Quadratic inequalities, Inequalities in two variables for SS 3

Class: SS 3
Term: First Term
Week: 11
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Topic: INEQUALITIES
Subject Matter: Quadratic inequalities; Inequalities in two variables
Previous Lesson: Variance of the Different Probability Distributions

Specific Objectives

By the end of the lesson, pupils/students should be able to:

Cognitive Domain

• Define quadratic inequalities.
• Identify the methods for solving quadratic inequalities.
• Solve given quadratic inequalities using appropriate methods.
• Explain what inequalities in two variables are.
• Graph inequalities in two variables on a coordinate plane.

Affective Domain

• Appreciate the importance of inequalities in real-world problem-solving.
• Develop a systematic approach to solving complex inequality problems.

Psychomotor Domain

• Demonstrate the steps for solving quadratic inequalities.
• Construct accurate graphs for inequalities in two variables.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools, Book 3
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Chart showing quadratic inequalities and inequalities in two variables.
• Whiteboard and markers.
• Graph paper.

Rationale for the Lesson

This lesson is important because it builds on students’ foundational knowledge of inequalities, extending it to more complex quadratic forms and two-variable systems. Mastering these concepts is crucial for advanced mathematical studies and for solving practical problems in fields such as engineering, economics, and science, where optimal solutions often involve inequality constraints.

Prerequisite/Previous Knowledge

Students should have prior knowledge of solving linear inequalities, factoring quadratic expressions, and plotting points on a Cartesian plane.

Lesson Content/Board Summary

INEQUALITIES

Quadratic Inequalities

A quadratic inequality is an inequality that contains a quadratic expression. It can be written in one of the following forms:

1. (ax^2 + bx + c > 0)
2. (ax^2 + bx + c < 0)
3. (ax^2 + bx + c ge 0)
4. (ax^2 + bx + c le 0)

Where (a, b, c) are real numbers and (a neq 0).

Methods for Solving Quadratic Inequalities

There are generally two methods to solve quadratic inequalities:

1. Using Critical Points and Test Intervals (Sign Table Method):
   1. Rewrite the inequality so that one side is zero.
   2. Find the critical points by setting the quadratic expression equal to zero and solving for (x). These are the roots of the quadratic equation.
   3. Plot the critical points on a number line. These points divide the number line into intervals.
   4. Choose a test value from each interval and substitute it into the original inequality to determine if the inequality is true or false in that interval.
   5. The solution set consists of the intervals where the inequality is true.
2. Using a Graph:
   1. Sketch the graph of the corresponding quadratic function (y = ax^2 + bx + c).
   2. Identify the (x)-intercepts (critical points).
   3. For (ax^2 + bx + c > 0), the solution is the (x)-values where the graph is above the (x)-axis.
   4. For (ax^2 + bx + c < 0), the solution is the (x)-values where the graph is below the (x)-axis.

Example 1

Question: Solve (x^2 – 5x + 6 > 0).

Solution:

Step 1: Find the critical points.

(x^2 – 5x + 6 = 0)

((x – 2)(x – 3) = 0)

(x = 2) or (x = 3)

Step 2: Plot critical points on a number line and test intervals.

Intervals: ((-infty, 2)), ((2, 3)), ((3, infty))

Test (x=0): ((0)^2 – 5(0) + 6 = 6 > 0) (True)

Test (x=2.5): ((2.5)^2 – 5(2.5) + 6 = 6.25 – 12.5 + 6 = -0.25 < 0) (False)

Test (x=4): ((4)^2 – 5(4) + 6 = 16 – 20 + 6 = 2 > 0) (True)

Step 3: Write the solution.

Answer: (x 3)

Example 2

Question: Solve (x^2 – x – 6 le 0).

Solution:

Step 1: Find the critical points.

(x^2 – x – 6 = 0)

((x – 3)(x + 2) = 0)

(x = 3) or (x = -2)

Step 2: Plot critical points on a number line and test intervals.

Intervals: ((-infty, -2]), ([-2, 3]), ([3, infty))

Test (x=-3): ((-3)^2 – (-3) – 6 = 9 + 3 – 6 = 6 notle 0) (False)

Test (x=0): ((0)^2 – (0) – 6 = -6 le 0) (True)

Test (x=4): ((4)^2 – (4) – 6 = 16 – 4 – 6 = 6 notle 0) (False)

Step 3: Write the solution.

Answer: (-2 le x le 3)

Inequalities in Two Variables

An inequality in two variables involves two variables, typically (x) and (y), and an inequality sign. Examples include (y > 2x + 1), (3x – y le 5).

Steps for Graphing Inequalities in Two Variables

1. Replace the inequality sign with an equality sign to get the equation of the boundary line.
2. Graph the boundary line:
   1. If the original inequality is (), draw a dashed line (indicating points on the line are not included).
   2. If the original inequality is (le) or (ge), draw a solid line (indicating points on the line are included).
3. Choose a test point not on the boundary line (e.g., ((0,0)) if it’s not on the line).
4. Substitute the test point coordinates into the original inequality.
5. Shade the appropriate region:
   1. If the test point makes the inequality true, shade the region containing the test point.
   2. If the test point makes the inequality false, shade the region not containing the test point.

Example 1

Question: Graph the inequality (y > 2x – 1).

Solution:

Step 1: Replace the inequality sign with an equality sign.

(y = 2x – 1)

Step 2: Graph the boundary line. Since it’s (>), use a dashed line.

When (x=0), (y = -1). Point: ((0, -1))

When (y=0), (0 = 2x – 1 implies 2x = 1 implies x = 0.5). Point: ((0.5, 0))

Step 3: Choose a test point. Let’s use ((0,0)).

Step 4: Substitute ((0,0)) into the original inequality.

