Visual Presentation in Algebra for First Year Students

CONTENTS

A premier university in CALABARZON, offering academic programs and related services designed to respond to the requirements of the Philippines and the global economy, particularly, Asian countries

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The University shall primarily provide advanced education, professional, technological and vocational instruction in agriculture, fisheries, forestry, science, engineering, industrial technologies, teacher education, medicine, law, arts and sciences, information technology and other related fields. It shall also undertake research and extension services, and provide a progressive leadership in its areas of specialization.

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In pursuit of the college vision/mission the College of Education is committed to develop the full potentials of the individuals and equip them with knowledge, skills and attitudes in Teacher Education allied fields to effectively respond to the increasing demands, challenges and opportunities of changing time for global competitiveness.


Produce graduates who can demonstrate and practice the professional and ethical requirements for the Bachelor of Secondary Education such as:1. To serve as positive and powerful role models in the pursuit of learning thereby maintaining high regards to professional growth.2. Focus on the significance of providing wholesome and desirable learning environment.3. Facilitate learning process in diverse types of learners.4. Use varied learning approaches and activities, instructional materials and learning resources.5. Use assessment data, plan and revise teaching-learning plans. 6. Direct and strengthen the links between school and community activities.7. Conduct research and development in Teacher Education and other related activities.


This Teacher’s Visual Presentation Hand-out entitled “Developing Skills in Algebra for First Year High School Students” is part of the requirements in Educational Technology 2 under the revised Education curriculum for based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies.

The students are provided with guidance and assistance of selected faculty members of the College through the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials.

The output of the group’s effort may serve as an educational research of the institution in providing effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students.

FOR-IAN V. SANDOVALComputer Instructor/ Adviser

Educational Technology 2

FLORANTE R. DE CASTROModule Consultant

LYDIA R. CHAVEZ

Dean, College of Education


This module that contributes to knowledge in algebra would not be possible without friends, families, teachers and the persons who encourage us to finish this module.

To Prof. Lydia R. Chavez, Dean of College of Education, for her support and motivation that lifts the spirit of the authors,

To Mr. Florante R. De Castro, our module consultant, for lending the authors his time and intelligence that helped a lot in finishing the module,

To Mr. For-Ian V. Sandoval, our adviser, for his guidance and help during the days that the authors find difficulties in completing the module,

To Mrs. Evangeline Cruz, the university librarian, in allowing us to borrow our reference books in the university library,

To Dr. Corazon San Agustin for her support and motivation that helped a lot in finishing this module,

To each member of our families who loves unconditionally and supports us financially,

And finally, all praises and glory be unto God whom we can’t thank enough for realizing the vision of our work.

THE AUTHORS


In pursuit of quality learning of high school student, we designed a module that will help the students develop their skills in Mathematics. This will also extend their learning and attain more knowledge about the topics. We have tried to bring out the basic ideas and techniques as simply and clearly as possible.

Most of the topics are introduced in every chapter. The authors believe that it will help the students to encourage themselves in studying the lessons. Numerous ILLUSTRATIVE EXAMPLES and ACTIVITIES are given in every topic. The authors believe that it will give the students opportunity to practice their mathematical abilities. A CHAPTER TEST and NOTES TO REMEMBER are included at the end of every chapter.

The authors’ aim is to develop the skills of the first year high school students in Algebra.


After reading, understanding and answering all the lessons and activities in this module, the students are expected to: 1. Understand what equality is,2. Apply the properties of equality in solving,3. Understand what inequality is,4. Apply the property of inequality in solving inequalities,5. Define what a linear function is,6. Get the x and y intercepts of the line,7. Identify what are the systems of linear equations,8. Learn ways of solving radical equations,9. Solve equations with two radical terms,10. Understand what matrices are,11. Identify the properties of matrices, and12. Learn how to add and multiply matrices.


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TABLE OF CONTENTS

VMGOTITLE

ACKNOWLEDGMENTINTRODUCTIONGENERAL OBJECTIVESTABLE OF CONTENTS

FOREWORD

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LESSON 1: EQUATIONSCHAPTER 1 - UNDERSTANDING EQUALITIES

LESSON 2: PROPERTIES OF EQUALITY

LESSON 3: SOLVING EQUALITIES IN ONE VARIABLE

CHAPTER 2 : UNDERSTANDING INEQUALITIES

LESSON 4: SOLUTION SET OF INEQUALITIES IN ONE VARIABLE

LESSON 5: PROPERTIES OF INEQUALITIES

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CHAPTER 4 : UNDERSTANDING RADICAL EQUATION

LESSON 11: PERFECT SQUARES AND PERFECT CUBES

LESSON 12: EVALUATING EQUATIONS USING RADICALS

LESSON 13: SOLVING RADICAL EQUATIONLESSON 14: SOLVING RADICAL EQUATION WITH TWO RADICAL EQ...

CONTENTS

LESSON 9 :GRAPHING LINEAR FUNCTION

CHAPTER 3 : UNDERSTANDING LINEAR FUNCTION

LESSON 8: GETTING THE X AND Y INTERCEPT OF THE LINE

LESSON 6: APPLYING THEPROPERTIES OF INEQUALITY

LESSON 7: DEFINING LINEAR FUNCTION

LESSON 10: SYSTEM OF LINEAR EQUATION

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REFERENCES

CONTENTS

CHAPTER 5 : UNDERSTANDING MATRICES

LESSON 16: ADDITION OF MATRICES

LESSON 18: BASIC PROPERTIES OF MATRICES

LESSON 17: MULTIPLICATION OF MATRICES

LESSON 19: PRODUCTS OF MATRICES

LESSON 15 : UNDERSTANDING MATRICES

CHAPTER I : UNDERSTANDING EQUALITIES


In this chapter, we will discuss the equalities and its properties. We will able to:

Understand what equality is, Identify the properties of equality that a given equation

illustrate,Show that properties of equality hold true for any real

number, andUse the properties of equality to transform equations to

equivalent equations.

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Equalities are mathematical sentences or mathematical statements having the symbol equal sign (=). It can be a closed sentence, like 6+2=8, which is a true mathematical sentence. However, it can be a false mathematical sentence like 2x6=8.Therefore, we can describe a closed sentence as a mathematical sentence which is either true or false but not both and having the equality symbol.

Equalities can be also be an open sentence,like3+y=21.Thismathematical sentence can neither be true nor false. We must first identify the value of y, which is called variable, before we can say that the open sentence is true or false.

To be able to understand further what equality is, look at the illustration below.

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In the illustration, we used the closed sentence 6+2=8. It is like a balance or weighing scale wherein fulcrum is represented by the equality sign or the equal symbol. Suppose that in the first box, we placed 6+2 kilos of lanzones and in the second box we placed 8 kilos of rambutan. We can say that the 6+2 kilos of lanzones in the first box and the 8 kilos of rambutan in the second box are equal and balance because 6+2 is equal to 8. This is what we call equality.

6+2 kilos 8 kilos

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In this lesson, we will be able to:Define statements and conditional equation,Determine the equation whether it is satisfied by the given number, andSolve equations with the given value.


