9 Mathematics

Quarter 1 - Module 5

Solving Quadratic Inequality

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Department of Education ● Republic of the Philippines

Math- Grade 9

Alternative Delivery Mode Quarter 1 - Module 5:Solving Quadratic Inequalities

First Edition, 2020

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Mathematics

Quarter 1 - Module 5

Solving Quadratic Inequality

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Department of Education ● Republic of the Philippines

9

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Table of Contents

What This Module is About i

What I Need to Know ii

How to Learn from this Module ii

Icons of this Module iii

What I Know ( Pre-Assessment) iii

Lesson 1:

(Solving Quadratic Inequalities)

What I Need to Know 1

What I know 1-3

What‟s In 4

What‟s New 5

What Is It 6-1 2

What‟s More 13-15

What I Have Learned 16-17

What I Can Do 18 -19

Summary 2 3 Assessment: (Post-Test) 20-22 Key to Answers 2 4 – 2 6

References 2 7

What This Module is About

This module talks about quadratic inequalities and their solution sets and graphs. The

module also provides opportunities to the learners to describe quadratic inequalities and their

solution sets using practical situations, mathematical expressions and their graphs. Moreover, it

provides opportunities to draw and describe the graphs of quadratic inequalities and to apply

the concepts by doing performance task.

What I Need to Know This lesson aims to assess your knowledge of previous mathematical concepts

and skills in performing mathematical operations. These prior knowledge and skills will help you understand and solve quadratic inequalities. As you go through this lesson you will also learn how to apply this and make decisions in real-life problems. You will be given a series of activities as guides to let you understand more about this topic.

How to Learn from this Module

To achieve the objectives cited above, you are to do the following:

• Take your time reading the lessons carefully.

• Follow the directions and/or instructions in the activities and exercises diligently.

• Answer all the given tests and exercises.

Icons of this Module

What I Need to This part contains learning objectives that Know are set for you to learn as you go along the module.

What I know This is an assessment as to your level of knowledge to the subject matter at hand, meant specifically to gauge prior related knowledge

What‟s In This part connects previous lesson with that of the current one.

What‟s New An introduction of the new lesson through various activities, before it will be presented to you

What is It These are discussions of the activities as a way to deepen your discovery and under- standing of the concept.

What‟s More These are follow-up activities that are in- tended for you to practice further in order to master the competencies.

What I Have Activities designed to process what you Learned have learned from the lesson

What I can do These are tasks that are designed to show- case your skills and knowledge gained, and applied into real-life concerns and situations.

Lesson

1

Solving Quadratic Inequalities

What I Need to Know

This lesson aims to assess your knowledge of previous mathematical concepts and skills in performing mathematical operations. These prior knowledge and skills will help you understand how to illustrate and solve quadratic inequalities. As you go through this lesson you will also learn how to apply this and make decisions in real-life problems.

What I Know

PRE- ASSESSMENT Directions: Find out how much you already know about this lesson. Encircle the letter that you think best answers the question. Please answer all items. If you were not able to answer correctly you can find out the right answer as you go through the lesson.

1. It is a polynomial of degree 2 that can be written in the form 02 cbxax , where a,

b, and c are real numbers and 0a . Symbols > , , and may also be used in place

of 0?

A. { x | x > 5 or x < –4} C. { x | –4 < x < 5}

B. { x | x > –5 or x < 4} D. { x | –4 < x < -5}

1

5. Solve the following quadratic inequality. x2 -x -4 2

A. -2 < x < 3 C. -3 < x < 2

B. -2 x 3 D. x -2 , x 3

6. What is the solution of x2 + 7x -8 > 0? A. x = -8 or x = 1 C. -8 < x < 1 B. x < -8 or x > 1 D. x < -1 or x > 8

7. Solve the inequality. 01072 xx

A. (0,-2)U(3,∞) C. (-∞,2)U(5 ,∞) B. (-∞,2]U[-2,3] D. [2,5]U[3,∞)

8. Which inequality is shown?

A. y< -x2 C. y≤ -x2

B. y> -x2 D. y≥ -x2

„9. Which inequality that describes the graph below ?

A. y-2x

2-8x-12

B. y≤-2x2-8x-12 D. y≥-2x2-8x-12

2

10. Using the graph of y= x2- 3x- 10 . What is the correct way to write the solution of x2- 3x- 10 0?

A. 52 xx C. 52 x B. 52 x D. 52 xx

What’s In

In grade 8, you studied linear inequalities and were able to identify and solve

problems involving it. In the previous lesson you studied quadratic equations. Now, you

will study quadratic inequalities. Start by doing the activity below.

