Linear Inequalities — Lesson
1) Hook — A Fun Real-Life Example
Imagine you are organizing a cultural fest in your school in Delhi. You have a budget of ₹50,000 for snacks and decorations. Each packet of snacks costs ₹200, and each decoration item costs ₹500. You want to buy at least 30 items in total but cannot exceed your budget. How many packets of snacks and decoration items can you buy without crossing your budget?
This situation can be modelled using linear inequalities, which help us find all possible combinations that satisfy the budget and quantity constraints.
2) Core Concepts — Linear Inequalities Explained
A linear inequality in one variable is an inequality which involves a linear expression. For example:
ax + b < 0, ax + b ≤ 0, ax + b > 0, ax + b ≥ 0 where a and b are real numbers and a ≠ 0.
Similarly, a linear inequality in two variables x and y is of the form:
ax + by + c < 0, ax + by + c ≤ 0, ax + by + c > 0, ax + by + c ≥ 0
Example 1: Solve the inequality 3x - 7 < 2
Solution:
| 3x - 7 < 2 |
| 3x < 9 (Adding 7 on both sides) |
| x < 3 (Dividing by 3, positive number so inequality sign remains same) |
Solution set: {x | x < 3}
Graphical Representation of Linear Inequalities in Two Variables
Consider the inequality 2x + 3y ≤ 6. The boundary line is 2x + 3y = 6. To graph:
- Find intercepts: When x=0, y=2; when y=0, x=3.
- Plot points (0,2) and (3,0) and draw the line.
- Since inequality is ≤, shade the region below the line (including the line).
This shaded region represents all solutions (x, y) satisfying the inequality.
| x | y = (6 - 2x)/3 |
|---|---|
| 0 | 2 |
| 1 | (6 - 2)/3 = 4/3 ≈ 1.33 |
| 2 | (6 - 4)/3 = 2/3 ≈ 0.67 |
| 3 | 0 |
3) Key Formulas / Rules
Rule 1: Adding or subtracting the same number on both sides of an inequality does not change the inequality sign.
Rule 2: Multiplying or dividing both sides by a positive number keeps the inequality sign same.
Rule 3: Multiplying or dividing both sides by a negative number reverses the inequality sign.
Rule 4: Graph of linear inequality ax + by + c < 0 or ≤ 0 is the half-plane on one side of the line ax + by + c = 0.
4) Did You Know?
Linear inequalities are used in Indian agriculture planning to optimize the use of limited resources like water and fertilizer. For example, farmers use inequalities to decide how much land to allocate to crops such that water usage stays within limits, maximizing yield and profit.
5) Exam Tips
- Always remember: When multiplying or dividing by a negative number, flip the inequality sign.
- Check if the inequality is strict (< or >) or inclusive (≤ or ≥) before graphing.
- For graphical problems, practice plotting intercepts carefully and shading the correct region.
- Previous CBSE questions often ask to solve linear inequalities and represent solutions on a number line or graph.
- Common mistake: Forgetting to reverse the inequality sign when multiplying/dividing by negative numbers.
- Board exam pattern example (2023): "Solve the inequality 4 - 5x ≥ 9 and represent the solution on a number line."
Linear Inequalities — Mcq
Linear Inequalities — Mnemonic
Mnemonic 1: "Inequality Signs Dance" 💃🕺
- < (Less than) – Think of a < sign as a "little mouth" wanting to eat the smaller number. Like a kid saying, "छोटा खाना पसंद है!" 🍽️
- > (Greater than) – The > sign is a "big mouth" ready to gobble the larger number. Like "बड़ा खाना खाओ!" 🍛
- Mnemonic phrase: "छोटा मुँह, बड़ा मुँह, हमेशा बड़ा खाएगा!" (The smaller mouth, the bigger mouth, always the bigger one wins!)
Mnemonic 2: "Flip the Sign, Flip the Mind!" 🔄
- When multiplying or dividing by a negative number, flip the inequality sign like flipping a pancake! 🥞
- Hindi rhyme: "नकारात्मक से गुणा करो, निशान पलटाओ!" (Multiply by negative, flip the sign!)
- Remember: Flip = Flip (sign flips when multiplying/dividing by negative)
Mnemonic 3: "Inequality Road Rules" 🛣️
- Rule 1: Keep the variable on one side, constants on the other. Like keeping your luggage on one side of the train! 🚂
- Rule 2: Solve step-by-step, don’t rush! Like crossing a busy Indian market street carefully. 🛒
- Rule 3: If you multiply/divide by a negative, flip the sign — like turning your scooter around on a narrow lane! 🛵
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