Word sentences, such as “The sum of five and two is seven” or “The sum of five and two is nine,” can be labeled true or false. Such sentences are called statements.We can draw a useful analogy between word sentences and the symbolic sentences of mathematics. For example;

5 + 2 = 7 (1) and5 + 2 = 9 (2)

are statements, because we can determine by inspection that (1) is true and (2) is false. x + 3 = 5 is an open sentence because we cannot make a judgment about the truth and falsity thereof until the variable x has been replaced with an element from its replacement set. Symbolic sentences involving only equality relationships, whether statements or open sentences are called equations.A conditional equation is an equation that is not true for every element in the replacement set of the variable.

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Name_________________________ Date:_____________ Year & Section____________ Teacher: ________________

Score: _________________

DIRECTION: Tell whether the equality is an open or closed equality. _______________1. 5x + 8 = 81

_______________2. 18 = 9 + 8

_______________3. 10 + 5 = 15

_______________4. 5x + 9 = 36

_______________5. 3y + 27 = 32

_______________6. 18 + 36 = 54 _______________7. 72x + 15 = 102 _______________8. 13 • 9 = 4 _______________9. 76 = 104 – 28 _______________10. 9 = x + 2

ACTIVITY1

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Score: _________________

DIRECTION: Tell whether the equality is balanced or not. If it is balance, draw a Ü. If not, draw a ü. ________ 1. 8 • 9 = 72 ________ 2. 13 + 26 – 6 = 17 + 15 ________ 3. 27 ÷ 9 + 8 = 16 – 2 • 2 ________ 4. 79 – 14 + 6 = 34 + 37 ________ 5. 11 + 24 ÷ 7 = 5 ________ 6. 42 ÷ 3 = 13 ________ 7. 125 = 25 • 5 ________ 8. 64 ÷ 4 +4 = 20 ________ 9. 68 = 32 • 3 – 32 ________ 10. 165 ÷ 5 • 52 – 1000 + 9 = 724

ACTIVITY2

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Score: _________________

DIRECTION: Tell whether the sentence is a statement or an open sentence. Then, identify if the statement is true statement or false statement.

__________________ 1. x + 9 = 15

__________________ 2. 5 + 5 = 10

__________________ 3. 3 + 9 = 6

__________________ 4. p + 19 = 23

__________________ 5. 6 – 3 = 3

__________________ 6. 10 – 5 = 6

__________________ 7. x – 4 = 10

__________________ 8. 1 - 5 = 4

__________________ 9. 17 – 5 = 12

__________________ 10. y + 9 = 17

ACTIVITY3

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Score: _________________

DIRECTION: Determine the equation if it is satisfied with the given number. Show your solution.

1. x – 3 = 1, by 4 2. 2x – 6 = 3, by 4 3. 3a = 2 = 8 + a, by 3 4. 0 = 6r – 24, by 4 5. 3x – 5 = 2x + 7, by -1 6. 3x + a = 5a, by -2a 7. 2x – 2a = x + a, by 3a 8. x + 2a – x = 2a, by -2a 9. 5x – 1 = 2x + 2, by 1 10. 6 – 2x + 6(2x+1), by 0

ACTIVITY 4

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In this lesson, we will be able to:Know the different properties of equalityName the properties of equalities,Define the different properties of equalities, andUnderstand the properties of equalities.


Equalities have seven (7) properties. and we will discuss it one by one.

REFLEXIVE PROPERTY

It is the first and simplest property of equality. It states that a number is always equal to itself.

a = aIllustrative Example:

3x = 3x 2 = 25x + 8 = 5x + 8 28 = 28

SYMMETRIC PROPERTY OF EQUALITY

This property states that interchanging the right side and the left side of the equality does not change the equality.

a = b, then b = a

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Illustrative Example: 15 = 3x, then 3x = 15 13 = 5 + 8, then 5 + 8 = 13 8 = 2(4), then 2(4) = 8

20 = (4)(5), then (4)(5) = 20

TRANSITIVE PROPERTY

This property states that if the left and right members of an equation are equal to the same quantity, then the two quantities are equal.

If a = b and b = c, then a = cIllustrative Example:

8 + 5 = 13 and 13 = 6 + 7, then 8 + 5 = 6 + 78 • 5 = 40 and 40 = 4 • 10, then 8 • 5 = 4 • 10x = y and y = -4, then x = -4

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ADDITION PROPERTY

This property states that adding the same number to the left and right members of equality does not affect the sum and the sums are equal.It also states that if equal quantities are added to equal quantities, the sums are equal quantities.

a = b then a + c = b + c Illustrative Example:

3 + 5 = 8 5 + 6 = 1110 + (3 + 5) = 8 + 10 12 + (5 + 6) = 11 + 12

18 = 18 23 = 23

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SUBTRACTION PROPERTY

This property states that if the same number is subtracted from the left and the right members of an equation, the differences are equal.

a = b then a – c = b – cIllustrative Example:

94 + 6 = 100 54 + 6 = 60

(94 + 6) – 9 = 100 – 9 (54 + 6 ) – 15 = 60 – 15 91 = 9145 = 45

MULTIPLICATION PROPERTY

This property states that multiplying both members of an equation by the same number does not affect the product.

a = b, then a = bc

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Illustrative Example:(4)(6) = 24 (9)(3) = 27(4)(6) • 3 = 24 • 3 (9)(3) • 5 = 27 • 572 = 72 135 = 135 DIVISION PROPERTY

This property states that if the same nonzero number is divided into the left and right members of an equation, the quotients are equal.

a = b, then a/b=b/c where c≠0 Illustrative Example:

(4)(6)=24 (10)(5)=50(4)(6)/3=24 (10)(5)/2=50 8=8 25=25

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Score: _________________

DIRECTION: Identify the property of equality illustrated in each statement below.

_______________ 1. 13 = 13_______________ 2. If 5 = 2 + 3, then 2 + 3 = 5_______________ 3. If 8 + 9 = 9 + 8, then 9 + 8 = 8 + 9_______________ 4. 16 –5 = 16 – 5_______________ 5. If 6 + 2 = 8 and 8 = 5 + 3, then 6 + 2 = 5 + 3_______________ 6. 18 • 0 = 0 • 18_______________ 7. If 5 = 9 • 4 and 9 • 4 = 5._______________ 8. If (3)(5) = 15, then (3)(5) • 4 = (15)4_______________ 9. If 5 • 6 = 30 and 30 = 3 • 10, then 5 • 6 = 3 • 10._______________ 10. 9 + 7 = (3 + 6) + 7_______________ 11. If (5)(6) = 30, then = _______________ 12. (25 + 8) + 0 = (25 + 8)_______________ 13. If (5)(9) = 45, then = _______________ 14. 15,908 = 15,908_______________ 15. (5+3) - 2 = 8 – 2

ACTIVITY 5

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Score: _________________

DIRECTION: Fill in the blanks the correct expression to make the statement true.1. 12 = ___________ , Reflexive Property 2. If 9 = 3 + 6, then 9 + ___________ = (3 + 6) +14, Addition Property 3. (13 • 5) 6 = 8 (___________), Subtraction Property 4. (12)(3) = 36 , Division Property _____ 4 5. If 8 • 7 = 56 and 56 , then _____________ , Transitive Property 6. (4)(5)(6) = (_____)(6), Multiplication Property 7. 3 + 15 = 18, then 18 = 3 + 5, ________________ Property 8. If (5)(10) = 50, then (5)(10)(3) = (______)(3), Multiplication Property 9. If 15 – 7 = 8, then (15 – 7) – 3 = 8-________ , Subtraction Property 10. 3x + 19 = ______ + 19, Reflexive Property

ACTIVITY 6

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In this lesson, we will be able to:Solve first degree equation,Use the properties of equalities in solving equations in one variable, andUnderstand further the properties of equalities through solving.


