Math9 - q1 - Mod2 - Solving Quadratic Inequality - v3-1
Math9 - q1 - Mod2 - Solving Quadratic Inequality - v3-1
Math9 - q1 - Mod2 - Solving Quadratic Inequality - v3-1
NOT
Mathematics
Quarter 1,Wk.6 - Module 2
Solving Quadratic Inequality
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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 23
Assessment: (Post-Test) 20-22
Key to Answers 24 – 26
References 27
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What This Module is About
This module is all about quadratic inequalities and their solution sets and graphs.
It 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
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.
b, and c are real numbers and a 0 . Symbols >, , and may also be used in place
of <.
A. Linear Inequality C. Quadratic Inequality
B. Linear Equation D. Quadratic Equation
1
2. Which of the following mathematical statements is a quadratic inequality?
A. 2p2 - 3p - 5 = 0 C. 3s2 + 7t - 2 0
B. 7n + 12 < 0 D. m2 + 8tm- 2 = 0
3. Which of the following coordinates of points belong to the solution set of the
inequality y < 2v2 + 5v - 1?
A. (-2,9) C. (3,1)
B. (-3,2) D. (1,6)
4. What is the solution set of the quadratic inequality x 2– x – 20 > 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}
2
‘9. Which inequality best describes the graph below?
A. y<-2x2-8x-12 C. y>-2x2-8x-12
B. y≤-2x2-8x-12 D. y≥-2x2-8x-12
10. Using the graph of y= x2- 3x- 10. What is the correct way to write the solution of
x2- 3x- 10 0?
A. x 2 x 5 C. 2 x 5
B. 2 x 5 D. x 2 x 5
3
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
2. n - 3 < 10, What are the possible values of n to make the statement true?
Show your solution.
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.
5. b2 + 5b + 6 = 0 ,
What do you call this mathematical statement?
How did you find the solution/s of this expression? How many solution did you
get?
4
What’s New
Activity 2
Directions: Use the table below to identify whether the following equations are
quadratic or not. Answer the questions that follow.
Questions
5
What Is It
Examples: 1. x2 + x -12 0 3. 5 ≥ x2 − x
2. 7x2 -28 < 0 4. 2y2 + 1 ≤ 7y
Solution
6
❏ Test a value from
each interval in the Intervals in x x2 + 3x - 10 = 0 True or False
inequality. the number
line
x < -5 -6 x2 + 3x - 10 > 0
(-6)2 + 3(-6) -10 > 0 True
36 - 18 -10 > 0
8>0
x>2 4 x2 + 3x - 10 > 0
(4)2 + 3(4) -10 > 0 True
16 + 12 - 10 > 0
18 > 0
x2 + 3x - 10 > 0 x2 + 3x - 10 > 0
(-5)2 + 3(-5) -10 > 0 (2)2 + 3(2) -10 > 0
25 - 15 - 10 > 0 4 + 6 - 10 > 0
10-10 > 0 10 - 10 > 0
0 > 0 False 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.
7
Method B: The Sign Graph
for x - 2
-------------- +++++++++++ -----------
❏ Apply the rules of signs for multiplying The product of x + 5 and x - 2 is positive if
sign numbers to determine which area the factors are both positive and negative.
satisfies the original inequality. These are possible to happen in the
regions where x < -5 or x > 2.
Solution
❏ Find the 3 test points The points at -3 and 1 can separate the real number line into
using -3 and 1. three intervals:
x < -3 , -3 < x < 1 , x > 1
8
❏ Test a value from
each interval in the Intervals in x x2 + 2x - 3 0 True or False
inequality. the number
line
x < -3 -4 x2 + 2x - 3 0
2
(-4) + 2(-4) -3 0 False
16 - 8 - 3 0
5 0
-3 < x < 1 0 x2 + 2x - 3 0
(0)2 + 2(0) -3 0 True
0+0-3 0
-3 0
x>1 2 x2 + 2x - 3 0
(2)2 + 2(2) -3 0 False
4+4-3 0
5 0
9
Method B
for x - 1
-------------------- --------- +++++++++
❏ Apply the rules of signs for multiplying The product of x + 3 and x - 1 is negative if
sign numbers to determine which area the factors have different signs. These are
satisfies the original inequality. possible to happen in the regions
where -3
Note: -3 and 1 are included in the solution
because it satisfies the equation.
