Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Linear inequalities
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Linear inequalities
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 2nd Term
Week: 3
Theme: Algebraic Processes
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Identify linear in equality in one variable Solve linear in equality in one variable Represent solutions of linear in equalities in one variable on number line Solve word problems in volving linear in equalities in one variable.
both sides by 3 (a positive number, so sign doesn't change): $3p/3 ≤ 15/3$ $p ≤ 5$
3. Number Line Representation: Draw a number line. Place a closed circle at 5 (because $p$ is less than or equal to 5). Draw an arrow pointing to the left from 5, indicating all numbers less than or equal to 5 are solutions. ``` 4 5 6 ``` Example 4: Solving and Representing (Multiple Steps, Negative Coefficient) Solve the inequality $10 - 2m 6/(-2)$ $m > -3$
3. Number Line Representation: Draw a number line. Place an open circle at -3 (because $m$ is greater than -3). Draw an arrow pointing to the right from -3. ``` -4 -3 -2 ``` Example 5: Solving a Word Problem (Nigerian Context) A Nigerian student needs to score at least 250 in JAMB to be considered for admission into a particular university. If the student scored $x$ marks in the exam, write an inequality to represent this situation and sketch the possible range of scores on a number line. * Solution:
1. Translate to Inequality: "At least 250" means the score must be 250 or higher. So, $x ≥ 250$.
2. Number Line Representation: Draw a number line. Place a closed circle at 250 (because the score can be 250). Draw an arrow pointing to the right from 250. ``` 240 250 260 ``` 2.1 Definition of an Inequality: An inequality is a mathematical statement that compares two expressions using an inequality symbol, indicating that one expression is not equal to the other, or is greater than, less than, greater than or equal to, or less than or equal to the other. 2.2 Inequality Symbols: The following symbols are used to represent inequalities: (greater than): Indicates that the expression on the left is larger than the expression on the right.
Example: $y > 2$ means $y$ can be any number larger than 2 (e.g., 3, 10, 2.01). ≤ (less than or equal to): Indicates that the expression on the left is smaller than or equal to the expression on the right.
Example: $z ≤ 7$ means $z$ can be any number smaller than or equal to 7 (e.g., 7, 6, 0, -5). ≥ (greater than or equal to): Indicates that the expression on the left is larger than or equal to the expression on the right.
Example: $m ≥ -3$ means $m$ can be any number larger than or equal to -3 (e.g., -3, 0, 10). 2.3 Linear Inequality in One Variable: A linear inequality in one variable is an inequality that involves a single variable (e.g., $x, y, p$) raised to the power of
1. It can be written in the form $ax + b c$, $ax + b ≤ c$, or $ax + b ≥ c$, where $a, b, c$ are constants and $a ≠ 0$. 2.4 Rules for Solving Linear Inequalities: Solving linear inequalities is similar to solving linear equations, with one crucial difference:
1. Adding or Subtracting: Adding or subtracting the same number to/from both sides of an inequality does not change the direction of the inequality sign.
Example: If $x - 3 12$, then $(-3x)/(-3) ' sign reversed to '$). It indicates that the endpoint value is not included in the solution set.
Closed Circle (or solid circle): Used for inclusive inequalities (≤ or ≥). It indicates that the endpoint value is included in the solution set.
Direction of Arrow: If the variable is less than the number (e.g., $x 2$), the arrow points to the right. Worked
Examples: Example 1: Identifying a Linear Inequality Which of the following is a linear inequality in one variable? a) $2x + 5 = 11$ b) $x^2 - 3 > 7$ c) $4y - 1 ≤ 9$ d) $3x + 2y ≥ 6$ Solution: a) is an equation. b) is an inequality, but not linear (due to $x^2$). c) is a linear inequality in one variable ($y$). d) is a linear inequality, but in two variables ($x$ and $y$).
Therefore, the correct answer is c) $4y - 1 ≤ 9$.
Example 2: Solving and Representing (Simple) Solve the inequality $x + 6 > 10$ and represent the solution on a number line.
Solution:
1. Subtract 6 from both sides: $x + 6 - 6 > 10 - 6$ $x > 4$
2. Number Line Representation: Draw a number line. Place an open circle at 4 (because $x$ is greater than 4, not equal to 4). Draw an arrow pointing to the right from 4, indicating all numbers greater than 4 are solutions. ``` 3 4 5 ``` Example 3: Solving and Representing (Multiple Steps, Positive Coefficient) Solve the inequality $3p - 7 ≤ 8$ and represent the solution on a number line.
