This is the Grade 9 Mathematics self-learning module for Quarter 1, Module 11: Illustrating Quadratic Inequality. It is set for Week 5 and covers learning code M9AL-Ie-7. It builds on the linear inequalities learners met in Grade 7. After working through it, learners should be able to recognize a quadratic inequality, write it in standard form, read it aloud, and use one to describe a real-life situation.
On this page 6 sections
What learners will learn
The module states one learning competency. The learners will be able to:
- illustrate quadratic inequality (M9AL-Ie-7)
What a quadratic inequality is
A quadratic inequality is an inequality that contains a polynomial of degree 2. It can be written in any of these forms, where a, b, and c are real numbers and a is not 0:
- ax² + bx + c > 0
- ax² + bx + c < 0
- ax² + bx + c ≥ 0
- ax² + bx + c ≤ 0
The module compares quadratic equations with quadratic inequalities:
| Quadratic equation | Quadratic inequality | |
|---|---|---|
| Highest exponent of the variable | 2 | 2 |
| Symbol used | = | <, >, ≤, or ≥ |
| Example from the module | x² + 9x + 20 = 0 | 3w² + 12w ≥ 0 |
Checking whether an inequality is quadratic
To check, expand any products. Then move every term to one side so the other side is zero. If the highest degree left is 2, the inequality is quadratic. The module’s worked examples show this:
- x² + 2x ≤ 9 becomes x² + 2x − 9 ≤ 0. It is quadratic and is read “x squared plus 2x minus nine is less than or equal to zero.”
- (x − 5)(5x + 7) > 7 expands to 5x² − 8x − 7 > 0. It is quadratic.
- 7x² − 15 > 7(x² + 5x) simplifies to 35x + 15 < 0. The x² terms cancel, so the degree is 1. This one is linear, not quadratic.
The last example is the key lesson: an inequality can look quadratic at first and turn out to be linear after you simplify it.
Writing situations as quadratic inequalities
The module turns word problems into inequalities by watching for keywords:
- Profit: Demz’s monthly profit from selling bicycles is P(x) = x² + 5x − 50. Demand is high, so the profit is taken to be positive: x² + 5x − 50 > 0.
- Height: An object’s height is h = −10x² + 72x. The keyword “below” a height of 144 feet calls for the < symbol: −10x² + 72x < 144.
Other items use “at most” (≤), “less than” (<), and area or volume formulas. Examples include a floor whose length is longer than its width and an open box folded from cardboard.
The module also gives real uses of quadratic inequalities. Roller coaster engineers use them to find the minimum speed that keeps cars on the track. Bicycle makers use them to find the maximum travel of the rear suspension.
Activities in the module
- What I Know: a 10-item multiple-choice pretest.
- What’s New: “Which Are Not Quadratic Equations?”, where learners sort five mathematical sentences and compare equations with inequalities.
- What’s More: “Test Yourself!” Learners label five items QI (quadratic inequality) or NQI (not a quadratic inequality). They also write the profit of Aling Tura’s bakery as an inequality, using Profit = Revenue − Cost.
- What I Can Do: “Quadratic Inequalities or Not?”, a 10-item identification activity.
- Assessment: a 10-item multiple-choice post-test.
- Additional Activities: “Make it Real!” (a short video on inequalities in real life) and “Create your Own!”, where learners write three real-life scenarios and trade them with a classmate.
- Problem-Based Worksheet: “Packaging Thing.” A sardine box is 20 inches longer than it is wide and 12 inches high, with a volume of at most 3,600 in³. Learners write the inequality and find the possible range for the width.
How to use this module
- Start with the pretest, What I Know. Write your answers on a separate sheet. After checking, note the items you missed and look for the right answers as you go through the module.
- Work through the parts in order: What’s In, What’s New, What is It, What’s More, What I Have Learned, and What I Can Do.
- Take the Assessment. If your answer is not among the choices, write the correct answer.
- Finish with the Additional Activities and the Problem-Based Worksheet.
The answer key at the end covers the pretest, the assessment, What’s More, What I Can Do, and the worksheet.
- Learners: Try each activity before you look at the answer key.
- Parents: Ask your child to explain why a sentence is or is not a quadratic inequality.
- Teachers: Several activities invite pair work, such as discussing What’s New and trading scenarios in Create your Own!
Frequently asked questions
What grade and quarter is this module for?
It is Grade 9 Mathematics, Quarter 1, Module 11, set for Week 5 under learning code M9AL-Ie-7.
