Inequalities — Lesson
1) Hook — A Fun Real-Life Story to Grab Attention
Imagine you are helping your family plan a trip from Delhi to Agra by car. Your parents say, "We must reach Agra in less than 4 hours to avoid the afternoon heat." The distance is 200 km. You quickly calculate the speed needed. You say, "We need to drive at a speed greater than 50 km/h." Here, you just formed an inequality to decide the speed! Inequalities help us make decisions when exact values are not enough — like speed limits, budgets, or even exam passing marks.
2) Core Concepts — Clear Explanation with Examples and Visual Tables
What is an Inequality?
An inequality compares two values or expressions showing that one is less than, greater than, less than or equal to, or greater than or equal to the other.
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | x < 5 (x is less than 5) |
| > | Greater than | y > 10 (y is greater than 10) |
| ≤ | Less than or equal to | a ≤ 7 (a is at most 7) |
| ≥ | Greater than or equal to | b ≥ 3 (b is at least 3) |
Solving Inequalities: To solve inequalities, treat them like equations but remember the special rule about multiplying or dividing by negative numbers.
Example 1: Solve 3x + 5 < 14
- Subtract 5 from both sides: 3x < 9
- Divide both sides by 3 (positive number): x < 3
Example 2: Solve -2x ≥ 6
- Divide both sides by -2 (negative number), flip the inequality sign: x ≤ -3
Graphical Representation on Number Line:
For x < 3, draw an open circle at 3 and shade all numbers to the left.
For x ≤ -3, draw a closed circle at -3 and shade all numbers to the left.
3) Key Formulas/Rules
Rules for Solving Inequalities:
- If you add or subtract the same number on both sides, the inequality sign does not change.
- If you multiply or divide both sides by a positive number, the inequality sign does not change.
- If you multiply or divide both sides by a negative number, the inequality sign must be reversed.
- Inequalities can be represented graphically on a number line using open (for < or >) or closed (for ≤ or ≥) circles.
4) Did You Know?
In India, the concept of inequalities was used in ancient times by mathematicians like Bhaskara II (12th century) who studied solutions to problems involving bounds and limits — a foundation for modern inequalities! Also, inequality symbols (<, >) were introduced in Europe only in the 17th century.
5) Exam Tips — Common Mistakes and Board Exam Patterns
- Always remember: Flip the inequality sign when multiplying or dividing by a negative number. This is the most common error.
- Check your solution: Substitute a test value from your solution set back into the original inequality to verify correctness.
- Graph answers clearly: Use open or closed circles correctly and shade the right region on the number line.
- Board Exam Pattern: Questions may ask to solve linear inequalities, represent solutions on number lines, or interpret inequalities in word problems (like speed, budget, or age).
- Mnemonic to remember sign flipping: "Multiply or Divide by Negative? Flip the Sign, be Creative!"
Inequalities — Mcq
Inequalities — Mnemonic
Memorable Mnemonics for Inequalities (IGCSE Class 10)
-
1. “Greater Than? Flip the Fan!” 🌀➡️
When multiplying or dividing an inequality by a negative number, flip the inequality sign like flipping a fan!
Hindi Hint: "Minus se multiply karo, inequality ko ulta karo!" (Multiply by minus, flip the inequality!) -
2. “Inequality Signs: < > are like Arrows 🏹”
Think of “<”“>” as an arrow pointing right, meaning “greater than” (bada).
Mnemonic: “< Less = Left, > Greater = Right” – Just like traffic signs on Indian roads! -
3. “Inequality Dance: Flip, Multiply, and Sign Switch!” 💃🕺
Remember the dance steps for inequalities:
Flip the sign when multiplying/dividing by negative,
Multiply carefully,
Switch the sign only on negative operations.
Hindi Rhyming Phrase: "Negative se kaam ho, toh sign ko palat do naam!"
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