What is a Compound Inequality (AND)?

S3-SA1-0558

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A compound inequality with 'AND' means we are looking for numbers that satisfy *two* different inequality conditions at the same time. Both conditions must be true for a number to be a solution. Think of it as finding the common ground between two rules.

Simple Example

Quick Example

Imagine you want to buy a cricket bat. The shop owner says, 'The bat must cost more than Rs 500 AND less than Rs 1200.' This means the price of the bat must be between Rs 500 and Rs 1200. A bat costing Rs 400 won't work (it's not more than Rs 500), and a bat costing Rs 1300 won't work (it's not less than Rs 1200).

Worked Example

Step-by-Step

Solve the compound inequality: x > 3 AND x < 7

1. First, let's understand the first condition: x > 3. This means x can be 4, 5, 6, 7, 8, and so on.
---2. Next, let's understand the second condition: x < 7. This means x can be 6, 5, 4, 3, 2, and so on.
---3. Now, we need to find numbers that satisfy *both* conditions. What numbers are greater than 3 AND also less than 7?
---4. Let's list some numbers: Is 2 > 3? No. Is 8 < 7? No.
---5. What about 4? Is 4 > 3? Yes. Is 4 < 7? Yes. So, 4 is a solution.
---6. What about 5? Is 5 > 3? Yes. Is 5 < 7? Yes. So, 5 is a solution.
---7. What about 6? Is 6 > 3? Yes. Is 6 < 7? Yes. So, 6 is a solution.
---8. The numbers that satisfy both are 4, 5, and 6. So, the solution is 3 < x < 7. This means x is any number between 3 and 7 (not including 3 or 7).
Answer: 3 < x < 7

Why It Matters

Compound inequalities are crucial for setting conditions in many fields. In Computer Science, they help define ranges for data or game levels. Data Scientists use them to filter information, like finding customers aged between 18 and 30 who spend more than Rs 1000 per month. Engineers use them to set safe operating limits for machines, ensuring a motor's temperature is above a minimum AND below a maximum.

Common Mistakes

MISTAKE: Thinking 'AND' means any number that satisfies *either* condition. | CORRECTION: Remember 'AND' means *both* conditions must be true at the same time. If even one condition is false, the number is not a solution.

MISTAKE: Confusing 'AND' with 'OR'. | CORRECTION: 'AND' means the solution is the *overlap* or intersection of the two conditions. 'OR' means the solution includes numbers from *either* condition.

MISTAKE: Not writing the final answer correctly as a single inequality (e.g., leaving it as two separate inequalities). | CORRECTION: For 'AND' inequalities where the solution is a range, combine them into a single inequality like 'a < x < b' or 'a <= x <= b'.

Practice Questions

Try It Yourself

QUESTION: Which whole numbers satisfy 5 < y AND y <= 9? | ANSWER: 6, 7, 8, 9

QUESTION: If a mobile data plan offers 'more than 1 GB AND less than 5 GB' daily data, write this as a compound inequality for 'd' (daily data in GB). | ANSWER: 1 < d < 5

QUESTION: A student needs to score 'at least 60 marks AND less than 85 marks' in a test to get a 'B' grade. If 'm' represents the marks, write this as a compound inequality. Then, list three possible whole number marks the student could get. | ANSWER: 60 <= m < 85. Possible marks: 60, 70, 80 (any three whole numbers between 60 and 84, inclusive of 60).

MCQ

Quick Quiz

Which of the following numbers is a solution to the inequality: x >= 10 AND x < 15?

The Correct Answer Is:

For x >= 10 AND x < 15, the number must be 10 or greater, AND less than 15. Only 12 fits both conditions. 9 is not >= 10. 15 is not < 15. 17 is not < 15.

Real World Connection

In the Real World

When you book train tickets online, the system often uses compound inequalities. For example, it might show available seats for 'adults older than 18 years AND younger than 60 years' for a specific concession, or 'tickets costing more than Rs 500 AND less than Rs 1500' for a certain class. These conditions help filter options quickly.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, >=, or !=. | COMPOUND INEQUALITY: Two or more inequalities joined by 'AND' or 'OR'. | AND: A logical connector meaning both conditions must be true. | SOLUTION SET: The set of all numbers that satisfy the inequality.

What's Next

What to Learn Next

Great job understanding 'AND' compound inequalities! Next, you should explore 'What is a Compound Inequality (OR)?'. It's similar but has a different meaning for combining conditions, which will further strengthen your understanding of solving real-world problems with inequalities.