What is a quadratic inequality?
It is an inequality that contains a polynomial of degree 2. It can be written as ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0, where a, b, and c are real numbers and a is not 0.
How is a quadratic inequality different from a quadratic equation?
Both have 2 as the highest exponent of the variable. A quadratic equation uses the equal sign, while a quadratic inequality uses <, >, ≤, or ≥.
Does the module have an answer key?
Yes. It has answers for the pretest, the assessment, What’s More, What I Can Do, and the Problem-Based Worksheet.
Is every inequality with x² quadratic?
No. The module shows that 7x² − 15 > 7(x² + 5x) simplifies to 35x + 15 < 0, which is linear because the x² terms cancel.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 1-Module 11 Illustrating Quadratic Inequality
Week 5 Learning Code – M9AL-Ie-7
[illustration: teacher at a screen with students learning at computers, tablets, and laptops, with speech bubbles]
GRADE 9 Learning Module for Junior High School Mathematics
Quarter 1 – Module 11 – New Normal Math for G9 First Edition 2020 Copyright © 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Writers: Kris Marie Odal- TI, Rowena F. Reyes – TI, Marvin G. Sollera – MTII
Editor: Sally C. Caleja – Head Teacher VI, Zacarias B. Rosales- Head Teacher VI, Melody P. Rosales- Head Teacher VI
Validators: Remylinda T. Soriano, EPS, Math; Angelita Z. Modesto, PSDS; George B. Borromeo, PSDS
Illustrator: Writers
Layout Artist: Writers
Management Team: Malcolm S. Garma, Regional Director; Genia V. Santos, CLMD Chief; Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator; Maria Magdalena M. Lim, CESO V, Schools Division Superintendent; Aida H. Rondilla, Chief-CID; Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator
Learning Module for Junior High School Mathematics
MODULE 11: ILLUSTRATING QUADRATIC INEQUALITY
When you were in grade 7, you have learned about linear inequalities and were able to find and apply its solution to some real-life problem situations. To have a broader range of application to real life problems, another type of inequality can also be studied. In this module, you will learn another type of inequality which is “quadratic inequality”.
What I Need to Know
LEARNING COMPETENCY
The learners will be able to:
- illustrate quadratic inequality. M9AL-Ie-7
What I Know
Find out how much you already know about the module. Write the letter that you think is the best answer to each question on a sheet of paper. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.
It is an inequality that can be written in any of the following forms: ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0 or ax² + bx + c ≤ 0.
A. Quadratic Equation
B. Quadratic Inequality
C. Linear Inequality
D. Linear EquationWhich of the following mathematical statements is a quadratic inequality?
A. 2r² − 3r − 5 = 0
B. 7h + 12 < 0
C. 3t² + 7t + 2 ≥ 0
D. s² + 8s + 15 = 0Which of the following is NOT true about quadratic inequality?
A. It always contains a linear expression
B. It contains a 2nd degree polynomial.
C. Two sides of the inequality separated by <, >, ≥ or ≤.
D. None of the aboveWhich of the following shows a relation of “greater than or equal to”?
A. (x + 4)² < 2x
B. (x + 4)² > 2x
C. x² + 4 ≥ 2x
D. x² + 4 ≤ 2x
Learning Module for Junior High School Mathematics
Which of the following is a quadratic inequality?
A. b > 9
B. c² − 4x ≥ c² + 8
C. v(v + 7) < v² + 8
D. x² + 8 < −4xThe monthly profit P of Karen’s business is represented by the equation P(x) = x² + 100. Which of the following quadratic inequalities is a representation of positive profit for Karen’s business?
A. x² + 100 ≥ 0
B. x² + 100 ≤ 0
C. −x² + 100 > 0
D. −x² + 100 < 0Referring to item #6, which of the following is a representation of negative or normal profit for Karen’s business?
A. x² + 100 ≥ 0
B. x² + 100 ≤ 0
C. x² + 100 > 0
D. x² + 100 < 0The floor of a conference hall can be covered completely with tiles. Its length is 36 ft longer than its width. The area of the floor is less than 2,040 square feet. What mathematical sentence would represent the given situation?
A. x² − 36 < 2040
B. x² + 36 < 2040
C. x² − 36x > 2040
D. x² + 36x < 2040Which of the following is NOT a quadratic inequality?
A. 3(2w² − 5) < w
B. 6(v² − 2) − 4v < 0
C. 4x² − 9 ≥ 0
D. (2x − 5)/(x + 2) ≤ 0An open box is to be formed out a rectangular piece of cardboard whose length is 12 cm longer than its width. To form the box, a square of side 5 cm will be removed from each corner of the cardboard. Then the edges of the remaining cardboard will be turned up. If the box is to hold at most 1900 cm³, what mathematical statement would represent the given situation?
