Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Inequalities
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Inequalities
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Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 1
Theme: Pure Mathematics
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This topic introduces students to solving inequalities beyond linear forms, focusing on quadratic inequalities and inequalities involving two variables. Understanding inequalities is fundamental in various fields, enabling individuals to model and solve problems involving constraints, limits, optimization, and resource allocation. In the Nigerian context, this knowledge is crucial for professions such as economics (budgeting, profit maximization), engineering (design constraints), and resource management (land use, water distribution), allowing for informed decision-making within specified boundaries.
Performance Objectives:
Definition of Inequality: An inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations that show exact equality, inequalities show a range of values.
Inequality Symbols: `>` : greater than ` 0`, then `ac bc` and `a/c > b/c`. This is a critical point often forgotten by students. A. Solving Quadratic Inequalities in One Variable A quadratic inequality is an inequality that can be written in one of the following forms: `ax2 + bx + c > 0`, `ax2 + bx + c 0` (or 0`, the parabola opens upwards (U-shape). If `a 0` or `ax2 + bx + c ≥ 0`, the solution consists of the x-values where the parabola is above the x-axis. For `ax2 + bx + c , 0`.
Solution: Rewrite: The inequality is already in the desired form: `-x2 + 6x - 5 > 0`.
Find roots: Solve `-x2 + 6x - 5 = 0`.
Multiply by -1: `x2 - 6x + 5 = 0` Factorize: `(x - 1)(x - 5) = 0` Roots are `x = 1` and `x = 5`.
Shape of parabola: `a = -1` (from `-x2`), so `a 0`, which means the part of the parabola above the x-axis. This occurs between `x = 1` and `x = 5`. The solution is `1 0` (or ` or `≥`, negative for ` 6`.
Create a sign table: | Interval | Test Value (x) | `(x - 2)` Sign | `(x - 6)` Sign | `(x - 2)(x - 6)` Sign | `x2 - 8x + 12 ≤ 0`? | | :----------- | :------------- | :------------- | :------------- | :-------------------- | :------------------ | | `x 6` | `7` | `+` | `+` | `(+)(+) = +` | No | Identify solution region: The inequality `x2 - 8x + 12 ≤ 0` is satisfied when the expression is negative or zero. This occurs in the interval `2 , ` or `0` mean for the parabola?").
Student Activity: Observe and take notes. Ask questions for clarification. Attempt similar problems as guided practice in their notebooks, comparing their steps with the teacher's.
Activity 3: Solving Inequalities in Two Dimensions (20 minutes)
Teacher Activity: Introduce inequalities in two dimensions, explaining that solutions are regions on a graph. Work through Example 2.3 step-by-step on the whiteboard, meticulously plotting the line, choosing a test point, and shading the correct region. Emphasize the difference between solid and dashed lines. Provide graph paper for students to follow along.
Student Activity: Actively participate by suggesting coordinates, test points, and shading directions. Draw the graph for Example 2.3 on their graph paper as the teacher demonstrates. Engage in peer-to-peer discussion about shading.
Activity 4: Group Problem Solving (15 minutes)
Teacher Activity: Divide the class into small groups (3-4 students). Assign each group one or two problems (e.g., one quadratic inequality and one 2D inequality). Circulate among groups, providing guidance, checking progress, and addressing misconceptions. Encourage group members to explain their reasoning to each other.
Student Activity: Collaborate within their groups to solve the assigned problems. Present their solutions on the board or to the class, explaining their steps. Critique and discuss solutions from other groups.
Activity 5: Consolidation and Wrap-up (5 minutes)
Teacher Activity: Summarize the key methods for solving quadratic inequalities and 2D inequalities. Reiterate the importance of clear steps, especially when dealing with signs and line types. Assign independent practice questions as homework.
Student Activity: Ask any remaining questions. Note down homework assignments. --- The teacher should present these questions and guide students through their solutions, allowing for student input before revealing the full solution. Question 1 (Quadratic Inequality - Graphical Method): Solve `x2 - 5x + 4 ≥ 0`.
Solution: Rewrite: The inequality is already in the form `ax2 + bx + c ≥ 0`.
Find roots: Solve `x2 - 5x + 4 = 0`.
Factorize: `(x - 1)(x - 4) = 0` Roots are `x = 1` and `x = 4`.
Shape of parabola: `a = 1` (from `x2`), so `a > 0`. The parabola opens upwards.
Sketch: Plot roots 1 and 4 on a number line. Sketch an upward-opening parabola passing through these points.
Identify solution region: We need `x2 - 5x + 4 ≥ 0`, meaning the part of the parabola above or on the x-axis. This occurs when `x` is less than or equal to 1, or `x` is greater than or equal to
4. The solution is `x ≤ 1` or `x ≥ 4`.
Commentary: Emphasize the "or" condition here, as the solution set is disjoint. Also, highlight the use of "or equal to" due to the `≥` sign, making the roots inclusive. Question 2 (Quadratic Inequality - Test Value Method): Solve `2x2 + 7x - 4 1/2`.
Create a sign table: | Interval | Test Value (x) | `(2x - 1)` Sign | `(x + 4)` Sign | `(2x - 1)(x + 4)` Sign | `2x2 + 7x - 4 1/2` | `1` | `+` | `+` | `(+)(+) = +` | No | Identify solution region: The inequality `2x2 + 7x - 4 `). Remind students to factorize carefully or use the quadratic formula if factorization is difficult.