By using the properties if equalities, we can solve first degree equation in one variable.

ADDITION PROPERTY OF EQUALITY

If the same number is added to the left and the right members of an equation, the equation remains unchanged.

To be able to get the value of the variable, the variable should be left alone in the left side of the equality sign.

Illustrative Example:1. x-5=8(x-5)+5=8+5 ● 5 is added to both sides of the

equation. In x+0=13 that way, we come up with the value

of x. x=132. x-12=-18 (x-12)+12=-18+12 Same thing with the first

equation. We add 12 x+0=-6 to both members of the equation.

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3. x-3.1=5.7 (x-3.1)+3.1=5.7+3.1 x+0=8.8 x=8.8 4. y-4=10 (y-4) + 4 = 10 + 4

y= 14 SUBTRACTION PROPERTY OF EQUALITY

If the same number is subtracted from the left and right members of an equation, the equation remains unchanged.

We will do the same step that we’ve done in Addition Property of Equality.

Illustrative Example:1. x+4=6 (x+4)-4=6-4 ● Subtract 4 from the left and x=2

right side of the x+0=2 equation to get the value of x

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2. x+12=25 (x+12)-12=25-12

x+0=13 x=13

3. x+1.9=2.6 (x+1.9)-1.9=2.6-1.9 x+0=0.7

x=0.7

4. y + 1.4 = 6.3 (y+1.4) – 1.4=6.3-1.4

y=4.9

MULTIPLICATION PROPERTY OF EQUALITY

If both members of an equation are multiplied by the same number, the equation remains unchanged.

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3. =8 (2.6) =8(2.6) x=96 D. DIVISION PROPERTY OF EQUALITY

If the left and right members of an equation are divided by the same number, the equation remains unchanged.

Illustrative Example:1. 5x=35

x= 7 Both members of the equation are divided by the numerical coefficient of x to change the

coefficient of x to 1. 2. 12y=-72

= y= -6

3. 3.6x=180

= x=50

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Score: _________________

DIRECTION: Solve for x.

1. x – 6 = 9 5. x – 18 = 32

2. x – 9 = -3 6. x – 25 = -42

3. x – 15 = -20 7. x – 5 = 30

4. x – 11 = 9 8. x – 8 =4

9. x – 6.2 = -8.1 13. x – 9=18

10. x – 1.9 = -45 14. x – 34 = 14

11. x – 72 = -95 15. 2x+44=42

12. x – 3.92 = -4.74

ACTIVITY 7

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Score: _________________

DIRECTION: Solve for the value of x.

1. x + 5 = -3 6. x +12 =36

2. x + 20 = 11 7. x +49 =14

3. x + 8 = -5 8. x + 3.7 = 2.2

4. x + 12 = 4 9. x + 1.94 = 0.7

5. x + 32 = 55 10. x + 9.4 = 2.18

ACTIVITY 8

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Score: _________________

DIRECTION: Solve for the unknown value of x.

1. x/2 = -4 6. x/4 = 3

2. x/5 = -2 7. x/18 = -9

3. x/36 = -6 8. x/12 = -8

4. x/19 = 14 9. x/42 = -6

5. x/33 = 3 10. x/11 = -11

ACTIVITY 9

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Score: _________________

DIRECTION: Solve for the unknown variable. 1.. -3x = 27 6. -8p = 112

2. 21x = 84 7. -9x = 54

3. 11x = -99 8. 24n = -120

4. -7y = 105 9. 3.2x = -96

5. 4m = -144 10. 1.1y = -12.1

ACTIVITY 10

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Score: _________________

DIRECTION: Find the value of the variables in each equation.

1. 3x – 4 = 5

2. x- 5=2

3. y+1=3

4. 3a+ 1.2=7a-2.4

5. -2(x-1)+4(y+3)=x+1

6. x-1=3

7. b+7=-2

8. 5x-1.7=5x+1.2

9. -3(x-1)+5(x+2)=x-2

10. c + 18 – 2c = 13c – 18 + 26

ACTIVITY 11

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Score: _________________

A. Identify the property of equality use in each statement. ___________________ 1. 6=6___________________ 2. 5 + 5 =10 and 10 = 9 + 1, then 5 + 5 = 9 + 1___________________ 3. 5(7) = 35 , then 35 = 5(7)___________________ 4. 7 + 8 =15, then, 5 + (7+8)= 15+5___________________ 5. 8 +7=15, then, (8+7) - 5 = 15 -5

B. Find the values of the variables in each equation.1. 3x - 4 = 52. x+5=2

3. y-1=3

4. 3a+ 1.2=7a-2.4

5. -2(x+1)+4(y+3)=x-1

CHAPTER TEST 1

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Properties of Equality1. Reflexive Property of Equality

A number is always equal to itself.2. Symmetric Property of Equality

Interchanging the left member and right member of an equation does not change the sense of equality.3. Transitive property of Equality

If one number is equal to a second number and the second number is equal to the third number, then the 1st and 3rd number are also equal.4. Addition Property of Equality

Adding same number to both sides of an equation does not change the sense of equality.5. Multiplication Property of Equality

Multiplying both sides of an equation by the same numberdoes not change the sense of equality.The properties of equality are very useful in transforming or rewriting an equation into an equivalent one.


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AL-KHOWARIZMI

Arithmetic, in its purest form, deals all the different kinds of real numbers, their properties, and the skills needed for calculating, manipulating, and utilizing them in practical situations. Algebra extends the range and power of elementary arithmetic to include not just the constant quantities called variables.

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CHAPTER II : UNDERSTANDING INEQUALITIES

CONTENTS

In this chapter, we will discuss the inequalities, its properties and finding the solution set of a given inequality. We will able to:

• Understand what inequality is,

• Identify the properties of inequality that a given inequality

illustrate, and

• Apply the property of inequality in solving inequalities.

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Inequalities are mathematical sentences having the unequal sign (≠) which is the symbol for inequality. If two expressions are unequal, then their relationship can be any of the following: greater than ( > ), greater than or equal to ( ≥ ), less than ( < ), less than or equal to (≤). 3 + 2 ≠ 4 is an example of inequality.Like what we have discussed in Chapter 1, when a mathematical sentence contains a variable, it becomes an open mathematical sentence.