Activity 1 Direction: Answer what is asked in each number given a mathematical sentence. 1. n – 3 = 10 , What is the value of n?

2. n - 3 < 10, What are the possible values of n to make the statement true?

Show your solution.

3. n – 3 10, What are the values of n?

4. 4f - 2 13, What is the meaning of the symbol ? You may illustrate your answer.

Find the value/s of f to make the statement true.

3

5. b2 + 5b + 6 = 0 ,

What do you call this mathematical statement?

How did you find the solution/s of this expression? How many solutions did you get?

What’s New

Activity 2 2x2 + 7x + 5 > 0 15 - 6x2 = 10 t2 = 6t - 7 4m2 - 25 = 0 p2 + 10p + 16 0 2t2 < 21 - 9t f2 + 9f + 20 = 0 3b2 + 12b 0

Quadratic Equation Not a Quadratic Equation

Questions

1. How do you describe a quadratic equation? a non quadratic equation?

2. How can you differentiate quadratic and non quadratic equation?

4

What Is It

Quadratic Inequality is an equality that contains a polynomial of degree 2 and

can be written with the symbols > , < , ≥ or ≤.

Examples: 1. x2 + x -12 0 3. 5 ≥ x2 − x

2. 7x2 -28< 0 4. 2y2 + 1 ≤ 7y

How to solve quadratic inequality?

Quadratic inequalities can be solved using A. Three Test Points; B. Sign Graph; and C. Graphing ( in two variables)

Study the examples given below. Example 1: Find the solution set of x2 + 3x > 10 Solution

5

❏ Test a value from each interval in the

inequality.

Intervals in the number

line

x x2 + 3x - 10 = 0 True or False

x < -5 -6 x2 + 3x - 10 > 0 (-6)2 + 3(-6) -10 > 0 36 - 18 -10 > 0 8 > 0

True

-5 < x < 2 1 x2 + 3x - 10 > 0 (1)2 + 3(1) -10 > 0 1 + 3 - 10 > 0 -6 > 0

False

x > 2 4 x2 + 3x - 10 > 0 (4)2 + 3(4) -10 > 0 16 + 12 - 10 > 0 18 > 0

True

❏ Test the roots x= -5 and x = 2 satisfy the inequality.

for x = -5 for x=2

x2 + 3x - 10 > 0 (-5)2 + 3(-5) -10 > 0 25 - 15 - 10 > 0 10-10 > 0 0 > 0 False

x2 + 3x - 10 > 0 (2)2 + 3(2) -10 > 0 4 + 6 - 10 > 0 10 - 10 > 0 0 > 0 False

Therefore, the inequality is true for any value of x in the interval < x < -5 or 2 < x < but points -5 and 2 does not satisfy the inequality x2 + 3x - 10 > 0.

The solution set of the inequality x2 + 3x > 10 is * +.

6

Method B The Sign Graph

❏ Write the quadratic inequality in standard form.

Given: x2 + 3x > 10 Standard Form: x2 + 3x - 10 > 0

❏ Factor the quadratic inequality. x2 + 3x - 10 > 0 ( )( )> 0

Illustrate a sign graph that shows the signs of each factor.

for x + 5 -------------- +++++++++++ -----------

for x - 2 -------------- +++++++++++ -----------

❏ Apply the rules of signs for multiplying sign numbers to determine which area satisfies the original inequality.

The product of x + 5 and x - 2 is positive if the factors are both positive or negative. These are possible to happen in the regions where x < -5 or x > 2.

Example 2: Find the solution set of x2 + 2x - 3 0 Solution Method A Three Test Points

❏ Write the quadratic inequality in standard form.

Given: x2 + 2x - 3 0

Already in standard form

❏ Find the roots of its corresponding equality.

x2 + 2x - 3 = 0 ( )( )= 0

( ) = 0 or ( ) = 0

❏ Find the 3 test points using -5 and 2.

The points at -3 and 1 can separate the real number line into three intervals: x < -3 , -3 < x < 1 , x > 1

7

❏ Test a value from each interval in the inequality.

