Mathematics - Senior Secondary 2 - Linear inequalities

Linear inequalities

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TERM: 2ND TERM

WEEK: 2

Class: Senior Secondary School 2

Age: 16 years

Duration: 40 minutes of 5 periods

Subject: Mathematics

Topic: Linear Inequalities

Focus:

  1. Linear inequalities in one variable
  2. Linear inequalities in two variables
  3. Range of values of combined inequalities
  4. Graph of linear inequalities in two variables
  5. Maximum and minimum values of simultaneous linear inequalities and their application in real-life situations

SPECIFIC OBJECTIVES:

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

  1. Define and solve linear inequalities in one variable.
  2. Define and solve linear inequalities in two variables.
  3. Understand and determine the range of values in combined inequalities.
  4. Plot and interpret the graph of linear inequalities in two variables.
  5. Identify the maximum and minimum values of simultaneous linear inequalities and apply them to real-life situations.

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Practice exercises
  • Discussion
  • Visual aids (scale balance, number line, graph)

INSTRUCTIONAL MATERIALS:

  • Scale balance
  • Number line chart
  • Graph board
  • Mathematical sets (rulers, compasses, etc.)
  • Worksheets for practice

 

PERIOD 1: Introduction to Linear Inequalities in One Variable

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of inequalities. Explains that inequalities are expressions that compare two values using symbols such as >, <, ≥, and ≤. Uses the scale balance to demonstrate the concept.

Students observe the balance and relate it to inequalities.

Step 2 - Solving Inequalities

Explains how to solve linear inequalities in one variable (e.g., 3x + 5 > 11). Demonstrates step-by-step solving.

Students listen attentively and take notes.

Step 3 - Example Problem

Solves an example inequality together with the class (e.g., 4x - 3 ≤ 9).

Students work through the example with the teacher.

Step 4 - Independent Practice

Assigns several practice problems for students to solve individually or in pairs.

Students solve inequalities on their own.

NOTE ON BOARD:

  • Linear inequality in one variable: 3x + 5 > 11
  • Solve by isolating x.

EVALUATION (5 exercises):

  1. Solve for x: 2x + 7 ≥ 15.
  2. Solve for x: 5x - 3 < 17.
  3. Solve for x: 7x + 4 ≤ 31.
  4. Solve for x: 6x - 8 > 10.
  5. Solve for x: 3x + 9 ≥ 18.

CLASSWORK (5 questions):

  1. Solve for x: 3x + 10 < 22.
  2. Solve for x: 4x - 5 ≥ 11.
  3. Solve for x: 2x + 8 ≤ 14.
  4. Solve for x: 5x - 2 > 18.
  5. Solve for x: 7x + 9 < 50.

ASSIGNMENT (5 tasks):

  1. Solve for x: 6x + 5 > 23.
  2. Solve for x: 3x - 4 ≤ 20.
  3. Write and solve an inequality for the following situation: “The sum of a number and 7 is greater than 15.”
  4. Solve for x: 4x + 3 ≥ 9.
  5. Solve for x: 2x - 5 > 12.

 

PERIOD 2: Linear Inequalities in Two Variables

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of linear inequalities in two variables (e.g., y < 2x + 3). Demonstrates using the graph of a line and how to shade the correct region.

Students observe the graph and understand the relation between inequality and graph.

Step 2 - Solving Inequalities in Two Variables

Explains the steps to graph linear inequalities in two variables by converting the inequality into an equation, graphing it, and then shading the region that satisfies the inequality.

Students watch the demonstration and take notes on the steps.

Step 3 - Example Problem

Solves an example inequality, e.g., y ≤ x + 4, using graphing.

Students follow the teacher's demonstration.

Step 4 - Guided Practice

Provides other inequalities for students to graph and shade the correct regions.

Students practice graphing inequalities in pairs.

NOTE ON BOARD:

  • To graph y < 2x + 3:
  1. Graph the line y = 2x + 3.
  2. Shade the region below the line for y < 2x + 3.

EVALUATION (5 exercises):

  1. Graph the inequality: y ≥ x + 2.
  2. Graph the inequality: y > -x + 4.
  3. Graph the inequality: y ≤ 3x - 5.
  4. Graph the inequality: y < 2x + 6.
  5. Graph the inequality: y ≥ -2x + 1.

CLASSWORK (5 questions):

  1. Graph the inequality: y ≥ x - 1.
  2. Graph the inequality: y ≤ 4x + 3.
  3. Graph the inequality: y < -x + 2.
  4. Graph the inequality: y ≥ -3x + 4.
  5. Graph the inequality: y < 5x - 2.