Illustrate Example:x – 4 < 3

The statement is neither true or false.But if x = 6, then x – 4 < 3 is true because 6 – 4 < 3.But if x = 10, then x – 4 < 3 is false because 10 – 4 < 3.

When a number replaces a variable to result in a true equation or inequality, that number is a solution. In the illustrative example above, {x = 6} is the solution to the equation x – 4 < 3. The solution set, on the other hand, is the set of all solutions for a given inequality or equation.

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In this lesson, we will be able to:Know that we can use a horizontal number line in illustrating the solutions of an inequality,Find the solution set of an inequality, andGraph the solution set of an inequality on a number line.


The solution of an inequality can be illustrated on a horizontal number line. We can use this number line to understand better how to find the solution set of an inequality.

Illustrative Example:A. Graph the inequality in the set of integers.

x – 4 < 3 x< 3 + 4 x < 7

This translates to: What number minus 4 is less than 3? The solution set is {. . . , 0, 1, 2, 3, 4, 5, 6}, shown with solid dots on the respective coordinates.

The replacement set for x in the previous problem is the set of integers. However, when the replacement set for x is the set of real numbers, the number line with its solution will appear as shown below,

x 7

The ray indicates the set of real numbers without gaps on the number line, and the open circle on 7 indicates that 7 is not included in the solution set.

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Score: _________________

DIRECTION: Tell whether each statement is true or false._____________ 1. 5 5_____________ 2. -3 0_____________ 3. -7 -7_____________ 4. -10 < 0_____________ 5. -4 -4_____________ 6. 3x =3, if x=1_____________ 7. 4+x=-6, if x=2_____________ 8. 2x=10, if x=8_____________ 9. x-3, if x=3_____________ 10. x 0, if x=5

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ACTIVITY11


Score: _________________

DIRECTION: Graph the solution in the set of integers. 1. x-32

2. 6x < 24

3. x+6 <1

4. x+8 >2

5. > 4

6. < 8

7. 5x > 25

8. x < 3

9. 2x > 14

10. 5 < x

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ACTIVITY12

In this lesson, we will be able to:Know that inequalities, like equalities, have properties too, Understand the different properties of inequality, andSolve inequalities using its different property.


Inequalities, like equalities, have properties too. Here are the two properties of inequalities.

A. ADDITION and SUBTRACTION PROPERTY of INEQUALITY

This property assures that addition or subtraction of any real number on both sides of an inequality will not change the sense of inequality.If a > b, then a+c > b+c and a-c > b-c

B. MULTIPLICATION and DIVISION PROPERTIES of INEQUALITY

This property indicates that the sense of an inequality will not change but when the inequality is multiplied or divided by a negative number, the inequality will change.

If a > b and c > 0, then ac > bc and > If a < 0 and c > 0, then ac <bc and < If a > b and c < 0, then ac < bc and <If a < b and c < 0, then ac < bc and >

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Score: _________________

DIRECTION: Solve and graph the following inequality.1. 4x + 12 < 36

2. 5y – 4 > 34

3. 4 (5-3x) ≥ 8x + 60

4. 3x – 6 > 27

5. 6 (r + 4) ≤ 44

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ACTIVITY13

In this lesson, we will be able to:Learn how to use the properties of the inequalities in solving,Apply the properties of inequality in solving, andSolve the equations using the properties of inequalities.


As in equalities, inequalities are solved using the properties of addition, subtraction, multiplication and division. This time, these are called properties of inequalities.

Illustrative Examples:A. For Addition C. For Multiplication

x – 2 > 6 x/6 ≥ 7x– 2 + 2 > 6 + 2 6 . x/6 ≥ 7.6x + 0 > 8 x ≥ 42x > 8

B. For Subtraction D. For Divisionx + 15 > - 7 2x ≤ 8x + 15 – 15 > - 7 – 15 2x/2 ≤ 8/2x + 0 > - 22 x ≤ 4x> - 22

Take note that when you divide/multiply on inequality by a negative number, the sign of inequality ( > or < ) is reversed. For example,-3x > 6 If x = 1, x > - 2-3x/3 >6/3 -3 ( -1 )> 6X < -2 3 > 6 is false.

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DIRECTION: Find the solution set of the following using the properties of inequalities.1. x + 7 > 12. x – 11 > 143. x + > 4. x – 25 > 28 5. 3x < 1.26. x+3.2 6.47. 0.68. x + 4.2 < 2.19. x – 0.6 1.110. > 511. 0.04 x > -2.812. 2.813. < 14. > 3.215. 4x 3.616. x + 4.2 < 2.417. x+ 18. 4.2x < 0.8419. x - 0.9 1.420. 1.6x

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Score: _________________

Chapter Test 2

DIRECTION: A. Graph each inequality on a real number line.1. x > -4

2. x ≥ 9

3. -3 < x < 3

4. x ≠ 8

5. x ≤ -11

6. x-8< -11

7. x + 7 ≥ 0

8. 5x-9 > -2x+5

9. 2 (x-3) ≤ 5x +12

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B. State the inequality describe in each problem.11. Twice a number increased by 15 is at least 7.12. Six less than a number is at most 3.13. The sum of two consecutive integers is greater than 28.14. A number decreased by 7 is greater than -6.15. Thrice the sum of two consecutive integers is at most 13.Solve each problem.16. The sum of two consecutive odd integers is greater then 57. Find the pair with the

least sum.17. If two less than thrice a number lies between -4 and 18 what is the number?18. Mike plans to spend at most Php. 45 for his projects in Makabayan and

Mathematics. He bought materials worth Php. 13.95 for Makabayan. How much can he spend for Mathematics?

19. Jack and Jill weigh at least 124 kg. Kack weighs 65 kg. What is the weight of Jill?20. The Principal’s List includes the name of students who got an average grade of 89,

90, 88 and 92. What must he get in the 5th academic subject to be included in the list?

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PROPERTIES OF INEQUALITY1. Addition/ Subtraction Property of InequalityIf the same quantity is added or subtracted on both sides

of an inequality, the resulting inequality is equivalent to the original inequality.

2. Multiplication Property of InequalityIf the same positive is multiplied to both sides of an

inequality, the resulting inequality is equivalent to the original inequality.

If the same negative quantity is multiplied to both sides of an inequality, the direction of the inequality should be reversed.


Leonhard Euler

Equations and inequalities are basic importance in science, technology, business and commerce. They are mathematical sentences of physical laws, logical relationships, or any other connection between quantities and objects.



In this chapter, we will discuss the linear functions. We will be able to:

Define what is a linear functionGet the x and y intercepts of the line, andKnow what the systems of linear equations

are.

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In this lesson, we will be able to:Define linear function,Understand what a linear function is, andLearn how to express a linear function in the form y=mx + b .


We have learned that a function is a relation. While a relation is a set of ordered pairs, a function is a set of ordered pairs have the same first component. Therefore, linear function (f) is a function who’s ordered pairs satisfy a linear equation and expressed as:{ ( x , y ) | y = mx + b }It may also be expressed f ( x ) = m x + bThe set of all possible x-values is called the domain of the function and the set of all possible y – values is the range of the function.In the notation y = mx + b, m can be determined if again we use table of values, and understand the finite differences in y.