Intervals in the number

line

x x2 + 2x - 3 0 True or False

x < -3 -4 x2 + 2x - 3 0 (-4)2 + 2(-4) -3 0 16 - 8 - 3 0 5 0

False

-3 < x < 1 0 x2 + 2x - 3 0 (0)2 + 2(0) -3 0 0 + 0 - 3 0 -3 0

True

x > 1 2 x2 + 2x - 3 0 (2)2 + 2(2) -3 0 4 + 4 - 3 0 5 0

False

❏ Test the roots x= -5 and x = 2 satisfy the inequality.

for x = -3 for x=1

x2 + 2x - 3 0 (-3)2 + 2(-3) -3 0 9 - 6 - 3 0 3 - 3 0 0 0 True

x2 + 2x - 3 0 (1)2 + 2(1) -3 0 1 + 2 - 3 0 3 - 3 0 0 0 True

The solution set of the inequality x2 + 2x - 3 0 is * +.

8

Method B The Sign Graph

❏ Write the quadratic inequality in standard form.

Given: x2 + 2x-3 Already in standard form

❏ Factor the quadratic inequality. x2 + 2x - 3 0 ( )( ) 0

Illustrate a sign graph that shows the signs of each factor.

for x + 3 --------------------- ++++++ ++++++++++

for x - 1 -------------------- --------- +++++++++

❏ Apply the rules of signs for multiplying sign numbers to determine which area satisfies the original inequality.

The product of x + 3 and x - 1 is negative if the factors have different signs. These are possible to happen in the regions

where -3 Note: -3 and 1 are included in the solution because it satisfies the equation.

❏ Solution The solution of the inequality x2 + 2x-3 is * +.

9

There are quadratic inequalities that involve two variables. These inequalities can be written in any of the following forms below, where a, b, and c are real numbers and a .

y > ax2 + bx + c y ax2 + bx + c

y ax2 + bx + c y ax2 + bx + c

The solution set of quadratic inequalities in two variables can be determined

graphically. To do this, write the inequality as an equation, then show the graph. Take note of the image of the following graphs for each quadratic inequality. y

Steps for graphing inequality:

1. Write the inequality to its corresponding equation.

2. Find the vertex of the parabola.

3. Construct table of values for x and y.

4. Sketch the parabola y= ax2 + bx + c (dotted line for < or > , solid line for ; opens up if the coefficient of x is positive or opens downward if negative) 5. Choose a test point and see whether it is a solution of the inequality.

6. Shade the appropriate region. (if the point is a solution, shade where the point is, if it‟s not a solution, shade the other region)

10

Example 1: Find the solution set of y - x2 + 4x - 3

❏ Write the inequality to its corresponding equation.

y - x2 + 4x - 3 y - x2 + 4x - 3

❏ Find the vertex of the parabola.

● Coordinates of the vertex

x =

= ( )

( )=

( ) = 2

to get y substitute 2 to the quadratic expression - x2 + 4x - 3

y = -( )2 + 4( ) - 3 = + 8 - 3 y = 4 -3 = 1 y= 1 V = (x, y) = (2 , 1)

❏ Construct table of values for x and y.

x 0 1 2 3 4

y -3 0 1 0 -3

❏ sketch the parabola y= - x2 + 4x - 3

Note: ● The parabola opens

downward because the coefficient of x is negative..

● Use dotted line because it has a > symbol

❏ Choose a test point and see whether it is a solution of the inequality.

Test point: (0, 0) Substitute (0,0) to the inequality y > - x2 + 4x - 3 y > - x2 + 4x - 3 0 > - (0)2 + 4(0) - 3 0 > 0 + 0 - 3 0 > -3 True

❏ Answer Therefore, the entire region containing (0,0) does not represent the solution set and we shade the outside region of the parabola.

11

Example 2: Find the solution set of y -x2 + 6x – 4

❏ Write the inequality to its corresponding equation.

y x2 + 6x - 4 y x2 + 6x - 4

❏ Find the vertex of the parabola.

● Coordinates of the vertex

x =

= ( )

( )=

( ) = -3

to get y substitute -3 to the quadratic expression x2 + 6x - 4

y = ( )2 + 6( ) - 4 = - 18 - 4 y = -9 -4 = -13 y= -13 V = (x, y) = (-3 , -13)

❏ Construct table of values for x and y.

x -1 -2 -3 -4 5

y -9 -12 -13 -12 -9

❏ sketch the parabola y= x2 + 6x - 4

Note: ● The parabola opens

upward because the coefficient of x is positive..