(0 > 2(0) – 1)

(0 > -1) (True)

Step 5: Shade the region containing ((0,0)).

Example 2

Question: Graph the inequality (x + 2y le 4).

Solution:

Step 1: Replace the inequality sign with an equality sign.

(x + 2y = 4)

Step 2: Graph the boundary line. Since it’s (le), use a solid line.

When (x=0), (2y = 4 implies y = 2). Point: ((0, 2))

When (y=0), (x = 4). Point: ((4, 0))

Step 3: Choose a test point. Let’s use ((0,0)).

Step 4: Substitute ((0,0)) into the original inequality.

(0 + 2(0) le 4)

(0 le 4) (True)

Step 5: Shade the region containing ((0,0)).

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Questioning

Teacher’s Activity: The teacher begins by asking students to recall how to solve simple linear inequalities like (2x + 3 > 7). The teacher then introduces that today’s lesson will cover more advanced types of inequalities: quadratic inequalities and inequalities in two variables.

Pupils’ Activity: Students respond to questions about linear inequalities and listen attentively to the introduction of the new topic.

Learning Point: Prior knowledge activation

Step 2: Quadratic Inequalities

Time: 10 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains what quadratic inequalities are, their general forms, and introduces the two main methods for solving them: using critical points and test intervals, and using graphs. The teacher uses the chart to illustrate the concept.

Pupils’ Activity: Students listen, observe the chart, and ask questions for clarification on the definition and methods.

Learning Point: Quadratic inequalities concept

Step 3: Worked Example 1 (Quadratic)

Time: 7 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher demonstrates how to solve the quadratic inequality (x^2 – 5x + 6 > 0) step-by-step on the board, emphasizing finding critical points and testing intervals. The teacher explains each step clearly.

Pupils’ Activity: Students observe the teacher’s demonstration, take notes, and follow the steps shown.

Learning Point: Solving quadratic inequalities

Step 4: Worked Example 2 (Quadratic)

Time: 5 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher guides students to solve another quadratic inequality, (x^2 – x – 6 le 0), encouraging them to participate in identifying critical points and testing intervals. The teacher provides support and corrects errors.

Pupils’ Activity: Students actively participate in solving the problem, providing answers for each step, and making corrections as guided by the teacher.

Learning Point: Practice quadratic inequalities

Step 5: Inequalities in Two Variables

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces inequalities in two variables and explains the steps for graphing them, including drawing solid or dashed boundary lines and using test points for shading. The teacher uses the chart to show examples.

Pupils’ Activity: Students pay attention to the explanation and observe the graphical representations on the chart.

Learning Point: Inequalities two variables

Step 6: Worked Example (Two Variables)

Time: 5 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher demonstrates how to graph an inequality in two variables, such as (y > 2x – 1), on the board or using graph paper. The teacher guides students through drawing the boundary line, choosing a test point, and shading the correct region.

Pupils’ Activity: Students follow along, drawing the graph in their notebooks, and participate in identifying the test point and shading region.

Learning Point: Graphing two-variable inequalities

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. What are the critical points for solving (x^2 – 4x + 3 < 0)?
2. Explain when to use a dashed line versus a solid line when graphing inequalities in two variables.
3. Which region would you shade for (y < x + 2) if the test point ((0,0)) makes the inequality true?
4. Solve (x^2 + 2x – 8 ge 0).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Understanding inequalities solutions

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on quadratic inequalities and inequalities in two variables into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson notes

Step 9: Conclusion

Time: 1 minute

Teaching Skill: Reinforcement

Teacher’s Activity: The teacher briefly summarizes the key methods for solving quadratic inequalities and graphing inequalities in two variables, reminding students of their practical applications.

Pupils’ Activity: Students listen and prepare for the next lesson.

Learning Point: Inequalities concept reinforced

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook.

1. Solve the quadratic inequality: (2x^2 + 7x + 3 le 0).
2. Solve the quadratic inequality: (x^2 – 4 > 0).
3. Graph the inequality: (y le -x + 3).
4. Graph the inequality: (3x – 2y < 6).

Lesson Keywords

• Quadratic inequality – An inequality containing a quadratic expression.
• Critical points – The roots of the quadratic equation used to divide the number line.
• Sign table – A method using test intervals to determine the solution of an inequality.
• Two variables – An inequality involving two unknown quantities, typically (x) and (y).
• Boundary line – The line representing the equality part of a two-variable inequality.
• Test point – A point chosen to determine which region satisfies a two-variable inequality.
• Shading – The process of indicating the region that satisfies a two-variable inequality.

Differentiation

For weaker learners, the teacher can provide simpler quadratic inequalities that are easily factorable and offer more guided practice with graphing basic two-variable inequalities. For advanced learners, the teacher can introduce problems involving systems of inequalities or real-world application problems that require setting up and solving inequalities.

Suggested Lesson Videos

For further understanding, students can search YouTube for: “Solving quadratic inequalities” or “Graphing inequalities in two variables”.

Teacher Guide for Using This Lesson Plan

Teachers should prepare thoroughly by reviewing the methods for solving quadratic inequalities and graphing inequalities in two variables, ensuring they can clearly explain each step. Have the instructional charts ready before class. Begin by activating prior knowledge of linear inequalities to build a foundation. Guide students through the examples, allowing them to participate actively in problem-solving. Pay close attention to common errors such as incorrect determination of critical points or improper shading of regions. Ensure students understand when to use solid versus dashed lines. Students should copy the Board Summary notes during the designated note-taking step to consolidate their learning. Check understanding frequently through questioning and provide immediate feedback. Offer additional support to students struggling with factoring or graphing, and challenge faster learners with more complex problems or real-world scenarios.