Illustrative Example:

1st differences in y

Consider the form y = mx + b. Substitute values for x and y from the table like:2 = m ( -1 ) + b3 = m ( -2 ) + b-1 = m ( 1 ) ●Subtracting the two -1 = m equations

X -2 -1 0 1 2 3

Y 3 2 1 0 -1 -2

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The same value will be obtained if the procedure above is repeated for any two

consecutive pairs of values. The value of m in the linear function is -1, while the b has the value y when x = 0. In this case, b = 1.

Since their first differences in y are equal, the pair of values in the table illustrate a linear function represented by the equation y = - x + 1.One characteristic of a linear function is that equal differences in x produce equal differences in y.


Since the first difference in y are equal, the values for x and y in the table illustrates a linear function.In the linear function, y=mx+b, m=3 while b takes the clue of y when x=0. in this example, b=2.

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x -2 -1 0 1 2 3

y -4 -1 2 5 8 11


Score: _________________

DIRECTION: Express the following as a linear function in the form y=mx+b.

1. x = y + 2

2. 2x + 5y = 10

3. 3 – 4 y = 8 x

4. 11 – 2y – 2x = 0

5. x = 2y

6. 3 = 15x – y

7. 12y = 24x – 2

8. – 8x + 4y = 2

9. 7x + ( -21y ) = 1

10. y – 1 = -8

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In this lesson, we will be able to:Get the x and y intercepts of the line,Solve for the slope of the line, andUse 0 as a substitute for the intercepts.


The equation y=mx+b is known as the slope-intercept formof the linear function, where m is the slope of the line and b is the y-intercept.The y-intercept is the point where the lines intersect the y-axis. Another way of finding the y-intercept is to solve for y when x=0 in the equation. Similarly, the x-intercept is the point where the line crosses the x-axis. To find the x-intercept, set y and 0 in the equation of the line and solve for x.


Given: 2x-y=3

If y=0, we have 2x-0=32x=3x=3/2 (x-intercept)

If x=0, we have 2(0)-y=30-y=3-y=3y=-3 (y-intercept)

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Score: _________________

DIRECTION: Find the x and y intercept of the following equations. 1. x+y=5 6. 5x-10y=20 2. 4x= -12 7. -36= -x+6y 3. 5x= -5y-10 8. 15y= -45 4. 2y= -11 9. -9x-4y= -36 5. 12y-4=3x 10. -20= -10x

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In this lesson, we will be able to:Graph the linear equations,Identify the x and y intercepts, andSolve the linear equations using graphs.


In grphing linear equations, we must know first how to identify the x and y-intercepts and use it as reference points.

Illustrative Example:Given: If y=0, we have x+0-2=0

x=2( x-intercept)(2,0)

If x=0, we have 0+y-2=0y=2(y-intercept)

(0,2)

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Score: _________________

DIRECTION: Solve for the value of x- and y- intercept and graph the following equations.1. 2x+3y-6=0

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2. 7y=14-2x

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3. -2y=x+4

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4. x-y=6

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5. 8y-10x=20

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In the lesson, we will be able to:Know the different systems of linear equations,Substitute the given value to the equation, andSolve the equations by graphical, substitution and elimination method.


The solution of a system of linear equations consists of an ordered pair or ordered pair of numbers that will satisfy the given equations in a system. A system may consist of two or more equations, and its solution can be arrived at either graphically or algebraically.

A. SOLUTION BY GRAPHING

The first most effective method or way of approaching the solutions of systems of linear equations is graphical method.

Illustrative Ex ample:Given: x-2y=7

x+y=-2

Substitute (1,-3) in the equation where x=1 and y=-3.

X-2y=7 x=y=-21-2(-3)=7 1+(-3)=-21+6=7 1-3=-27=7 -2=-2

In general, two nonparallel lines on a plane intersect at a point. This is consistent system of linear equations whose solution is unique.

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B. SOLUTION BY SUBSTITUTION

An algebraic method used to obtain the solution of systems of linear equations is the substitution method.Using this method, solve for one variable in one equation and substitute that value of the variable in the other equation.


Given: x-2y=7 Equation 1 x+y=-2 Equation 2

Solution: From Equation 1: x=2y+7

Replace x in Equation 2 by 2y+7(2x+7)+y=-2

Solve for y: 2y+7+y=-23y=-2-73y=-9y=-3

With y=-3, solve for x either in Equation 1 or in Equation 2.

in Equation 1: x-2(-3)=7x+6=7x=7-6x=1

The solution is x=1 and y=-3, and the corresponding ordered pair is (1,-3).

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C. SOLUTION BY ELIMINATION

Solving a system of equations by elimination is also known as the Addition/Subtraction method and is most convenient when the coefficients of the corresponding variables have the same absolute value.However, if this is not the case, then, some algebraic manipulation is undertaken to translate either of the equations involved so that the coefficients of x and y will have the same value in the system.

Illustrative Example:Given: x-2y=7 Equation1

x+y=-2 Equation2Solution:

Multiply Equation2by -1: -1(x+y)=-2-x-y=2

Add Equation 1: x-2y=7-3y=9 y=-3

replace y in Equation 1 by -3: x-2(-3)=7x+6=7x=7-6x=1

The solution is x=1 and y=-3, and the corresponding ordered pair is (1,-3).

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Score: _________________

DIRECTION: Graph each equation, then, classify each system of linear equations.1. x+3y=6 x-2y=6 2. 2x+3y=10

2x=8-3y 3. y= -5x+1

y= 4-5x 4. 2y-x+4=0

-2x= -4y-8 5. 5x=5y=15

3y+5x=15

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Score: _________________

DIRECTON: Find the solution of each of the following systems of equations using the substitution method.6. x+y=4

x-y=2 7. 4x+2y=28

4x-2y=4 8. 2x-2y=7

x+y=11 9. 2a=3b

3a-b= -15 10.x-3y=7

X=12+3y

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Score: _________________

DIRECTION: Find the solution for each equation using elimination method.11. 4x+2y=28

4x-2y=4 12. x+y=5

2x-y=4 13. 4x+y=49

3x=2y-10 14. 3x-2y=12

3y= -2x-5 15. x-2y= -13

3x +y= -4

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Score: _________________

DIRECTION: A. When solving linear equations, show all work and show the calculations that check or test your solution.

1. Solve the (scrambled) triangular system3x + 2y + z = 65

3y = 45 6x + y = 75

2. Solve by Gaussian Elimination

3x + 2y = 17 2x - y = 2

3. Solve by Gaussian Elimination x + y + z=12

2x + 3y + 5z = 32 6x + 7y + z =64

4. (a) Solve for x and y

ax = p bx + cy = q

5. Suppose w = x + y + z + 12 and x+y+z= 20. Find the value of w.

Chapter Test 3

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•Function is a relation, while a relation is a set of ordered pairs.So function is a setoff ordered pairs have the same first component.•Linear function is a function who’s ordered pairs satisfy a linear equation and expressed as: y=mx+b.X-values are called domain and y-values are called the range of the function.•Using an Equation to Find an Intercept

To find the y-intercept, you can either put the equation into y=mx+b form in which case b is the y-intercept or you can just plug x =0 into the equation and solve for y and vice versa.