A. x² − 12x ≤ 360
B. x² − 12x ≤ 380
C. x² − 8x ≤ 400
D. x² − 17x ≤ 310
Learning Module for Junior High School Mathematics
What’s In
Communication, character building and collaboration
“Racism”, “discrimination”, “being biased”, “favoritism”. These are some words synonymous to the word “INEQUALITY”. We cannot deny that this really happens in real life but despite of that, you have to remember that in every action you do or goal you reach, do your best and do not let someone discourage you. Do not forget that the “Creator” will be the one who will give you the prize.
Mathematically, there are other relevant use of Quadratic INEQUALITY.
Engineers of roller coasters (or caterpillars) must know the minimum speed required for the cars to stay on the track. It is important that the engineer is sure that the speed of the car is in the solution region.
A bicycle manufacturer must know the maximum distance the rear suspension will travel when going over rough terrain. Note that the movement of the rear wheel of bicycles is described by a quadratic equation. If the manufacturer will be made aware of this maximum distance, then it can set a specific month/date for the warranty so that it can reduce warranty claims.
What’s New
Communication, Critical Thinking and Collaboration
Which Are Not Quadratic Equations?
Use the mathematical sentences below to answer the questions that follow. You may discuss your ideas with your classmate.
- x² + 9x + 20 = 0
- 2t² < 21 − 9t
- r² + 10r ≤ −16
- 3w² + 12w ≥ 0
- 15 − 6h² = 10
Questions:
- Which of the given mathematical sentences are quadratic equations?
- How do you describe quadratic equations?
- Which of the given mathematical sentences are not quadratic equations? Why?
- What is the similarity between the mathematical sentences that are quadratic equations and the quadratic inequality? How are they different from each other?
Learning Module for Junior High School Mathematics
What is It
Communication, Critical Thinking, and Collaboration
A quadratic inequality is an inequality that contains a polynomial of degree 2 and can be written in any of the following forms.
ax² + bx + c > 0 ax² + bx + c < 0 ax² + bx + c ≥ 0 ax² + bx + c ≤ 0
where a, b, and c are real numbers and a ≠ 0.
In the activity done in “What’s new” part, to determine which mathematical sentences are quadratic equations and which are not quadratic, we simply make use of equality and inequality symbols.
Which of the given mathematical sentences are quadratic equations?
- x² + 9x + 20 = 0
- 15 − 6h² = 10
How do you describe quadratic equations?
➢ A quadratic equation is a polynomial equation of degree 2.Which of the given mathematical sentences are not quadratic equations? Why?
- 2t² < 21 − 9t
- r² + 10r ≤ −16
- 3w² + 12w ≥ 0
➢ The mathematical sentences contain inequality symbols.
What is the similarity between the mathematical sentences that are quadratic equations and the quadratic inequality? How are they different from each other?
➢ The highest exponent of the variable in each mathematical sentence is 2.
➢ The quadratic inequality makes use of inequality symbols while the quadratic equations make use of equality symbol.
Example 1: Which of the following is quadratic inequality? For those that are, read it loudly. a. x² + 2x ≤ 9 b. (x − 5)(5x + 7) > 7 c. 7x² − 15 > 7(x² + 5x)
Solutions: a. The inequality x² + 2x ≤ 9 is quadratic but not in the form ax² + bx + c ≤ 0. If we move all the terms to the left side, we have: x² + 2x − 9 ≤ 0 and can be read as “x squared plus 2x minus nine is less than or equal to zero”
Learning Module for Junior High School Mathematics
b. To check if the inequality is quadratic, we have (x − 5)(5x + 7) > 7 5x² + 7x − 25x − 35 > 7 5x² + 7x − 25x − 35 − 7 > 0 5x² − 8x − 7 > 0
Since the degree of the expression on the left of the inequality is 2, it is quadratic inequality. It is read as “Five x squared, minus eight x, minus seven is greater than zero”.
c. To check if the inequality is quadratic, expand the product and make one side of the inequality be zero. 7x² − 15 > 7(x² + 5x) 7x² − 15 > 7x² + 35x 0 > 7x² − 7x² + 35x + 15 0 > 35x + 15 35x + 15 < 0
Since the resulting expression on the left side has a degree of 1, then it is not quadratic. The inequality is linear. It is read as “Thirty-five x plus fifteen is less than zero”.