Question 3 (Inequality in Two Dimensions): A trader at Tejuosho market wishes to sell `p` bags of rice and `q` bags of beans. Due to space constraints, the total number of bags cannot exceed
1
0. Represent the inequality `p + q ≤ 10` graphically, assuming `p ≥ 0` and `q ≥ 0`.
Solution: Replace inequality with equality: `p + q = 10`.
Plot the boundary line: If `p = 0`, then `q = 10`.
Point: `(0, 10)` If `q = 0`, then `p = 10`.
Point: `(10, 0)` Since the inequality is `≤`, draw a solid line connecting `(0, 10)` and `(10, 0)`.
Choose a test point: Let's use `(0, 0)`.
Substitute the test point: Substitute `p = 0`, `q = 0` into `p + q ≤ 10`. `0 + 0 ≤ 10` `0 ≤ 10` This statement is True.
Shade the region: Since `(0, 0)` satisfies the inequality, shade the region that contains `(0, 0)`. This is the region below the line `p + q = 10`. Also, consider `p ≥ 0` and `q ≥ 0`, so the shading is confined to the first quadrant.
Commentary: Ensure students understand that in real-world contexts, variables like `p` and `q` (number of bags) cannot be negative, hence the restriction to the first quadrant. --- Differentiation and Remediation (Supporting Struggling Learners): Revisit Prerequisite Skills: Factoring Quadratics: Provide extra practice on factorizing quadratic expressions. Students often struggle with quadratic inequalities due to weak factoring skills. Use simple quadratic expressions (e.g., `x2 - 4`, `x2 - 5x + 6`).
Solving Linear Inequalities: Review the rules for multiplying/dividing by negative numbers, as this principle is vital for both linear and quadratic inequalities.
Plotting Points and Lines: Provide direct instruction and practice on plotting points and drawing straight lines from equations.
Visual Aids and Step-by-Step Scaffolding: Graphical Method Emphasis: For quadratic inequalities, strongly emphasize the graphical method as it is often more intuitive. Use pre-drawn parabolas to illustrate the regions above/below the x-axis.
Colour-Coding: Use different colours to represent positive and negative regions in sign tables or on the number line.
Checklist: Provide a step-by-step checklist for solving each type of inequality.
Simplified Problems: Start with quadratic inequalities that have integer roots and `a=1` (e.g., `x2 - 3x + 2 > 0`). For 2D inequalities, begin with simple forms like `x + y x`. Work through one problem extensively together as a class, with students dictating each step.
Peer Tutoring: Pair struggling learners with more advanced students for one-on-one or small-group support during practice sessions. Extension and Enrichment (Challenging High-Achieving Learners): Inequalities Involving Absolute Values: Introduce solving inequalities of the form `|ax + b| 0`). This requires considering critical points from both the numerator and denominator, and handling restrictions (denominator cannot be zero).
Example: Solve `(x + 3)/(x - 2) ≥ 0`. (Solution: `x ≤ -3` or `x > 2`) Systems of Linear Inequalities in Two Variables (Introduction to Linear Programming): Present problems requiring the graphing of multiple linear inequalities on the same plane to find a feasible region.
Example: Graph the feasible region for: `x + y ≤ 5` `y ≥ x` `x ≥ 0, y ≥ 0` This lays a strong foundation for future topics in linear programming, which has significant real-world applications in optimization (e.g., maximizing profit, minimizing cost).
Word Problems with Complex Formulations: Provide word problems that require students to formulate the inequalities themselves from more complex real-life scenarios before solving. Pure Mathematics
Budgeting and Resource Allocation in Nigerian Households/Communities: Scenario: A Nigerian household has a monthly income of NGN 150,
0
0
0. They allocate `x` for food and `y` for school fees and transportation. The sum of these two expenses must not exceed their income, and food expense is often prioritized.
Application: This can be modeled as `x + y ≤ 150,000`, with `x ≥ 0` and `y ≥ 0`. Graphing this inequality helps visualize the permissible combinations of spending, allowing the household to plan effectively without overspending. Community leaders could use similar models to allocate funds for projects like boreholes (`B`) and healthcare centers (`H`) within a budget `B + H ≤ Total Fund`. Agricultural Planning and Profit Optimization in Rural Nigeria: Scenario: A farmer in Benue State cultivates yam (`x` hectares) and cassava (`y` hectares). The total land available is 10 hectares. Also, due to market demand, the area planted with yam must be at least twice the area planted with cassava.
Application: This creates a system of inequalities: `x + y ≤ 10` (land constraint) `x ≥ 2y` (market demand constraint) `x ≥ 0, y ≥ 0` (non-negative land area) Solving this graphically helps the farmer identify the feasible region for planting both crops, allowing them to make decisions that maximize their yield or profit based on other factors like fertilizer availability and labor. The quadratic inequalities might come into play when considering the yield per hectare, which might not be linear (e.g., too much fertilizer causing diminishing returns). Small Business Profitability and Pricing Strategy (e.g., tailor, baker in Lagos): Scenario: A local baker in Lagos produces loaves of bread. The profit `P(x)` (in Naira) from selling `x` loaves is modeled by a quadratic function, perhaps `P(x) = -2x2 + 100x - 800`. The baker wants to determine the range of loaves to bake to ensure a profit of at least NGN
5
0
0. Application: The problem translates to solving ` -2x2 + 100x - 800 ≥ 500`.
Rearranging yields a quadratic inequality: ` -2x2 + 100x - 1300 ≥ 0`. Solving this inequality helps the baker understand the minimum and maximum number of loaves they need to sell to achieve their profit target, informing their production and pricing strategies. ---