Two nonparallel lines on a plane intersect at a point. This is consistent system of linear equations whose solution is unique.•The pair of linear equations whose graphs consist of distinct parallel lines and which have no common solution is called an inconsistent system of linear equations.•The pair of equations whose graphs are coincident lines has an infinite number of ordered pair as its solution. Every solution of one equation is the solution of the other. Such pair of equations is called a dependent system.


Rene Descartes

Undoubtedly one of the most ingenious and useful inventions of mathematics is the xy-coordinate system, which is formally called the Cartesian Coordinate System, named after its inventor, Rene Descartes. A thorough study of the graphical representation of linear equations in two variables using the Cartesian coordinate system.


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CHAPTER IV : UNDERSTANDING RADICAL EQUATIONS

CONTENTS

In this chapter, we will discuss radical equations. We will able to:

Evaluate if the given values of the variables is a solution of radical equation,

Learn ways of solving radical equations,Solve equations with two radical terms, andUse the power rule twice.

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LESSON 11: PERFECT SQUARES AND PERFECT CUBES

In this lesson, we will be able to:Determine whether the expression is square or

cube,Learn how to read the number representation,

andSolve for the square and cube root.


The square root of a number a is the solution of the equation x2=a. Every positive number a have two square roots: the positive square or principal square root and the negative square root. It is written as x = + to represent the 2 roots. This is read as, “x equals plus and minus the square roots of a.”

Illustrative Examples:

Solve: x2 = 64x = x =

.

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Solve: 4x2 + 16 =0 This equation has no real solution because a < 0.

x2+4=0 x2= -4 -4 has no real square roots.

The cube root of a number a is the solution of the equation x3 =a. It is denoted by .

Illustrative Examples: Solve: x3 = 27 x =

x = 3 Solve: 2x3 – 16 =0

x3 – 8 = 0x3 = 8x = x = 2

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Score: _________________

DIRECTION: Determine whether each expression is a perfect square or perfect cube.

__________________ 1. 1__________________ 2. 125 x6y6

__________________ 3. 27__________________ 4. 9__________________ 5. 64__________________ 6. 1000 x3y12

__________________ 7. 144 x4y2

__________________ 8. 25__________________ 9. 16__________________ 10. 125__________________ 11. 625__________________ 12. 512 x6y12

__________________ 13. 225 x2y8

__________________ 14. 4913__________________ 15. 343

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Score: _________________

DIRECTION: Solve the following.1. 6. 2. 7. 3. 8. 4. 9. 5. 10.

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LESSON 12: EVALUATING EQUATIONS USING RADICALS

In this lesson, we will be able to:Show if the equation is true or false,Evaluate the equation for given value, andSubstitute the given value and to simplify.


We can show an equation is true or false by evaluating the equation for given values of the variables for a given replacement set.Illustrative Example:1. 4x+2=3x when x=(2,3,-2)

Substitute x by each value in the replacement set and simplify.If x=2 4(2)+2=3(2)

8+2=6 10 6 False

If x=3 4(3)+2=3(3)

12+2=9 14 9 False

If x= -2 4(-2)+2=3(-2) -8+2=-6 -6= -6 True

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Therefore, {-2} is the solution of the equation and {2,3} are not solutions of the equation.

2. = 3 when x=(-2,2)Substitute x by the given value and simplify

If x= -2 =3 =3 2.24 3 False

If x=2 =3

=3 3=3 True

Therefore, {2} is the solution of the equation and {-2} is not the solution of the equation.

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Other Examples:1. x= when x= (-2,3)

If x= -2 -2= -2= -2=2 False

If x=3 3= 3= 3=3 True

2. x+3= when x= (-1,-2)

If x= -1 -1+3= 2= 2= 2=2 True

If x= -2 -2+3=

1= 1= False

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Score: _________________

DIRECTION: Evaluate the following statements and identify which value is true.1. a = when a = ( -1, 1 , 2 , 6 ) 2. = when x = ( -7 , 7 ) 3. = 3 when x = ( 0 , 8 , 11 ) 4. 6 + = y when y = ( 4, 9 ) 5. = 8 when x = ( 4 , 20 )

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LESSON 13: SOLVING RADICAL EQUATION

In this lesson, we will be able to:Identify the steps in solving radical equations,How to solve radical equations, andFollow the guide in solving radical equations.


An equation in which the variable appears under a radical sign is called a radical equation.

To be able to solve radical equations, follow this step/guide.

1.Square both sides of equation and let x be left alone on the left side of the equation.

Illustrative Example:a. = 9( )2 = ( 9 )2

X = 81

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b. 6 - 5 = ( 6 – 5 )2 = ( )2

1 = x or x = 1 c. ( )2 = ( 3 )2

= 9 x = 63

REMEMBER!

To solve radical equations,

1. Isolate the radical, they should be on the left side of the equation,

2. Apply the power rule,

3. Solve the resulting equation, and

4. Check it.

REMEMBER!

To solve radical equations,

1. Isolate the radical, they should be on the left side of the equation,

2. Apply the power rule,

3. Solve the resulting equation, and

4. Check it.

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Illustrative Examples:1. + 6 = a

a. Add -6 to both sides6 + ( - 6 ) = a + ( -6 )

= a – 6b. Square both sides

( )2 = ( a – 6 )2

a= a2-12 a+36c. Add –a to both sides

a+(-a)=a2-12a+(-a)+36d. Simplify the equation

0= a2-13a+36e. Solve the quadratic equation

a2-13a+36=0 by factoring( a-9 ) ( a – 4 ) = 0

f. Apply the principle ab = 0 if and onlya – 9 = 0 | a – 4 = 0

if a = 0 or b = 0 a = 9 | a = 4

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Checking:Substitute a = 9 in the original equation same with a = 4 + 6 = a

a = 9 b. a = 4 + 6 = 9 + 6 = 4 3 + 6 = 9 2 + 6 = 4

9 = 9 True 8 ≠ 4 False

Therefore { 9 } is the equation and { 4 } is not the solution of the equation.

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- 2 = 0a. Add +2 to both sides - 2 + 2 = 0 + 2b. Simplify c. Square both sides

=(2)2

d. Add -5 to both sides 3x+5=4

3x+5+(-5)=4+(-5)e. Simplify 3x= -1f. Multiply both sides by

3x( )= -1( )

x= -

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Checking:Substitute x= - to the original

equation - 2 + 2 = 0 -2=0 2-2=0

0=0

Therefore the solution set is {- }.

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Score: _________________

DIRECTION: Fill in the blanks with the missing data. Perform the shortcut method. (5pts each)

1. Solvea. Add _____________ to both sides

________________and simplify ________________b. ________________ both sides ________________c. Simplify the equation ________________

Checking:

Substitute ____________ ________________in the original equation.Did it make the equation true? Yes NoTherefore the solution set is _____________________.

