What is a Solution Set of an Inequality?

S3-SA1-0458

Grade Level:

Class 8

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

Definition

What is it?

The solution set of an inequality is the collection of all possible values for the variable that make the inequality true. It's not just one answer, but often a range of numbers.

Simple Example

Quick Example

Imagine you need to score more than 70 marks in your Maths exam to get an 'A' grade. If 'x' represents your marks, the inequality is x > 70. The solution set includes all marks like 71, 72, 85, 99, 100 – basically any mark greater than 70.

Worked Example

Step-by-Step

Let's find the solution set for the inequality: 3x - 5 < 10

1. Add 5 to both sides of the inequality:
3x - 5 + 5 < 10 + 5
3x < 15

2. Divide both sides by 3:
3x / 3 < 15 / 3
x < 5

3. This means any number less than 5 will make the original inequality true.

4. So, the solution set includes numbers like 4, 3, 0, -2, -100, etc. It can be written as {x | x < 5}, meaning 'all x such that x is less than 5'.

Why It Matters

Understanding solution sets helps scientists and engineers set limits and boundaries in their work. For example, in AI/ML, it helps define acceptable ranges for data, or in Physics, it can describe the conditions for a reaction to occur. It's a fundamental skill for many future careers!

Common Mistakes

MISTAKE: Treating an inequality like an equation and only finding one value for the variable. | CORRECTION: Remember that inequalities often have many solutions, forming a range, not just a single point. Think about 'greater than' or 'less than' a number.

MISTAKE: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. | CORRECTION: Always flip the inequality sign (e.g., < becomes >, >= becomes <=) if you multiply or divide both sides by a negative number.

MISTAKE: Including the boundary number in the solution set when the inequality sign is strict (< or >). | CORRECTION: If the sign is < or >, the boundary number is NOT part of the solution. If it's <= or >=, then the boundary number IS included.

Practice Questions

Try It Yourself

QUESTION: Find the solution set for x + 7 >= 12. | ANSWER: x >= 5

QUESTION: Find the solution set for 2x - 3 > 11. | ANSWER: x > 7

QUESTION: Find the solution set for -4x + 6 <= 18. | ANSWER: x >= -3

MCQ

Quick Quiz

Which of the following numbers is NOT part of the solution set for 5x - 1 < 19?

3.9

4.1

The Correct Answer Is:

First, solve the inequality: 5x - 1 < 19 => 5x < 20 => x < 4. Any number less than 4 is in the solution set. 4.1 is not less than 4.

Real World Connection

In the Real World

Think about mobile data plans in India. If your plan gives you 'up to 2 GB' of data per day, this is an inequality! If 'd' is data used, then d <= 2 GB. The solution set includes all amounts of data you can use without extra charges, from 0 to 2 GB. Companies like Jio and Airtel use this concept to define limits for their services.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement showing that two expressions are not equal | VARIABLE: A symbol (usually a letter) representing a quantity that can change | SOLUTION: A value that makes an equation or inequality true | SOLUTION SET: The collection of all values that satisfy an inequality

What's Next

What to Learn Next

Great job understanding solution sets! Next, you can explore how to represent these solution sets on a number line, which gives a visual picture of all the possible answers. This will help you understand more complex inequalities.