the inequality 2 3 𝑥 − 1 ≥ 𝑥 − 3. Solution. 3 2 3 𝑥 − 1 ≥ 𝑥 − 3 3 Multiplication Property (𝑖) 2 𝑥 − 1 ≥ 𝑥 − 3 3 2𝑥 − 2 ≥ 3𝑥 − 9 Distributive Property 2𝑥 − 2 + 2 − 3𝑥 ≥ 3𝑥 − 9 + (2 − 3𝑥) Addition Property (2𝑥 − 3𝑥) + −2 + 2 ≥ (3𝑥 − 3𝑥) + (−9 + 2) Associative Property −𝑥 + 0 ≥ 0 − 7 Additive Inverse −𝑥 ≥ −7 Additive Identity Thus, 𝑥 ≤ 7 Multiplication Property (𝑖𝑖)
• 21. Writing the solution in a set, 𝑥ȁ𝑥 ≤ 7 Read: The set of all elements 𝑥 such that 𝑥 is less than or equal to 7. To illustrate the solution in a number line, To check, choose one value in the solution, say 𝑥 = −2 . Substituting this value for 𝑥, 2 3 −2 − 1 ≥ −2 − 3 2 3 −3 ≥ −5 and −2 ≥ −5 which is true.
• 22. Compound Inequalities A compound inequality is formed by joining two inequalities with the connective word and or or. The solution set of a compound inequality with the connective word 𝑜𝑟 is the union of the solution sets of the two inequalities. The solution set of a compound inequality with the connective word 𝑎𝑛𝑑 is the intersection of the solution sets of the two inequalities.
• 23. Example 1.11 Solve the following of compound inequalities: a) 2𝑥 < 8 𝑜𝑟 𝑥 − 3 > 4 b) 𝑥 + 3 > 5 𝑎𝑛𝑑 3𝑥 − 7 < 14 Solution a) 2𝑥 < 8 𝑜𝑟 𝑥 − 3 < 4 Solving each inequality, we have 2𝑥 < 8 𝑥 − 3 > 4 𝑥 < 4 𝑥 > 7 𝑥ȁ𝑥 < 4 𝑥ȁ𝑥 > 7 Write the union of the solution sets. 𝑥ȁ𝑥 < 4 ∪ 𝑥ȁ𝑥 > 7 = 𝑥ȁ𝑥 < 4 𝑜𝑟 𝑥 > 7
• 24. Illustrating the solution in a number line, 𝑥ȁ𝑥 < 4 ∪ 𝑥ȁ𝑥 > 7 = 𝑥ȁ𝑥 < 4 𝑜𝑟 𝑥 > 7
• 25. b) 𝑥 + 3 > 5 𝑎𝑛𝑑 3𝑥 − 10 > 14 Solving each inequality, we have 𝑥 + 3 > 5 𝑥 > 2 𝑥ȁ𝑥 > 2 3𝑥 − 10 > 14 3𝑥 > 24 𝑥 > 8 𝑥ȁ𝑥 > 8 Write the intersection of the solution sets. 𝑥ȁ𝑥 > 2 ∩ 𝑥ȁ𝑥 > 8 = 𝑥ȁ 𝑥 > 8 .
• 26. The inequality given by 𝑎 < 𝑏 < 𝑐, is equivalent to the compound inequality 𝑎 < 𝑏 𝑎𝑛𝑑 𝑏 < 𝑐. Example 1.12 Solve the inequality 5 < 2𝑥 − 3 < 13. Solution. This can be solved by either of the following methods. Method 1 5 < 2𝑥 − 3 < 13 is equivalent to 5 < 2𝑥 − 3 𝑎𝑛𝑑 2𝑥 − 3 < 13. 5 < 2𝑥 − 3 𝑎𝑛𝑑 2𝑥 − 3 < 13 8 < 2𝑥 2𝑥 < 16 4 < 𝑥 𝑜𝑟 𝑥 > 4 𝑥 < 8 𝑥ȁ𝑥 > 4 ∩ ȁ 𝑥 𝑥 < 8 = { ȁ 𝑥 4 < 𝑥 < 8}
• 27. Method 2. Solve 5 < 2𝑥 − 3 < 13 5 < 2𝑥 − 3 < 13 Add 3 to all parts of the inequality. 8 < 2𝑥 < 16 Multiply the parts by ½. 4 < 𝑥 < 8 Thus, the solution set is { ȁ 𝑥 4 < 𝑥 < 8}.
• 28. Example 1.13 Solve the inequality 2𝑥 − 5 < 9. Solution The definition of the absolute value of a number 𝑎, denoted by 𝑎 , is the real number such that 𝑎 = 𝑎 when 𝑎 ≥ 0. 𝑎 = −𝑎 when 𝑎 < 0. From the definition, we have 2𝑥 − 5 < 9 and − 2𝑥 − 5 < 9 Note that − 2𝑥 − 5 < 9 when multiplied by −1, becomes 2𝑥 − 5 > −9 Thus, we can write the inequalities as 2𝑥 − 5 < 9 and 2𝑥 − 5 > −9
• 29. Thus, we can write the inequalities as 2𝑥 − 5 < 9 and 2𝑥 − 5 > −9 Or −9 < 2𝑥 − 5 < 9 Add 5, −4 < 2𝑥 < 14 Multiply by 1 2 (or divide by 2), −2 < 𝑥 < 7 Thus, the solution set is { ȁ 𝑥 − 2 < 𝑥 < 7}.
• 30. Solving the two inequalities, 2𝑥 − 5 < 9 and − 2𝑥 − 5 < 9 𝑜𝑟 2𝑥 − 5 > −9. Or −9 < 2𝑥 − 5 < 9 Add 5. −4 < 2𝑥 < 14 Multiply the parts by ½. −2 < 𝑥 < 7. Thus, the solution set is { ȁ 𝑥 − 2 < 𝑥 < 7}. Solving the two inequalities, 2𝑥 − 5 < 9 and − 2𝑥 − 5 < 9 𝑜𝑟 2𝑥 − 5 > −9. Or −9 < 2𝑥 − 5 < 9 Add 5. −4 < 2𝑥 < 14 Multiply the parts by ½. −2 < 𝑥 < 7. Thus, the solution set is { ȁ 𝑥 − 2 < 𝑥 < 7}.
• 31. An inequality that is true for all values of the variable is called an absolute inequality. An inequality that is true for some values of the variables but not true for other values is called a conditional inequality.
• 32. Example 1.14 The inequality 𝑥 + 5 < 8 is true only if 𝑥 < 3 and thus, is a conditional inequality. In the inequality 𝑥2 + 1 > 0, the lowest possible value of 𝑥2 is 0, that is, when 𝑥 = 0. For all other real values of 𝑥, 𝑥2 will always be positive. Thus, 𝑥2 + 1 > 0 is true for all real values of 𝑥 and is an absolute inequality.
Current LanguageEnglish
Español
Portugues
Français
Deutsche