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2. Solve +2=na. Add _____________

to both sides ________________and simplify ________________

b. ________________ both sides ________________

c. Simplify the equation ________________ Checking:

Substitute ____________ ________________in the original equation.Did it make the equation true? Yes NoTherefore the solution set is _____________________.

3. Solve = 10a. ______________ ________________ b. ______________ ________________c. ______________ ________________d. _____________ ________________

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Checking:Substitute ____________ ________________in the original equation.Did it make the equation true? Yes NoThe solution set is _____________________.

4. Solve =18a. ______________ ________________ b. ______________ ________________c. ______________ ________________d. _____________ ________________

Checking:Substitute ____________ ________________in the original equation.Did it make the equation true? Yes NoThe solution set is _____________________.

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LESSON 14: SOLVING RADICAL EQUATION WITH TWO RADICAL

EQUATION

In this lesson, we will be able to:Use the power rule twice,Solve radical equation with two radical

terms, andSimplify the radical equations.


In solving radical equation with two radical terms, we will use the power rule twice, that is squaring both twice, that is, squaring both sides of the equation twice. But there are some things that we should remember. Do not omit the middle term when the radical equation requires squaring the binomial.

Illustrative Example:(x+2)2 x2+4 but (x+2)2= x2+4x+4(2)(a+b)2 is not the same as a2+b

If a=3 and b=3(3+3)2=62 (3)2+(3)2=9+9

=36 =18

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Illustrative Example:1. Solve the radical equation =

a. Square both sides( =

b. Simplify (6n+5)=(2n+10)

c. Add -5 to both sides 6n+5+(-5)=2n+10+(-5)

d. Simplify 6n=2n+5e. Add -2n to both sides

6n+(-2n)= 2n+(-2n)+5f. Simplify 4n=5g. Multiply both sides by

4n ( )= 5 ( )h. Simplify

n=

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Score: _________________

DIRECTION: Simplify the following radical equations and show your solutions.1. =

2. =

3. - = 3

4. =

5. =

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Score: _________________

DIRECTION: A. Simplify the following radical expressions.

1. Simplify

2. Simplify

3. Simplify

4. Simplify

5. Simplify

CHAPTER TEST 4

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DIRECTION: B. Simplify the following radical expressions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

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Radical is an expression consisting of a radical sign and the radicand. Radical sign is the sign which indicates the root of a number. Radicand is a number inside the radical sign or the number whose root is being

considered. Index is a small number written in the upper left of the radical sign. An equation in which variable occurs under a radical sign is called a radical

equation.In solving radical equations, examine the equation carefully before raising both sides

to an exponent. Applying the property of power may produce a more complex equation that still contains a radical.

If there is one radical term, isolate the radical or place it on the left side of the equation.

If there is two radical terms, place one radical on the left side and the other radical on the right side of the radical equation.

Checking the obtained values or potential solutions with the original equation is a must to be able to identify extraneous solutions.

Extraneous solutions are solutions that satisfy the derived equation but not the original equation.


Niels Henrik AbelIn 1824, Norwegian mathematician Abel proved that equations of

degree higher than four cannot be generally solved using four basic arithmetical operations and radicals. These facts clearly contradict the mentioned above statement by the author of the review. Moreover, even though Cardan, Ferrari, and Abel made their discoveries while in Christian-dominated countries, no one ever calls them "algebraists of medieval Christianity." Religion and mathematics have been rather.


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CHAPTER V : UNDERSTANDING MATRICES

CONTENTS

In this chapter, we will discuss matrices and its basic properties. We will be able to:

Understand what matrices are,Identify the properties of matrices, andApply the basic properties of matrices in solving

matrices.

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LESSON 15: INTRODUCTION TO MATRICES

In this lesson, we will be able to:Know what matrices are,How to use the ordered pair, andHave an idea about matrices.


If you were asked for your weight in pounds, you would use a real number such as 140 to answer the question. If you were asked for your height in inches, you would answer with another real number such as 66.5. If we asked these questions to everyone in the class, we would want some way to know which weight goes with which height. One way to organize this data is to use an ordered pair. We could represent your weight and height with the ordered pair (140, 66.5). This is called an ordered pair because we always list the information in the same order. In other words, we list weight first and then height in every pair of numbers, so (140, 66.5) would be different from (66.5, 140). The elements are the individual pieces of information.

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Elements are also referred to as entries or components. In this book, we will only use real numbers as elements. The elements of this ordered pair are 140 and 66.5. We could also ask you for your age in years and append that information so that we have the ordered triple (140, 66.5, 18). We could ask you for n pieces of information, where n is any counting number. If we arrange the n pieces of information in a specific order, we call it an ordered n-tuple. In general, lists of ordered information are called vectors. If we write them in rows, as we did above, we call them row vectors. If we write them in columns, such as

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and we call them column vectors.

Definition 1.1 A real n-vector is an ordered n-tuple of real numbers. The real numbers are called the elements of the vector. Since we are only working with real numbers in this book, we will

drop the word real when referring to vectors. When it is not important to specify how many elements are in the vector, we drop the qualifier n.

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Remark 1 Did you notice that we used parentheses on some vectors and brackets on others? Actually, both are accepted notations, but we will use brackets for consistency throughout the rest of the book.

Remark 2 Sometimes you will see the elements of a row vector separated by commas. Commas are not necessary unless confusion can arise without the use of commas.

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Score: _________________

DIRECTION: Answer the following questions.1. For a matrix A, what is the transpose of AT?2. Does a symmetric matrix have to be square?3. Are all square matrices symmetric?

B. Answer the following.1.

a. Form a 4 by 5 matrix, B, such that bij = i*j, where * represents multiplication.

b.What is BT?

c. Is B symmetric? Why or why not?

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2. Using matrix A below, spell words by replacing each element requested with the letter in that position of the matrix. For example, a52a21a32represents cat.

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1.a53a21a32a24a14a52a12a43a21a24a12a12a21a43a33

2.a34a24a14a12a35a55a43

3.a52a21a13a52a44a13a21a32a41a24

LESSON 16: ADDITION OF MATRICES

In this lesson, we will be able to:Know how to add matrices,Follow the rules on how to add matrices, andLearn how to add matrices in the right way.


If the Cardinals won 7 games in the first half of the regular season and won 8 in the second half, how many games did they win during the regular season? You know that the answer is 15 because 7 + 8 = 15. The Eagles lost 8 games in the first half and lost 6 in the second half of the season. How many games did the Eagles lose all season? They lost 14 games. We know how to answer these questions using real numbers because we have represented our data by real numbers, and addition, subtraction, and multiplication are all defined and well-known operations for real numbers. However, how would we add when our information is represented by matrices?

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Let the matrix A represent the statistics from the first half of the season, and let the matrix B represent the statistics from the second half of the season.

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Look carefully at how you answered the questions above. Then look at where those numbers appear in the matrices. How would you add A + B?

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Definition 2.1 Matrices of the same dimensions are added by adding corresponding elements.

For instance, aij corresponds to bij because they both lie in the ith row and jth column of their respective matrices. Therefore, we would add, aij +bij to obtain the (i,j)th element of A + B.