Example 2: Represent each situation using quadratic inequality. a. Despite of implementing general community quarantine, public transports are still prohibited. Because of this, there was a high demand on bicycles. The monthly profit that Demz makes on selling bicycles is represented by a mathematical model, P(x) = x² + 5x − 50. Using quadratic inequality, represent the current profit of Demz. b. An object is launched directly upward from the ground at the rate of 72 feet per second. The equation for the height of the object is: h = −10x² + 72x Give a quadratic inequality representation of the scenario if the object is below a height of 144 feet.
Solutions: a. Since it was stated that there was a high demand in bicycles, we assume that the profit is positive. Therefore, it can be represented by: x² + 5x − 50 > 0. b. The keyword is “below”. So, the inequality symbol must be “<“. Therefore, the scenario can be represented by: −10x² + 72x < 144
Learning Module for Junior High School Mathematics
What’s More
Critical Thinking
Test Yourself!
I. Tell whether the following is a quadratic inequality or not. If it is, write QI. Otherwise, write NQI. a. −2x² ≤ 8 b. 5(5x + 7) > 7 c. 7x² − 15 > 7(2x² + 5x) d. 0.1x² + 2.3x < −4 e. −3x − 5 ≤ 2x²
II. Represent the following scenario below using quadratic inequality. a. The revenue of Aling Tura’s bakery in selling bread is determined by the formula, R(x) = 40x − x² where x is the number of breads sold. The cost for producing those breads is given by a formula C(x) = x + 50. Using quadratic inequality, represent the positive or normal profit of Aling Tura. (Note: Profit = Revenue − Cost)
What I Have Learned
Quadratic inequality is an equality that contains a polynomial of degree 2 and can be written in any of the following forms.
ax² + bx + c > 0 ax² + bx + c < 0 ax² + bx + c ≥ 0 ax² + bx + c ≤ 0
where a, b and c are real numbers and a is not equal to zero.
Quadratic inequalities make use of inequality symbols such as <, >, ≤, ≥.
What I Can Do
Activity: “Quadratic Inequalities or Not?”
Directions: Determine whether the mathematical sentence is quadratic inequality or not.
- x² + 9x + 14 > 0
- 3s² − 5s = 1
- 4t² − 7t + 2 ≤ 0
- x² < 10x − 3
- 12 − 5x + x² = 0
- 3m + 20 ≥ 0
- (2r − 5)(r + 4) > 0
- x² − 1 < x + 1
- (4h² − 9)(2h + 3) ≥ 0
- 15 − 2x = 3x²
ASSESSMENT
Write the letter of the correct answer on your answer sheet. If your answer is not found among the choices, write the correct answer.
- Which of the following is in the form of quadratic inequality?
A. ax² + bx + c > 0
B. bx + c < 0
C. ax² + bx + c = 0
D. bx + c = 0 - Which of the following mathematical statements is a quadratic inequality?
A. 5t² − 3r − 20 = 0
B. 4x² + 9x − 2 ≥ 0
C. 9h + 18 < 0
D. m² + 9m + 16 = 0 - Which of the following is true about quadratic inequality?
A. It contains a 2nd degree polynomial
B. It contains a 3rd degree polynomial.
C. Two sides of the inequality separated by “=” symbol.
D. None of the above - Which of the following shows a relation of “greater than”?
A. (x + 4)² < x
B. (x + 4)² ≥ x
C. x² + 4 > x
D. x² + 4 < x - Which of the following is NOT a quadratic inequality?
A. (2w⁴ − 5) < 2w⁴ + 3x²
B. 4v² − 4v < 0
C. 4x² − 9 ≥ 3 + 4x²
D. n² + 1 < 0 - A field planted with palay has a length of (x + 3) feet and a width of (x + 2) feet. Its area is at most 2000 square feet. Which of the following quadratic inequalities represents the area?
A. (x + 3)(x + 2) ≥ 2000
B. (x + 3)(x + 2) > 2000
C. x² + 5x + 6 ≤ 2000
D. x² + 5x + 6 < 2000 - The weekly revenue R of Kevin’s Laundry Shop is represented by the equation R(x) = x² + 250 while the cost expenses C is represented by C(x) = 2x² − 400.If x is the number of laundry services rendered, which of the following is the quadratic inequality representation of positive profit for Kevin’s laundry shop?
