Inequalities — Lesson
1) Hook — A Fun Real-Life Story to Grab Attention
Imagine you are helping your family plan a wedding feast in Delhi. The caterer says, "We can serve up to 200 guests comfortably." If you invite more than 200 people, the food and seating will be insufficient. So, you must ensure the number of guests does not exceed 200. This is a perfect example of an inequality in real life — a condition that restricts a value to be less than or greater than another.
2) Core Concepts — Understanding Inequalities
An inequality compares two values or expressions and shows 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 ≥ 7 (b is at least 7) |
Example 1: If the temperature in Mumbai is more than 30°C, people prefer to drink cold drinks. Write this as an inequality.
Answer: Temperature (T) > 30
Example 2: A shopkeeper sells at least 50 mangoes daily. Write the inequality.
Answer: Number of mangoes sold (m) ≥ 50
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.
- Important: If you multiply or divide both sides by a negative number, reverse the inequality sign.
Example: Solve 3x - 5 < 10
Step 1: Add 5 to both sides: 3x - 5 + 5 < 10 + 5 ⇒ 3x < 15
Step 2: Divide both sides by 3 (positive): x < 5
Example with negative multiplication: Solve -2x ≥ 6
Divide both sides by -2 (negative), reverse inequality:
x ≤ -3
4) Did You Know?
In India, the concept of inequalities was used in ancient times by mathematicians like Bhāskara II (12th century) when solving real-life problems such as land measurement and trade. Inequalities help us understand limits and constraints, which is why they are essential in economics, engineering, and computer science today!
5) Exam Tips — Avoid These Common Mistakes!
- Do not forget to reverse the inequality sign when multiplying or dividing by a negative number.
- Write the solution clearly using inequality symbols or interval notation if asked.
- Check your solution by substituting a value from the solution set back into the original inequality.
- For board exams, expect questions like: “Solve the inequality and represent the solution on a number line.”
- Practice graphing inequalities on number lines — this is a common question.
Mnemonic to remember sign reversal: "Multiply or divide by negative? Flip the sign to be positive!"
Inequalities — Mcq
Inequalities — Mnemonic
Mnemonic 1: "Greater Than, Less Than - The Hungry Crocodile 🐊"
- Remember: The crocodile always wants to eat the bigger number!
- So, “>” means “greater than” and the open mouth faces the larger number.
- Hindi Tip: "Bada khaata hai, chhota bachata hai!" (The bigger number gets eaten!)
Mnemonic 2: "Inequality Signs - G.L.L. Trick" 🎯
- G for Greater ( > ) — Mouth opens to the right (bigger number on right)
- L for Less ( < ) — Mouth opens to the left (bigger number on left)
- L for Less or Equal ( ≤ ) — Small line under the less than sign means “can be equal too”
- Hindi rhyme: "Bada dikhata bada, chhota dikhata chhota"
Mnemonic 3: "Inequality Direction - 'DOST' Rule" 🤝
- D for Direction of the sign’s mouth (always open to the bigger number)
- O for Open sign means strict inequality ( > or < )
- S for Solid line under sign means inclusive ( ≥ or ≤ )
- T for Think of the number line: smaller numbers on left, bigger on right
- Hindi phrase: "Dost bade ko pakadta hai, chhote ko nahi!"
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