What is a Compound Inequality?

S3-SA1-0137

Grade Level:

Class 7

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

Definition

What is it?

A compound inequality is like having two simple inequalities joined together by words like 'and' or 'or'. It describes a range of values that satisfy both or either of the conditions. Think of it as setting boundaries for a number.

Simple Example

Quick Example

Imagine you want to buy a mobile phone. You want one that costs at least Rs 8,000 but no more than Rs 12,000. This means the price (P) must be P >= 8000 AND P <= 12000. This is a compound inequality.

Worked Example

Step-by-Step

Let's solve the compound inequality: x + 3 > 7 AND x - 2 < 5.

Step 1: Solve the first inequality: x + 3 > 7.
Subtract 3 from both sides: x > 7 - 3.
So, x > 4.

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Step 2: Solve the second inequality: x - 2 < 5.
Add 2 to both sides: x < 5 + 2.
So, x < 7.

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Step 3: Combine the solutions using 'AND'.
We need values of x that are greater than 4 AND less than 7.

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Step 4: Write the combined solution.
The solution is 4 < x < 7.

Answer: 4 < x < 7

Why It Matters

Compound inequalities are super useful! In Data Science, they help filter data within specific ranges, like finding customers aged between 18 and 30. Engineers use them to set safe operating limits for machines, ensuring a motor's speed is neither too low nor too high. This helps in fields like AI/ML and Physics.

Common Mistakes

MISTAKE: Forgetting to apply the operation (like adding or subtracting) to all parts of an 'and' inequality (e.g., 5 < x + 2 < 10, only operating on x + 2 and 10). | CORRECTION: When you have 'a < x + b < c', remember to subtract 'b' from 'a', 'x + b', AND 'c' to isolate 'x'.

MISTAKE: Confusing 'AND' with 'OR' conditions. For 'AND', the solution must satisfy BOTH inequalities. For 'OR', it can satisfy EITHER one. | CORRECTION: Always read the connecting word carefully. 'AND' means intersection (overlap), 'OR' means union (all parts combined).

MISTAKE: Flipping the inequality sign when multiplying or dividing by a positive number. | CORRECTION: You ONLY flip the inequality sign when multiplying or dividing BOTH sides by a NEGATIVE number. If the number is positive, the sign stays the same.

Practice Questions

Try It Yourself

QUESTION: Solve: y - 5 < 2 AND y + 1 > 4 | ANSWER: 3 < y < 7

QUESTION: Find the values of z that satisfy: 2z >= 10 OR z/3 < 2 | ANSWER: z >= 5 OR z < 6

QUESTION: A delivery drone can fly for a minimum of 30 minutes and a maximum of 75 minutes on one charge. If it has already flown for 10 minutes, what is the range of additional minutes it can fly? Let 'm' be the additional minutes. | ANSWER: 20 <= m <= 65

MCQ

Quick Quiz

Which of these represents 'The temperature is at least 25 degrees Celsius but less than 35 degrees Celsius'?

25 > T > 35

T >= 25 OR T < 35

25 <= T < 35

T < 25 AND T >= 35

The Correct Answer Is:

The phrase 'at least 25' means T >= 25, and 'less than 35' means T < 35. Combining these with 'but' (which implies 'and') gives 25 <= T < 35.

Real World Connection

In the Real World

Compound inequalities are used in many places! When you book a train ticket online, the system might show available seats only for passengers aged between 5 and 60 years. Or, when your mobile data plan states you get 'up to 1 GB per day, but not less than 500 MB', that's a compound inequality in action, controlling your usage limits.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, or >= | AND: A logical connector meaning both conditions must be true | OR: A logical connector meaning at least one of the conditions must be true | SOLUTION SET: The set of all values that make an inequality true | VARIABLE: A symbol (usually a letter) representing an unknown number

What's Next

What to Learn Next

Great job understanding compound inequalities! Next, you can explore how to graph these inequalities on a number line. This will give you a visual understanding of the solution sets and make solving more complex problems even easier. Keep up the good work!