● Use dotted line because it has a > symbol

❏ Choose a test point and see whether it is a solution of the inequality.

Test point: (0, 0) Substitute (0,0) to the inequality y x2 + 6x - 4 y x2 + 6x - 4

❏ Answer Therefore, the entire region containing (0,0) represents the solution set and we shade it.

12

test point

What’s More

Activity 3 Direction: Fill-in the table below to find the solution set of each of the following quadratic inequalities.

1) x2 - 9x + 14 0 (Use three test point method.)

Write the quadratic inequality in standard form.

Find the roots of its corresponding equality

Find the 3 test points

Test a value from each interval in the inequality.

Test the roots

Solution set

13

2) 0122 xx (Use the sign graph method.)

Write the quadratic inequality in standard form.

Factor the quadratic inequality.

Illustrate a sign graph that shows the signs of each factor.

Apply the rules of signs for multiplying sign numbers to determine which area satisfies the original inequality.

Solution Set

14

Activity 4 Directions: Find the solution set of each of the following quadratic inequalities then graph and explain your answer.

1. 0132 2 xx

2. x2 - 2x 15

15

What I Have Learned

Activity 5 Direction: Determine whether the indicated ordered pair is a solution to the quadratic inequality y < x2 + 4x - 5 . Justify your answer.

1. A (-2, 5 ) 2. B ( 6, -2 ) 3. C (-3, 2 ) 4. D (1, -1 ) 5. E ( 2, -2 )

Activity 6 Direction: Match from the list of mathematical sentences the inequality that is described by the given graphs. Answer the questions that follow.

1. 2. 3. __________________ __________________ ___________________

y x2 - 4x + 1 y x2 - 4x + 1 y x2 - 4x + 1 y -x2 - 4x + 1

Questions:

1. What are your hints to determine the quadratic inequality that is described by a given graph?

2. How do you know if the graph opens upward or downward? 3. In each graph, what does the shaded region represent? What does the dashed line

and solid line represent? 4. How would you describe the graphs of quadratic inequalities in two variables

involving “less than”? “greater than”? “less than or equal to”? “greater than or equal to”?

5. How are you going to graph if you are given a quadratic inequality in two variables?

16

Activity 7 Direction: Answer the questions that follow.

a. Graph the inequality y x2 - 7x + 10?

b. Is the point (-4 , -8 ) in the solution set ? Justify your answer.

c. Is the boundary line drawn solid or dashed? Explain?

17

What I Can Do

Activity 8 Directions: Read the situation below then answer the questions that follow. Questions:

1. How would you represent the width and the length of the floor?

2. Write a mathematical sentence that would represent the given situation?

3. What are the possible lengths and widths of the floor?

4. What are the possible areas of the floor?

18

T The floor of a house can be covered completely

with tiles. It’s length is 38 ft. longer than its width. The area of the food is less than 2 040 square feet.

Activity 9 Direction: Perform the following activity.

1. Look for a rectangular floor in your house. Find its dimensions and indicate the measure (in meters) obtained in the table below.

Length Width

2. Seek help to determine the measure and costs of your preferred tile that is available in the nearest hardware store or advertised on the internet. Write your answer in the table below.

Tile Length Width Cost

3. Formulate a quadratic inequality involving the dimensions of the floor, and the measure and cost of the tile. Find, then graph the solution sets of the inequalities.

19

Activity 10 Post Test Directions: Circle the letter that corresponds to the correct answer.

1. Which of the following mathematical statements is a quadratic inequality?

A. 0532 2 pp C. 23s + 7s - 2 0

B. 7k + 12 < 0 D. b2 + 8b + 16= 0

2. Which of the following coordinate of points belong to the solution set of the inequality

532 2 xxy ?

A. (-2, 9) C. (-1, 5) B. (-3, 2) D. (1, 6)

3. What is the solution set of 0122 xx ?

A. 4/{ xx or }3x C. 4/{ xx or }3x

B. 4/{ xx or }3x D. 4/{ xx or }3x

4. Which of the following mathematical sentence is not a quadratic inequality?

A. 274 2 tt C. 23215 xx

B. 3102 xx D. 0452 rr

5. What is the solution set of the inequality 1492 xxy ?

A. 2/{ xx or }7x C. 2/{ xx or }7x

B. 2/{ xx or }7x D. 2/{ xx or }7x

6. Which inequality is shown in the graph?

20

A. y< x2 B. y> x2

C. y≤ x2

D. y≥ x2

7. What is the solution set of 5112 2 xxy

A. 5/{ xx or }5.0x C. 5/{ xx or }5.0x

B. 5/{ xx or }5.0x

D. 5/{ xx or }5.0x

8. Which of the following shows the graph of 672 xxy ?

.

