Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Linear inequalities
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Linear inequalities
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Subject: Further Mathematics
Class: Senior Secondary 1
Term: 3rd Term
Week: 2
Theme: Pure Mathematics
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
An inequality is a mathematical statement comparing two expressions that are not necessarily equal. Unlike equations which use an equals sign (=), inequalities use one of the following symbols: (greater than) ≤ (less than or equal to) ≥ (greater than or equal to) A linear inequality is an inequality that involves a linear expression. For example, $2x + 3 0$, then $ac bc$ and $a/c > b/c$.
Representing Solutions on a Number Line: For , use an open circle (or an unshaded circle) at the critical value, indicating that the value itself is not included in the solution set.
exceed ₦500: $50x + 100y \le 500$ (Also, $x \ge 0$ and $y \ge 0$ since quantity cannot be negative).
Step 2: Simplify the inequality (optional but good practice): Divide by 50: $x + 2y \le 10$ Step 3: Convert to equation: $x + 2y = 10$ Step 4: Determine line type: Solid line (due to $\le$).
Step 5: Find points for the boundary line. If $x = 0$: $0 + 2y = 10 \implies 2y = 10 \implies y = 5$.
Point: $(0, 5)$. If $y = 0$: $x + 2(0) = 10 \implies x = 10$.
Point: $(10, 0)$. Plot $(0,5)$ and $(10,0)$ and draw a solid line.
Step 6: Choose a test point. Use $(0,0)$.
Step 7: Substitute into the original inequality: $0 + 2(0) \le 10$ $0 \le 10$ (TRUE)
Step 8: Shade the unwanted region. Since $(0,0)$ makes the inequality TRUE, the region containing $(0,0)$ is the solution. Shade the region above the line. The unshaded region, confined to the first quadrant (due to $x \ge 0, y \ge 0$), represents the possible combinations of snacks. (Visual representation needed: A coordinate plane showing only the first quadrant. A solid line passes through (0,5) and (10,0). The region above this line (and outside the first quadrant) is shaded. The unshaded triangular region in the first quadrant, including the boundaries, is the solution region.) Linear inequalities in two variables (e.g., $Ax + By , ≤, ≥) with an equality sign (=) to obtain the equation of the boundary line.
Example: $2x - y (strict inequalities), the boundary line is broken (dashed). This indicates that points on the line are not part of the solution. If the original inequality contains ≤ or ≥ (non-strict inequalities), the boundary line is solid. This indicates that points on the line are part of the solution.
3. Draw the boundary line: Find at least two points that satisfy the equation. The x- and y-intercepts are usually the easiest: To find the x-intercept, set $y=0$ and solve for $x$. To find the y-intercept, set $x=0$ and solve for $y$. Plot these points and draw the appropriate line (broken or solid).
4. Choose a test point: Select any point not on the boundary line. The origin $(0,0)$ is the most convenient choice, unless the line passes through it.
5. Substitute the test point coordinates into the original inequality: If the substitution results in a true statement, the region containing the test point is the solution region. If the substitution results in a false statement, the region opposite to the test point is the solution region.
6. Shade the region: In many Nigerian educational contexts, it is common and often preferred to shade the UNWANTED region. This leaves the solution region (the feasible region) clear and unshaded, making it easier to identify. If the test point makes the inequality true, shade the region not containing the test point. If the test point makes the inequality false, shade the region containing the test point. Alternatively, one can shade the WANTED region and label it clearly as the "solution region." Consistency is key. For this lesson note, the method of shading the unwanted region will be applied in examples.
Worked Example (Two Variables): Graph the inequality $x - 2y \ge -2$.
Step 1: Convert to equation: $x - 2y = -2$ Step 2: Determine line type: Solid line (due to $\ge$).
Step 3: Find points for the boundary line. If $x = 0$: $0 - 2y = -2 \implies -2y = -2 \implies y = 1$.
Point: $(0, 1)$. If $y = 0$: $x - 2(0) = -2 \implies x = -2$.
Point: $(-2, 0)$. Plot $(0,1)$ and $(-2,0)$ and draw a solid line through them.
Step 4: Choose a test point. Use $(0,0)$.
Step 5: Substitute into the original inequality: $0 - 2(0) \ge -2$ $0 \ge -2$ (This is TRUE)
Step 6: Shade the unwanted region. Since $(0,0)$ makes the inequality TRUE, the region containing $(0,0)$ is the solution region.
Therefore, shade the region not containing $(0,0)$, which is the region above the line. The unshaded region (below and including the line) is the solution. (Visual representation needed: A coordinate plane. A solid line passes through (-2,0) and (0,1). The region above this line is shaded, leaving the region below and on the line unshaded as the solution region.)
Example (Nigerian Context): A student has a weekly budget of ₦500 for snacks. A sachet of juice (J) costs ₦50 and a packet of biscuits (B) costs ₦
1
0
0. Let 'x' be the number of sachets of juice and 'y' be the number of packets of biscuits bought. Formulate an inequality and graph it to show possible combinations of snacks the student can buy.
Step 1: Formulate the inequality.
Cost of juice: $50x$ Cost of biscuits: $100y$ Total cost must not exceed ₦500: $50x + 100y \le 500$ (Also, $x \ge 0$ and $y \ge 0$ since quantity cannot be negative).
Step 2: Simplify the inequality (optional but good practice): Divide by 50: $x + 2y \le 10$ Step 3: Convert to equation: $x + 2y = 10$ Step 4: Determine line type: Solid line (due to $\le$).
Step 5: Find points for the boundary line.** If $x = 0$: $0 + 2y = 10 \implies 2y = 10 \implies y = 5$.
