How to Solve an Inequality
Understanding inequalities is an important part of mathematics. If you are a student preparing for exams or someone who wants to strengthen basic math skills, learning how to solve inequalities is very useful. In this guide, you will find simple explanations, step-by-step methods, examples, and common mistakes to avoid.
What Is an Inequality?
An inequality shows a comparison between two values. Instead of using the equal sign (=), inequalities use:
- Greater than: >
- Less than: <
- Greater than or equal to: ≥
- Less than or equal to: ≤
For example:
- 5 > 2 means 5 is greater than 2.
- x < 7 means x is less than 7.
Why Are Inequalities Important?
Inequalities help us understand ranges of possible answers instead of just one answer. They are used in:
- Mathematics
- Physics
- Economics
- Daily life situations like budgeting time, money, and resources
Basic Rules of Solving Inequalities
To solve inequalities, we use steps similar to solving equations, but there is one important rule you must always remember.
1. You Can Add or Subtract the Same Number on Both Sides
Example:
x + 5 > 12
x > 7
2. You Can Multiply or Divide by the Same Positive Number
Example:
3x < 18
x < 6
3. Important Rule: When You Multiply or Divide by a Negative Number, Flip the Sign
Example:
-2x > 10
x < -5 (inequality flips from > to <)
This is the most common mistake students make.
Step-by-Step Method to Solve Inequalities
Follow these steps to solve any inequality easily.
Step 1: Move Numbers to One Side
Keep the variable (x) on one side and numbers on the other.
Step 2: Simplify the Expression
Combine like terms.
Step 3: Apply the Rules (Add, Subtract, Multiply, or Divide)
Make sure to flip the sign if multiplying or dividing by a negative number.
Step 4: Write the Final Answer
This shows all possible values of the variable.
Step 5: Represent the Answer on a Number Line (optional but helpful)
This helps you visualize the solution.
Examples Every Student Should Know
Example 1: Solve the inequality
2x + 3 < 11
Solution:
2x < 8
x < 4
Example 2: Solve the inequality
5 – x ≥ 2
Solution:
-x ≥ -3
x ≤ 3 (sign flips because we divided by -1)
Example 3: Solve the inequality
3(x – 2) > 12
Solution:
3x – 6 > 12
3x > 18
x > 6
Common Mistakes to Avoid
- Forgetting to flip the sign when dividing by a negative number.
- Mixing up inequality symbols.
- Not simplifying properly.
- Leaving the final answer unsimplified.
Tips to Master Inequalities
- Practice different types of questions.
- Draw number lines to understand the range.
- Review the rules regularly.
- Start with easy problems and slowly move to harder ones.
Final Thoughts
Solving inequalities becomes simple when you remember the basic rules and practice regularly. This topic builds the foundation for advanced mathematics, so understanding it clearly will help you in future chapters as well.
This guide was written in a simple, clear, and natural style so that any student can understand it easily. Keep practicing, and you will become confident with inequalities.