Solution:
1. Add 7 to both sides: $3p - 7 + 7 ≤ 8 + 7$ $3p ≤ 15$
2. Divide both sides by 3 (a positive number, so sign doesn't change): $3p/3 ≤ 15/3$ $p ≤ 5$
3. Number Line Representation: Draw a number line. Place a closed circle at 5 (because $p$ is less than or equal to 5). Draw an arrow pointing to the left from 5, indicating all numbers less than or equal to 5 are solutions. ``` 4 5 6 ``` Example 4: Solving and Representing (Multiple Steps, Negative Coefficient) Solve the inequality $10 - 2m 6/(-2)$ $m > -3$
3. Number Line Representation: Draw a number line. 3.1 Introduction (10 minutes)
Teacher Activity: Begin by reviewing linear equations and how they represent exact values. Introduce the concept of "inequality" by posing real-life scenarios familiar to students: "The minimum age to vote in Nigeria is 18 years." (Ask: Can someone who is 17 vote? Can someone who is 20 vote? What mathematical symbol fits 'at least 18'?) "The speed limit on this road is 100 km/h." (Ask: Can a car go 110? Can it go 90? What symbol means 'not more than 100'?) Introduce the inequality symbols (, ≤, ≥) and explain their meanings with simple numerical examples (e.g., 5 3, 7 ≤ 7, 7 ≥ 5). Define a linear inequality in one variable.
Student Activity: Engage in a brief discussion on the real-life scenarios, identifying whether a value is included or excluded. Identify and state other examples of inequalities from their daily lives (e.g., "money in my pocket is less than NGN 500," "number of students in class is greater than 20"). 3.2 Development of Concepts (40 minutes)
Teacher Activity: Rule 1 & 2 (Adding/Subtracting, Multiplying/Dividing by Positive): Explain and demonstrate the rules for adding/subtracting a number from both sides and multiplying/dividing by a positive number. Emphasize that the inequality sign remains unchanged. Work through Example 2 ($x + 6 > 10$) and Example 3 ($3p - 7 ≤ 8$) step-by-step on the board. Guide students on representing solutions on a number line, explaining the use of open and closed circles and the direction of the arrow.
Rule 3 (Multiplying/Dividing by Negative): Highlight the critical rule: when multiplying or dividing by a negative number, the inequality sign MUST be reversed. Demonstrate this with a clear example like Example 4 ($10 - 2m -3).
Word Problems: Explain strategies for translating word problems into mathematical inequalities. Identify key phrases (e.g., "at least," "at most," "minimum," "maximum," "more than," "less than"). Work through Example 5 (JAMB score problem) to demonstrate the translation and solution process.
Student Activity: Work along with the teacher, solving problems in their notebooks. Participate in Q&A sessions, asking clarifying questions. Practice representing solutions on individual number lines. In pairs or small groups, discuss the sign reversal rule and try to explain it to each other. Attempt to translate simple verbal statements into inequalities. 3.3 Guided Practice (20 minutes)
Teacher Activity: Provide scaffolded practice questions (as outlined in section 4) for students to solve. Circulate around the classroom, monitoring student progress, providing immediate feedback, and correcting misconceptions. Call on students to present their solutions on the board and explain their steps.
Student Activity: Work individually or in small groups to solve the guided practice questions. Present solutions and actively participate in peer review and correction. 3.4 Conclusion (10 minutes)
Teacher Activity: Summarise the key points of the lesson: identification of linear inequalities, rules for solving (especially the sign reversal), and number line representation. Emphasise the importance of inequalities in real-world problem-solving. Assign independent practice/homework.
Student Activity: Ask any final questions. Note down homework assignment.
Question 1: Identify which of the following expressions is a linear inequality in one variable: a) $x + 7 = 12$ b) $5m^2 - 2 ≥ 0$ c) $3y - 4 5$ Solution 1: Correct Answer: c) $3y - 4 6 7 8 ```
Commentary: The solution includes 7, so a closed circle is used. The values are less than or equal to 7, so the arrow points left.
Question 3: Solve the inequality $15 - 4k > 3$ and represent its solution on a number line.