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Definition 2.2 Matrices of the same dimensions are subtracted by subtracting correspondingelements. Suppose Y represents the wins, losses, and ties for these teams for the entire season (regular season and the playoffs together). Consider the following data

How would you find the number of wins, losses, and ties for the playoffs? We would subtract the number of wins, losses, and ties for the regular season from the number of wins, losses, and ties for the entire season.


Score: _________________

DIRECTION: Answer the following problems.1. Using the following matrices, perform the operation indicated when it is defined and

state that the operation is not defined for the particular matrices when that is the case:

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1. A + C

2. D + E

3. F – D

4. F + B

5. B - (A + C )

6. D - (E + F)

7. B + C – B

8. A – D

9. A + DT

10. D + E - BT

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In this lesson, we will be able to:Know how to multiply matrices,Follow the steps in multiplying matrices, andLearn the right way to multiply matrices.


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We have three recipes for breakfast foods. Each recipe feeds three people. The ingredients are as follows:

Pancakes: 2 cups baking mix, 2 eggs, and 1 cup milk.

Biscuits: cups baking mix and cups milk.

Waffles: 2 cups baking mix, 1 egg,

cups milk, and 2 tablespoons

Let's write this in the form of a labeled matrix so that it is easier to read.

vegetable oil.

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If we want to feed 6 people instead of 3, what do we need to do? We double each recipe. That means we need twice asmuch of each ingredient, so we multiply every element of the matrix by the number 2.

When we multiply a matrix by a real number, we call the real number a Scalar and call the operation scalar multiplication.

Scalar multiplication consists of multiplying each element of a matrix by a given scalar.

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We use the terms scalar and scalar multiplication because, in abstract algebra, we often have the need to consider more general scalars than real numbers. However, in this book, we restrict our attention to scalars that are real numbers.

Definition 3.1 If c is a real number and A is a matrix whose (i,j)th element is aij, then the scalar product cA is the matrix whose (i,j)th element is caij.


Score: _________________

This is a good place to use your calculator if it handles matrices. Do enough examples of each to convince yourself of your answer to each question. If your calculator does not handle matrices, or if you want a more mathematical argument, use generic matrices and carry out these operations like we did in the addition section.

DIRECTION: A. Answer these questions on your own before you read beyond this paragraph. Remember to consider the dimensions of the matrices.

1. Consider . Does AB = BA for all B for which matrix multiplication is defined?

2. In general, does AB = BA?3. Does A(BC) = (AB)C?4. Does A(B + C) = AB + AC?5. Does (AB)T = BTAT?

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Score: _________________

DIRECTION: Answer the following problems.1. The matrix below expresses the approximate distance, in miles, between any of the

following two cities: Houston, Los Angeles, New York, and Washington DC.

a. What special kind of matrix is this (other than square and 4 by 4)?b. If we want to know the same information in kilometers, what should we do?

Remember, for our purposes here, one mile is equal to 1.6 kilometers.c. What is the resulting matrix when you perform the operation that you suggested in

part (b)?

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2. Perform the operations requested below if they are possible using these matrices.

a.4C f. C (A + B)

b. AD g. AB

c. DA h. BA

d. BC i. CAD

e. 3CB j. DBC

LESSON 18: BASIC PROPERTIES OF MATRICES

In this lesson, we will be able to:Know what are the basic properties of

matrices,Determine if the two matrices are equal, andApply these basic properties in solving

matrices.


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When we say matrix, we mean any rectangular array of numbers, for example

The numbers 1,2,-3 and so on appearing in the matrix are called entries or the elements of the matrix. In this chapter, we will confine out attention to those matrices in which the entries are real numbers, the horizontal lines of numbers are called rows, while the vertical lines of numbers are called columns. In general, if a rectangular array has m rows and n columns. So in the examples above, the first is a 2 by 3 matrix, while the other one is 3 by 1 matrix.

or

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We shall use capital letters to denote matrices, and shall enclose the actual matrix in square brackets.Illustrative Example:

or

In the examples above, we can’t say that D=E. To determine whether two matrices are equal, we should consider the following conditions:Two matrices are equal if and only if1. The two rectangular arrays have the same numbers of rows and columns, and2. Their corresponding entries are equal.

D=

E=


Score: _________________

DIRECTION: Find the values of a,b,c and d in the Problems 1-5.

1.

2. =

3. =

4. =

5. =

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=

ACTIVITY31

In this lesson, we will be able to:Define an operation of multiplication of one matrix by another,Presenting single linear equation in a product form,andSolving for the products of matrices.


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In the previous lesson, we defined the operation of addition and subtraction of matrices with the same numbers of rows and columns, and we defined an operation of scalar multiplication of a matrix by real number. In this lesson, we will define an operation of multiplication of one matrix by another. First of all we present a way of representing a single linear equation in a product form, and then show how matrices may be used to represent systems of equations. To represent a system of equations in matrix form, we write

=


Score: _________________

DIRECTION: Carry out the indicated matrix multiplication in each of the following problems.1.

2.

3.

4.

5.

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ACTIVITY32


Score: _________________

DIRECTION: Solve the following matrices.

1. +

2.

3.

4.

5. +

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CHAPTER TEST 5

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B. Solve. 6. 3A 7. B+C

8. DE 9. CB

10. E+D

A = B = C = D = E =

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A matrix is a rectangular array of numbers or elements of a ring. One of the principal uses of matrices is in representing systems of equations of the first degree in several unknowns. Each matrix row represents one equation, and the entries in a row are the coefficients of the variables in the equations, in some fixed order.

Addition and multiplication of matrices can be defined so that certain sets of matrices form algebraic systems. Let the elements of the matrices considered be arbitrary real numbers, although the elements could have been chosen from other fields or rings. A zero matrix is one in which all the elements are zero; an identity matrix, Im of order m, is a square matrix of order m in which all the elements are zero except those on the main diagonal, which are 1. The order of an identity matrix may be omitted if implied by the text, and Im is then shortened to I.

CONTENTS

James Joseph SylvesterThe term "matrix" for such arrangements was

introduced in 1850 by James Joseph Sylvester. Sylvester, incidentally, had a (very) brief career at the University of Virginia, which came to an abrupt end after an enraged Sylvester hit a newspaper-reading student with a sword stick and fled the country, believing he had killed the student!


BOOKSAlferez, M. S. Quick Math Review. Gepress Printing. Benigno, Ph. D., G. D. Basic Mathematics for College Students (Revised Ed). Rex Bookstore. Bernabe, J. G. Elementary Algebra, Textbook for First Year. JTW Corporation. Dasco, N. T. Intermediate Algebra (Mathematics II). Academic Publication. Marquez, L. Mathematics beyond 2000. Vibal Publishing House. Orines, F. B. Elementary Algebra. Phoenix Publishing House. Padua, R. N. , Adanza, E. G.Contemporary College Algebra with Applications. Rex Bookstore. Vance-Addison, E. P.. Modern Algebra (3rd Ed.) Addison-Wesley Publishing Company, Incorporated.


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