Point: $(0, 5)$. If Linear inequalities in two variables (e.g., $ax + by c$, $ax + by \le c$, $ax + by \ge c$) have solution sets that are regions on a Cartesian plane. Steps for Graphing a Linear Inequality in Two Variables: Replace the inequality sign with an equality sign to obtain the equation of the boundary line. For example, $3x + 2y \le 6$ becomes $3x + 2y = 6$.
Draw the boundary line: Find at least two points that satisfy the equation (e.g., intercepts). To find x-intercept, set $y = 0$. To find y-intercept, set $x = 0$. If the original inequality is strict (), draw a broken (dashed) line to indicate that points on the line are not part of the solution. If the original inequality is non-strict (≤ or ≥), draw a solid line to indicate that points on the line are part of the solution. Choose a test point not on the line (the origin $(0,0)$ is often the easiest, if it's not on the line). Substitute the coordinates of the test point into the original inequality. If the test point makes the inequality true, then the region containing the test point is the solution region. If the test point makes the inequality false, then the region opposite to the test point is the solution region.
Shade the region: In many Nigerian contexts and for clarity, it is common practice to shade the unwanted region, leaving the solution region (the feasible region) unshaded. This makes the solution region clearly visible. Alternatively, one can shade the wanted region and label it as the "solution region." Ensure consistency in the chosen method. This lesson note will use the method of shading the unwanted region.
Worked Example (Two Variables): Graph the inequality $3x + 2y \le 6$.
Step 1: Convert to equation: $3x + 2y = 6$ Step 2: Find points for the boundary line. If $x = 0$: $3(0) + 2y = 6 \implies 2y = 6 \implies y = 3$.
Point: $(0, 3)$. If $y = 0$: $3x + 2(0) = 6 \implies 3x = 6 \implies x = 2$.
Point: $(2, 0)$.
Step 3: Draw the line. Since the inequality is $3x + 2y \le 6$ (non-strict), draw a solid line connecting $(0, 3)$ and $(2, 0)$.
Step 4: Choose a test point. Use $(0, 0)$.
Step 5: Substitute into the original inequality. $3(0) + 2(0) \le 6$ $0 \le 6$ (This is TRUE)
Step 6: Shade the unwanted region. Since $(0,0)$ makes the inequality TRUE, the region containing $(0,0)$ is the solution region.
Therefore, shade the region not containing $(0,0)$ (the region above the line). The unshaded region is the solution. (Graphical representation needs to be visualised: A coordinate plane with a solid line passing through (0,3) and (2,0). The region above this line is shaded, leaving the region below the line (which includes the origin) unshaded as the solution region.)
Example (Nigerian Context): A farmer wants to plant maize (M) and yam (Y) on a total land area not exceeding 10 hectares. Each hectare of maize requires 2 bags of fertilizer, and each hectare of yam requires 1 bag. The farmer has a maximum of 15 bags of fertilizer. Let 'x' be the number of hectares for maize and 'y' for yam.
Formulate the inequalities: Land area constraint: $x + y \le 10$ Fertilizer constraint: $2x + y \le 15$ Non-negativity: $x \ge 0, y \ge 0$ (hectares cannot be negative) This problem involves a system of inequalities, which is an extension. For the purpose of this lesson (graphing a single inequality), the teacher can focus on just one of them initially, e.g., $x+y \le 10$. Graphing $x+y \le 10$: Equation: $x + y = 10$ Points: $(0, 10)$ and $(10, 0)$.
Line: Solid (due to $\le$). Test point $(0,0)$: $0 + 0 \le 10 \implies 0 \le 10$ (TRUE).
Shade: Region above the line (unwanted region). The solution is the region below the line. This section outlines the activities for a typical 40-minute lesson period. If the school runs a double period, the teacher can extend activities, perhaps by introducing systems of inequalities. | Teacher Activities | Student Activities | Activities for students to engage actively. | 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Senior Secondary 1 Further Mathematics
Budgeting and Financial Planning (Household/Small Business): Description: Families or small business owners in Nigeria constantly face limitations on income and resources. Linear inequalities can model these constraints. For instance, a household budget of ₦50,000 for food for a month, buying rice (x kg) at ₦800/kg and beans (y kg) at ₦700/kg, can be represented by $800x + 700y \le 50000$. Graphing this inequality helps to visualise all possible combinations of rice and beans that can be purchased within the budget.
Local Context: Useful for "Mama Put" (local food vendors) to plan purchases of ingredients, or for artisans (e.g., shoemakers, tailors) to manage raw material costs. Resource Allocation (Agriculture/Production): Description: Farmers often have limited land, fertilizer, or labor hours. If a farmer has 5 hectares of land and wants to plant maize (x hectares) and cassava (y hectares), then $x + y \le 5$. Additionally, if maize requires 2 bags of fertilizer per hectare and cassava 1 bag per hectare, and the farmer has 8 bags of fertilizer, then $2x + y \le 8$. Graphing these inequalities helps the farmer determine the feasible planting combinations that satisfy both land and fertilizer constraints.
Local Context: Directly applicable to small-scale farmers in rural and peri-urban areas of Nigeria to optimize yield based on available resources. Health and Safety Regulations (Public Transport/Infrastructure): Description: Many public services and infrastructure have capacity or weight limits for safety. For example, a commercial bus might have a maximum passenger capacity (e.g., not more than 18 passengers: $P \le 18$) or a bridge might have a maximum weight limit (e.g., total weight of vehicles $W \le 10,000 \text{ kg}$). Speed limits on roads are also inequalities (e.g., speed $S \le 100 \text{ km/h}$).
Local Context: Relevant for enforcing road safety, managing passenger load in commercial vehicles like "Danfo" or "Molue" buses, and ensuring structural integrity of bridges.