ASSIGNMENT (5 tasks):

  1. Graph the inequality: y < x + 1.
  2. Graph the inequality: y ≥ 2x - 3.
  3. Graph the inequality: y > -x + 5.
  4. Graph the inequality: y ≤ -3x + 2.
  5. Describe a real-life scenario where linear inequalities in two variables can be applied.

 

PERIOD 3 & 4: Combined Inequalities and Application in Real Life

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces combined inequalities (e.g., 3 < 2x + 4 ≤ 10). Explains how to solve combined inequalities step-by-step.

Students observe the process and take notes.

Step 2 - Solving Combined Inequalities

Demonstrates solving an example of combined inequalities, e.g., 3 < 2x + 4 ≤ 10. Shows how to break the problem into parts and solve each.

Students follow along with the teacher and work out the solution.

Step 3 - Real-Life Application

Guides students to solve a problem involving the application of combined inequalities in a real-life situation, such as budgeting.

Students work through a real-life example in pairs.

Step 4 - Guided Practice

Provides combined inequalities for students to solve in class.

Students practice solving the inequalities individually or in groups.

NOTE ON BOARD:

  • Combined Inequality: 3 < 2x + 4 ≤ 10.
  • Solve each part separately and combine the results.

EVALUATION (5 exercises):

  1. Solve: 5 ≤ 3x - 1 < 8.
  2. Solve: -2 < 4x + 1 ≤ 6.
  3. Solve: 7 ≤ 2x + 5 < 12.
  4. Solve: 4 ≤ 3x - 2 < 9.
  5. Solve: -3 < 2x + 5 ≤ 7.

CLASSWORK (5 questions):

  1. Solve: 2 ≤ 3x - 1 < 7.
  2. Solve: 5 < 2x + 3 ≤ 8.
  3. Solve: 1 ≤ 2x + 4 < 9.
  4. Solve: -2 ≤ x + 5 < 3.
  5. Solve: 4 ≤ 2x + 1 < 10.

ASSIGNMENT (5 tasks):

  1. Solve: 6 ≤ 4x - 2 < 10.
  2. Solve: 2 ≤ x + 3 < 6.
  3. Solve: -4 ≤ 3x + 1 < 5.
  4. Apply linear inequalities to a real-life situation involving time management or budgeting.
  5. Solve a word problem involving linear inequalities in two variables.

 

PERIOD 5: Maximum and Minimum Values of Simultaneous Linear Inequalities

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces simultaneous linear inequalities and explains how to find the maximum and minimum values that satisfy both inequalities.

Students observe and take notes.

Step 2 - Solving Simultaneous Inequalities

Demonstrates how to solve simultaneous linear inequalities graphically by finding the intersection of shaded regions.

Students follow the teacher’s steps and participate in the graphing activity.

Step 3 - Application

Solves an example involving the maximum and minimum values of simultaneous inequalities.

Students practice solving similar problems.

Step 4 - Real-Life Application

Guides students to apply simultaneous inequalities in real-life situations such as optimizing profit or resource allocation.

Students work in pairs to solve a real-life problem.

NOTE ON BOARD:

  • Simultaneous Inequalities: Solve for maximum and minimum values by finding the intersection of shaded regions.

EVALUATION (5 exercises):

  1. Find the maximum and minimum values that satisfy the inequalities: x + y ≤ 5 and x ≥ 2.
  2. Find the maximum and minimum values that satisfy the inequalities: 2x + y ≤ 8 and `x - y ≥
  3. Find the maximum and minimum values for x + y ≤ 6andx ≥ 1.
  4. Find the maximum and minimum values for 2x + 3y ≤ 12andx + y ≥ 4`.
    5. Apply simultaneous inequalities to a real-life problem, such as determining the number of units to produce in a factory with limited resources.

CLASSWORK (5 questions):

  1. Find the maximum and minimum values for x + y ≤ 7 and x ≥ 3.
  2. Find the maximum and minimum values for 3x + y ≤ 10 and x - y ≥ 2.
  3. Find the maximum and minimum values for x + y ≤ 5 and x ≥ 2.
  4. Solve a real-life application problem using simultaneous inequalities.
  5. Identify the optimal solution for a business scenario using simultaneous inequalities.

 

SUMMARY AND CONCLUSION

  1. Recap of all topics covered: Linear inequalities in one variable, two variables, and simultaneous inequalities.
  2. The importance of inequalities in real-life situations, such as budgeting, resource allocation, and optimization.
  3. Reinforcement of the maximum and minimum value problem-solving approach.