Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define and explain linear inequalities.
• Solve simultaneous linear inequalities using graphical methods.
• Model real-life problems involving linear inequalities.
• Identify the solution region on a graph.

Lesson summary

This lesson explores how we can use algebra, specifically inequalities, to model and solve real-life problems. In Ghana, from the market woman managing her stock to a farmer planning their planting season, people constantly make decisions based on constraints like limited money, time, or resources. Algebra gives us a powerful tool to represent these constraints mathematically and find the best possible solutions. We will focus on using graphs to visualise these problems, making complex situations easier to understand. We will begin with systems of linear inequalities and then apply similar modelling principles to situations involving quadratic inequalities.

Lesson notes

This topic is broken down into two main parts: solving simultaneous linear inequalities graphically, and modelling problems that lead to quadratic inequalities. Part 1: Solving Simultaneous Linear Inequalities Graphically

When a problem has multiple conditions or constraints that must all be satisfied at the same time, we have a system of simultaneous inequalities. A graphical approach is an excellent way to see all possible solutions.

The Process: Identify the variables: Determine the unknown quantities in the problem (e.g., number of items to buy, amount of land to cultivate). Assign variables like `x` and `y` to them. Formulate the inequalities: Translate each constraint or condition from the word problem into a mathematical inequality involving your variables. Graph each inequality: For each inequality: Treat it as an equation: First, pretend the inequality sign (` `, `≤`, `≥`) is an equals sign (`=`) to find the boundary line. Draw the boundary line: Find two points on the line (the x- and y-intercepts are often easiest). If the inequality is `≤` or `≥` (includes "or equal to"), draw a solid line `(_______)`. If the inequality is ` ` (strictly less/greater than), draw a dashed line `(- - - - -)`. Shade the correct region: Pick a test point that is NOT on the line (the origin `(0, 0)` is usually the easiest). Substitute its coordinates into the *original* inequality. If the test point makes the inequality TRUE, shade the side of the line that contains the test point. If it makes the inequality FALSE, shade the other side of the line. Identify the Feasible Region: After graphing all the inequalities, the area where ALL the shaded regions overlap is called the feasible region. Every single point within this region is a possible solution to the problem.

Evaluation guide

• Use a graphical approach (by hand and technology) to solve simultaneous linear inequalities.
• Collaborative Learning: Learners to work in convenient groups (ability, mixed -ability, mixed gender,
• etc.) to model real life problems involving linear inequalities and present answers.
• Talk for Learning Approaches –learners engage in brainstorming, think -pair-share, building on what