A. −x² + 650 > 0
B. x² + 650 < 0
C. x² − 650 > 0
D. −x² − 650 < 0 - Referring to item #7, which of the following inequalities refers to the negative profit for Kevin’s Laundry shop?
A. x² + 650 > 0
B. x² − 650 < 0
C. −x² − 650 > 0
D. −x² − 650 ≤ 0 - The floor of a conference hall can be covered completely with tiles. Its length is 25 ft. longer than its width. The area of the floor is less than 1,800 square feet. What mathematical sentence would represent the given situation?
A. w² − 25w < 1800
B. w² − 25w > 1800
C. w² + 25w < 1800
D. w² + 25w > 1800 - An open box is to be formed out a rectangular piece of cardboard whose length is 16 cm longer than its width. To form the box, a square of side 5 cm will be removed from each corner of the cardboard. Then the edges of the remaining cardboard will be turned up. If the box is to hold at most 2100 cm³, what mathematical statement would represent the given situation?
A. w² + 4w ≤ 480
B. w² − 4w ≤ 420
C. w² + 4w ≥ 420
D. w² − 4w ≤ 480
ADDITIONAL ACTIVITIES
Collaboration, Creativity and Critical Thinking
A. “Make it Real!”
✓ This short video clip shows how quadratic inequalities are used in solving real-life problems and in making decisions.
Watch in youtube: https://www.youtube.com/watch?v=PFwwbLIS0jg
[photo: screenshot of the video “One Variable Inequalities in Real-Life Situations”, showing the text “Maximum allowed speed at a highway is 100 km/hour. Represent this situation as an inequality:” with the sign “MAXIMUM 100 km/h” and “Let s be the speed of a car, then s ≤ 100.” Caption printed below: “One Variable Inequalities in Real Life Situations | Infinite-Math.com”]
B. Create your Own!
Make 3 real life scenarios that can be illustrated by quadratic inequality. Trade your constructed scenarios to your classmate and let them represent it using quadratic inequality.
E-Search
You may also check the link for your reference and further learnings on illustrating quadratic inequality:
- https://www.khanacademy.org
REFERENCES
Dugopolski, Mark. 2006. Elementary and Intermediate Algebra 2nd edition. McGraw-Hill. New York City
https://www.depednegor.net/uploads/8/3/5/2/8352879/math_9_lm_draft_3.24.2014.pdf
https://www.depednegor.net/uploads/8/3/5/2/8352879/math_9_tg_draft_3.24.2014.pdf
PROBLEM – BASED WORKSHEET
Packaging Thing
A company decided to increase the size of the box for the packaging of their canned sardines. The length of the original packaging box was 20 inches longer than its width, the height was 12 inches, and the volume was at most 3, 600 in³.
[photo: open empty cardboard box]
- If w represents the width of the box, what mathematical sentence would represent the volume of the original packaging box?
- What could be the range of the length of the width of the original packaging box?
[transcriber’s note: this page is printed upside down; it has been transcribed as it reads after turning the page 180°]
ANSWER KEY
WHAT I KNOW
- B
- C
- A
- C
- D
- C
- B
- D
- C
- C
ASSESSMENT
- A
- B
- A
- C
- C
- C
- A
- B
- C
- D
WHAT’S MORE
I.
- QI
- NQI
- QI
- QI
- QI
II. −x² + 39x − 50 ≥ 0 or x² − 39x + 50 ≤ 0
WHAT I CAN DO
- Quadratic inequality
- Not Quadratic inequality
- Quadratic inequality
- Quadratic inequality
- Not Quadratic inequality
- Not Quadratic inequality
- Quadratic inequality
- Quadratic inequality
- Quadratic inequality
- Not Quadratic inequality
PROBLEM – BASED WORKSHEET
Let w = the width of the box, length = w + 20, height = 12 inches.
Since, Volume(box) = (width)(length)(height)
Then, the mathematical sentence that will help us solve the problem is,
(w)(w + 20)(12) ≤ 3 600 or w² + 20w − 300 ≤ 0To solve the resulting mathematical sentence, apply the rule on solving quadratic inequality, that is, replacing the inequality symbol by an equal sign to solve for the value of w.
w² + 20w − 300 = 0 → (w + 30)(w − 10) = 0 → w = −30 or w = 10We only consider the positive value of w since the situation involves measures of length. Thus, the range for the width of the box is 10 inches or less. And if the dimensions of the box is in whole inches, the greatest possible dimensions of the box are: width = 10 inches, length = 30 inches, and height = 12 inches.
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