Inequalities — Lesson
1) Hook — A Fun Real-Life Story to Grab Attention
Imagine you are helping your family plan a trip to the famous Taj Mahal. The budget for travel and food is ₹5,000. You know the train ticket costs ₹1,200 and food expenses per day are about ₹800. You want to find out how many days you can spend without exceeding the budget. This situation naturally leads to an inequality — a mathematical way to express limits and constraints.
2) Core Concepts — Understanding Inequalities
An inequality compares two 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 > 3 (y is greater than 3) |
| ≤ | Less than or equal to | a ≤ 10 (a is at most 10) |
| ≥ | Greater than or equal to | b ≥ 0 (b is at least 0) |
Example 1: Solve the inequality 3x + 5 < 20.
Step 1: Subtract 5 from both sides: 3x < 15
Step 2: Divide both sides by 3: x < 5
Interpretation: x can be any number less than 5.
Example 2: A shopkeeper sells mangoes. He wants to sell at least 100 mangoes in a day to make a profit. If he sells x mangoes, write an inequality to represent this.
Answer: x ≥ 100
3) Key Formulas / Rules
Rule 1: Adding or subtracting the same number on both sides does not change the inequality.
If a < b, then a + c < b + c and a - c < b - c
Rule 2: Multiplying or dividing both sides by a positive number keeps the inequality direction same.
If a < b and c > 0, then ac < bc and a/c < b/c
Rule 3: Multiplying or dividing both sides by a negative number reverses the inequality sign.
If a < b and c < 0, then ac > bc and a/c > b/c
4) Did You Know?
In ancient India, mathematicians like Brahmagupta worked with inequalities while studying astronomy and calculating planetary positions. Inequalities helped them estimate ranges for celestial events, a practice still important in modern space science!
5) Exam Tips — Avoid These Common Mistakes
- Sign reversal error: Remember to reverse the inequality sign when multiplying or dividing by a negative number.
- Not simplifying completely: Always isolate the variable on one side before stating the solution.
- Misreading symbols: Pay attention to ≤ and ≥ signs; they include equality.
- Graphing mistakes: Use open circles for < or > and closed circles for ≤ or ≥ on number lines.
Board Exam Pattern: Expect questions like solving linear inequalities, representing solutions on number lines, and word problems involving inequalities with real-life contexts such as budgeting, speed limits, or population estimates.
Inequalities — Mcq
Inequalities — Mnemonic
Memorable Mnemonics for Inequalities (IGCSE Class 9)
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1. "Greater Than" Direction Trick 👉
Remember: The inequality symbol always points to the smaller number, like a hungry crocodile 🐊 eating the bigger number!
Mnemonic: "Crocodile Always Eats Bigger" 🐊
Example: 7 > 5 means crocodile mouth opens towards 5 (smaller), eats 7 (bigger).
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2. Hindi Rhyming Phrase for Inequality Signs 🔄
"छोटा छोटा तीर, बड़ा बड़ा शेर" (Chhota chhota teer, bada bada sher)
Translation: The small arrow (< or >) points to the smaller number, the big number is like a lion 🦁.
This helps remember that the arrow points to the smaller number.
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3. Acronym for Solving Inequalities: "S.I.M.P.L.E" ✔️
- S - Sign flip when multiplying/dividing by negative
- I - Isolate the variable
- M - Move terms to one side
- P - Perform operations carefully
- L - Look at inequality direction
- E - Express solution set clearly
Helps in remembering the stepwise approach to solve inequalities.
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