Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Predict and identify regions for solutions to systems of linear inequalities.
• Solve systems of linear inequalities and represent solutions graphically.
• Analyze real-life situations modeled by linear inequalities.
• Use tools like GeoGebra to visualize solutions.

Lesson summary

In our daily lives in Ghana, we constantly make decisions based on limitations or constraints. A market woman must decide how many bags of rice and maize to stock with her limited capital. A driver must manage fuel and time to complete deliveries. A student must balance study time for Mathematics and English within the few hours they have after school. Algebra, specifically the study of inequalities, gives us a powerful tool to model these real-life situations with constraints and find all possible solutions. This lesson will teach you how to translate these problems into mathematical language and visualise the solutions on a graph.

Lesson notes

Part 1: Recap of Linear Inequalities in One Variable

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another. When solving them, we follow the same rules as for equations, with one critical exception: Golden Rule of Inequalities: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. `>` becomes ` 12 / -3` `x > –4`

Combining the Solutions: We need `x` to satisfy both conditions: `x -4`. We can write this as a single statement: `–4 `y = 4`. The point is (0, 4). When `y = 0`, `2x = 12` -> `x = 6`. The point is (6, 0). Plot these two points and draw a line through them. Solid or Dashed Line? Use a solid line if the inequality is `≤` (less than or equal to) or `≥` (greater than or equal to). This means points on the line are part of the solution. Use a dashed (or broken) line if the inequality is ` ` (greater than). This means points on the line are *not* part of the solution. For our example, `2x + 3y ≤ 12`, we use a solid line. Shade the Solution Region (Half-Plane): The boundary line divides the plane into two halves. We need to find which half is the solution. The Test Point Method: Pick a simple point that is *not* on the line. The origin, (0, 0), is the best choice unless the line passes through it. Substitute the test point into the *original inequality*. `2x + 3y ≤ 12` `2(0) + 3(0) ≤ 12` `0 ≤ 12` Is this statement true? Yes, 0 is less than or equal to 12. Conclusion: Since the statement is true, the region containing the test point (0, 0) is the solution. We shade this entire region. If the statement were false, we would shade the other side of the line.

Visual Representation:

Evaluation guide

• Predict and identify the region representing the solution to systems of linear inequality.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -Based
• Learning and Talk for Learning
• Activity: Regions representing solutions of systems of equations and inequalities.