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What is the Solution Set of an Inequality?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The solution set of an inequality is the collection of all possible values for the variable that make the inequality statement true. Unlike equations that often have one or a few exact answers, inequalities usually have a range of values that satisfy them.
Simple Example
Quick Example
Imagine you need to score at least 80 marks in your Maths exam to get an A grade. If 'M' represents your marks, the inequality is M >= 80. The solution set includes all marks from 80 upwards (80, 81, 82, ... 100), because any of these marks will get you an A grade.
Worked Example
Step-by-Step
Find the solution set for the inequality 2x + 5 < 15, where x is a natural number.
1. Start with the inequality: 2x + 5 < 15
---2. Subtract 5 from both sides: 2x + 5 - 5 < 15 - 5
---3. Simplify: 2x < 10
---4. Divide both sides by 2: 2x / 2 < 10 / 2
---5. Simplify: x < 5
---6. Since x must be a natural number (1, 2, 3, ...), the values less than 5 are 1, 2, 3, and 4.
---Answer: The solution set is {1, 2, 3, 4}.
Why It Matters
Understanding solution sets is crucial for fields like AI/ML to set parameters, in Physics to define ranges for variables, and in Engineering to ensure designs operate within safe limits. Doctors use this concept to determine safe dosage ranges for medicines, ensuring patient safety.
Common Mistakes
MISTAKE: Treating inequalities like equations and only finding one value. | CORRECTION: Remember inequalities represent a range. Always consider all values that make the statement true, often showing them on a number line or as a set.
MISTAKE: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. | CORRECTION: If you multiply or divide both sides of an inequality by a negative number, always flip the direction of the inequality sign (e.g., < becomes >, >= becomes <=).
MISTAKE: Confusing natural numbers, integers, and real numbers when defining the solution set. | CORRECTION: Always check the domain of the variable (e.g., 'x is an integer', 'x is a real number') before writing down the final solution set.
Practice Questions
Try It Yourself
QUESTION: Find the solution set for x - 7 >= 3, where x is a whole number. | ANSWER: {10, 11, 12, ...}
QUESTION: Solve the inequality -3x + 4 > 16, where x is an integer. | ANSWER: {..., -6, -5}
QUESTION: A mobile data plan gives you 20 GB per month. If you've already used 8 GB and want to keep your remaining usage 'y' below 10 GB, write and solve the inequality for 'y'. | ANSWER: Inequality: 8 + y < 20 and y < 10. Combining gives y < 10. Since y must be positive, the solution set is 0 <= y < 10.
MCQ
Quick Quiz
Which of the following numbers is NOT part of the solution set for 4x - 2 <= 10, if x is a natural number?
1
2
3
4
The Correct Answer Is:
D
Solving 4x - 2 <= 10 gives 4x <= 12, so x <= 3. Since x is a natural number, the solution set is {1, 2, 3}. Therefore, 4 is not part of the solution set.
Real World Connection
In the Real World
When you book a train ticket on IRCTC, the system might have an inequality like 'age >= 60' for senior citizen discounts. Or, when a food delivery app like Zomato calculates delivery time, it uses inequalities to ensure 'delivery_time <= estimated_time' to keep customers happy.
Key Vocabulary
Key Terms
INEQUALITY: A mathematical statement showing that two expressions are not equal, using symbols like <, >, <=, >=. | SOLUTION SET: The collection of all values that make an inequality true. | VARIABLE: A symbol, usually a letter, representing an unknown quantity. | NATURAL NUMBERS: Counting numbers: {1, 2, 3, ...}. | INTEGERS: Whole numbers and their negatives: {..., -2, -1, 0, 1, 2, ...}.
What's Next
What to Learn Next
Great job understanding solution sets! Next, explore 'Graphing Inequalities on a Number Line'. This will help you visually represent these solution sets, especially for real numbers, and make them even clearer to understand.