Solution 3: Subtract 15 from both sides: $15 - 4k - 15 > 3 - 15$ $-4k > -12$ Divide both sides by -
4. Crucially, reverse the inequality sign. $(-4k)/(-4) 2 3 4 ```
Commentary: The inequality sign was reversed because division by a negative number occurred. The solution does not include 3, hence an open circle and arrow pointing left for values less than
3. Question 4: A tailor in Kaduna needs to buy fabric. Each meter of fabric costs NGN
8
0
0. If the tailor has a budget of NGN 12,000 for fabric, what is the maximum number of meters of fabric, $m$, the tailor can buy? Write an inequality and solve it.
Solution 4: Translate to Inequality: The cost of $m$ meters of fabric is $800m$. The tailor's budget is NGN 12,000, meaning the cost must be less than or equal to NGN 12,
0
0
0. So, $800m ≤ 12000$.
Solve the inequality: Divide both sides by 800 (positive number, sign remains the same): $800m/800 ≤ 12000/800$ $m ≤ 15$
Commentary: The tailor can buy a maximum of 15 meters of fabric. This is a practical application of "less than or equal to" in a budgeting context.
Example 1: Identifying a Linear Inequality
Which of the following is a linear inequality in one variable?
a) $2x + 5 = 11$
b) $x^2 - 3 > 7$
c) $4y - 1 ≤ 9$
d) $3x + 2y ≥ 6$
Solution:
a) is an equation.
b) is an inequality, but not linear (due to $x^2$).
c) is a linear inequality in one variable ($y$).
d) is a linear inequality, but in two variables ($x$ and $y$).
Therefore, the correct answer is c) $4y - 1 ≤ 9$.
Example 2: Solving and Representing (Simple)
Solve the inequality $x + 6 > 10$ and represent the solution on a number line.
Solution:
Subtract 6 from both sides:
$x + 6 - 6 > 10 - 6$
$x > 4$
Number Line Representation:
Draw a number line. Place an open circle at 4 (because $x$ is greater than 4, not equal to 4). Draw an arrow pointing to the right from 4, indicating all numbers greater than 4 are solutions.
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3 4 5
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Example 3: Solving and Representing (Multiple Steps, Positive Coefficient)
Solve the inequality $3p - 7 ≤ 8$ and represent the solution on a number line.
Solution:
Add 7 to both sides:
$3p - 7 + 7 ≤ 8 + 7$
$3p ≤ 15$
Divide both sides by 3 (a positive number, so sign doesn't change):
$3p/3 ≤ 15/3$
$p ≤ 5$
Number Line Representation:
Draw a number line. Place a closed circle at 5 (because $p$ is less than or equal to 5). Draw an arrow pointing to the left from 5, indicating all numbers less than or equal to 5 are solutions.
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Transport and Load Limits (Safety): Many lorries and vehicles in Nigeria operate on roads and bridges with specific weight or passenger capacity limits. For instance, a small commercial bus (Danfo in Lagos) might have a capacity of 18 passengers. This can be represented as $P ≤ 18$, where $P$ is the number of passengers. Understanding this helps prevent overloading, which is a common cause of accidents and road damage. Similarly, bridge load limits ($W ≤ 5000$ kg) ensure structural integrity and public safety. Age Restrictions and Eligibility (Citizenship/Education): Various activities and civic responsibilities in Nigeria have age restrictions.
Voting: A citizen must be at least 18 years old to vote ($A ≥ 18$).
JAMB/Tertiary Education: Candidates usually need to be a minimum age (e.g., 16 years) to sit for JAMB examinations ($A ≥ 16$).
Driving License: The minimum age for obtaining a driver's license in Nigeria is 18 years ($A ≥ 18$). These are clear examples of "greater than or equal to" inequalities influencing daily life and access to opportunities. Business and Financial Management (Economy/Personal Finance): Profit Margins: A small business owner might aim for a profit of "at least NGN 50,000" per month ($P ≥ 50,000$) to sustain the business and personal needs.
Budgeting: A household might set a budget for groceries of "not more than NGN 20,000" per week ($E ≤ 20,000$), using inequalities to manage expenditure and avoid debt.
Savings: An individual saving for a project needs to accumulate "more than NGN 100,000" ($S > 100,000$) before starting. These scenarios demonstrate how inequalities are fundamental to financial planning and decision-making in the Nigerian economy.