10
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 .
Take note of the image of the following graphs for each quadratic inequality.
y ax 2 bx c y ax 2 bx c y ax 2 bx c y ax 2 bx c
y
11
Example 1: Find the solution set of y - x2 + 4x - 3
❏ Construct table of
values for x and y. x 0 1 2 3 4
y -3 0 1 0 -3
12
Example 2: Find the solution set of y -x2 + 6x – 4
❏ Construct table of
values for x and y. x -1 -2 -3 -4 5
12
What’s More
Activity 3
Direction: Fill-in the table below to find the solution set of each of the following quadratic
inequalities.
Solution set
13
2) x 2 x 12 0 (Use the sign graph method.)
Solution Set
14
Activity 4
Directions: Find the solution set of each of the following quadratic inequalities then graph
and explain your answer.
1. 2 x 2 3x 1 0
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
17
What I Can Do
Activity 8
Directions: Read the situation below then answer the questions that follow.
T
The floor of a house can be covered completely
with tiles. Its length is 38 ft. longer than its width. The
area of the floor is less than 2 040 square feet.
Questions:
1. How would you represent the width and the length of the floor?
18
Activity 9
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.
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
A. 2 p 2 3 p 5 0 C. 3s 2 + 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
y 2 x 2 3x 5 ?
A. (-2, 9) C. (-1, 5)
B. (-3, 2) D. (1, 6)
A. {x / x 4 or x 3} C. {x / x 4 or x 3}
B. {x / x 4 or x 3} D. {x / x 4 or x 3}
A. 4t 2 7t 2 C. 15 2 x 3 x 2
B. x 2 10 x 3 D. 2r 5r 4 0
A. {x / x 2 or x 7} C. {x / x 2 or x 7}
B. {x / x 2 or x 7} D. {x / x 2 or x 7}
A. y< x2
B. y> x2
C. y≤ x2
D. y≥ x2
20
7. What is the solution set of y 2 x 2 11x 5
A. {x / x 5 or x 0.5} C. {x / x 5 or x 0.5}
B. {x / x 5 or x 0.5} D. {x / x 5 or x 0.5}
A.
B.
C.
D.
21
9. The figure below shows the graph of y 2 x 2 4 x 1. 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.
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.B
2. C 7. C
3. C 8. D
4. A 9. A
5. B 10. B
Activity 1
Activity 2
t2 = 6t -7 p2 + 10p + 16 0
f2 + 9f + 20 =0 3b2 + 12b 0
Activity 3
1. 2.
24
Activity 5
1. Not a solution
2. Solution
3. Not a solution
4. Solution
5. Solution
Activity 6
1. y -x2 - 4x + 1
2. y x2 - 4x + 1
3. y< x2 - 4x + 1
Activity 7
a.
b. Yes, (-4 , -8 ) is a solution because it satisfies the inequality y x2 - 7x + 10
c. Dashed because the inequality use the symbol <
Activity 8
3.
possible width 29 ft. 28 ft. 27 ft
...
possible
length 67 ft 66 ft 65 ft ...
4.
possible areas 1943 ft2 1846 ft2 1755 ft2
...
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/1YUPmlvFkE10wsoypNCRGoai8H0zXxn8nZFXukI
vsOfM/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/
27
References
https://owl.purdue.edu/owl/research_and_citation/chicago_manual_17th_edition/cmos_formatting_and_st
yle_guide/chicago_manual_of_style_17th_edition.html
you can also use citation machine generators: citethisforme.com and citefast.com
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