21

A.

B.

C.

D.

9. The figure below shows the graph of .142 2 xxy Which of the following is true

about the solution set of the inequality? I. The coordinates of all points along the parabola as shown by the broken line

belong to the solution set of the inequality.

II. The coordinates of all points on the shaded region belong to the solution set of

the inequality.

III. The coordinates of all points along the parabola as shown by the broken line

do not belong to the solution set of the inequality.

A. I and II B. I and III C. II and III D. I, II and III 10. Choose a possible dimension of a rectangle with a width = 2x-1 and length = 3x + 2 so that its area is greater than 153 sq. cm. A. W= 13 L= 15 C. W= 11 L=21 B. W=11 L=20 D. W= 13 L= 22

22

Summary The lesson was about quadratic inequalities, their solution sets and graphs. The lesson equipped you to solve, describe, and graph quadratic inequalities using your mathematical skills and concepts learned in the previous topics. Furthermore, you were given the opportunity to determine what method to apply in solving quadratic inequalities and test your understanding of the lesson by doing a practical task.

23

Answer keys: Pre- Assessment

1. C 6. A 2. C 7. C 3. C 8. D 4. A 9. A 5. B 10. B

Activity 1

1. n=13 3. 13n 5. varied answer 2. n0

t2 = 6t -7 p2 + 10p + 16 0

4m2 -25 = 0 2t2 < 21 - 9t

f2 + 9f + 20 =0 3b2 + 12b 0

Activity 3

1) solution set 72/ orxxx

2) solution set 34/ orxxx

Activity 4

1. 2.

24

Activity 5

1. Not a solution 2. Solution 3. Not a solution 4. Not a solution 5. Solution

Activity 6

1. y -x2 - 4x + 1

2. y x2 - 4x + 1

3. y< x2 - 4x + 1 Activity 7

1. 2. . dashed

3. Yes, (-4 , -8 ) is a solution because it satisfies the inequality y x2 - 7x + 10 4. Dashed because the inequality use the symbol <

Activity 8

1. Let x be the width of the rectangle Length= 38 + x

2. (38 + x) (x) < 2040 x2 + 38 x -2040 < 0

3.

possible width 29 ft. 28 ft. 27 ft

... possible length

67 ft

66 ft

65 ft ...

4.

possible areas 1943 ft2 1846 ft

2 1755 ft

2

...

25

Activity 9 Varied answers Activity 10 1. C 6. A 2. B 7. D 3. D 8. C 4. C 9. C 5. B 10. B .

26

References:

https://quizizz.com/admin/quiz/58fe1628ddf8911200aec51d/quadratic-inequalities https://study.com/academy/practice/quiz-worksheet-1-variable-quadratic-inequalities.html https://www.varsitytutors.com/algebra_ii-help/quadratic-inequalities https://docs.google.com/document/d/1YUPmlvFkE10wsoypNCRGoai8H0zXxn8nZFXukIvsOfM/edit# https://mathsmadeeasy.co.uk/wp-content/uploads/2017/11/Quadratic-Inequalities-Questions.pdf https://www.slideshare.net/swartzje/58-graphing-quadratic-inequalities https://slideplayer.com/slide/10769827/

Equals Worktext in Mathematics 9 Mathematics Learner‟s Material 9

27

https://quizizz.com/admin/quiz/58fe1628ddf8911200aec51d/quadratic-inequalitieshttps://study.com/academy/practice/quiz-worksheet-1-variable-quadratic-inequalities.htmlhttps://www.varsitytutors.com/algebra_ii-help/quadratic-inequalitieshttps://docs.google.com/document/d/1YUPmlvFkE10wsoypNCRGoai8H0zXxn8nZFXukIvsOfM/edithttps://mathsmadeeasy.co.uk/wp-content/uploads/2017/11/Quadratic-Inequalities-Questions.pdfhttps://www.slideshare.net/swartzje/58-graphing-quadratic-inequalitieshttps://slideplayer.com/slide/10769827/

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