Inequalities — Lesson
1) Hook — Real-Life Story: The Budget Shopping Challenge
Imagine you have ₹500 to buy snacks for your cricket team. You want to buy packets of chips costing ₹30 each and bottles of juice costing ₹50 each. How many packets and bottles can you buy without exceeding your budget? This is where inequalities help us decide the possible combinations!
2) Core Concepts — Understanding Inequalities
An inequality compares two values showing that one is less than, greater than, or not equal to the other. It uses symbols like <, >, ≤, ≥.
| 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 ≥ 7 (b is at least 7) |
Example 1: Solve and graph the inequality x + 3 < 7.
Solution:
- Subtract 3 from both sides:
x < 7 - 3 - So,
x < 4
On a number line, shade all points less than 4 (open circle at 4).
Example 2: If ₹30 per packet of chips and ₹50 per juice bottle, and total money is ₹500, write the inequality for the number of packets (x) and bottles (y) you can buy.
Solution:
30x + 50y ≤ 500
Here, x, y ≥ 0 because you cannot buy negative items.
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, reverse the inequality sign.
- Example: If
-2x > 6, dividing both sides by -2 givesx < -3.
4) Did You Know?
In ancient India, mathematicians like Bhāskara II worked on solving inequalities and quadratic equations centuries ago! Inequalities are fundamental to optimization problems in economics, engineering, and even cricket strategy, such as deciding the minimum runs needed to win.
5) Exam Tips — Avoid These Common Mistakes!
- Do not forget to reverse the inequality sign when multiplying or dividing by a negative number.
- Always write the solution set clearly, using interval notation or set-builder notation if required.
- When graphing, use an open circle for
<or>and a closed circle for≤or≥. - Check your solution by substituting values back into the inequality.
- Board exams often ask for both solving inequalities and representing solutions on number lines.
Inequalities — Mcq
Inequalities — Mnemonic
Memorable Mnemonics for Inequalities (IGCSE Class 10)
-
1. "Greater Than? G-T, Go To the Bigger Side!" 😎
Remember: The inequality symbol always points to the smaller number, and the open end faces the bigger number. Think of the symbol as a hungry crocodile 🐊 that always wants to eat the bigger number!
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2. Hindi Rhyming Trick: "Chhota Kam, Bada Jyada" (छोटा कम, बड़ा ज्यादा) 🔢
This means: "Smaller is less, bigger is more." Use this phrase to recall that < means 'less than' and > means 'greater than'. Example: 5 < 10 (5 कम है 10 से), 10 > 5 (10 ज्यादा है 5 से)
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3. Acronym: "S.A.L.T." for Solving Inequalities 🧂
- S - Switch the inequality sign when multiplying/dividing by a negative number.
- A - Always keep the variable on one side.
- L - Look for the solution set on the number line.
- T - Test a number to verify the inequality.
Remember: Multiplying/dividing by negative flips the sign! Mnemonic to remember flipping sign: "Flip the sign, or it won’t be fine!"
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