Inequalities
Inequalities help us compare values instead of finding one exact answer. They are used when a quantity can be greater than, less than, or within a range, which is very common in real life, exams, and decision-making.
In this lesson, you will learn all inequality symbols, how to solve inequalities, how to represent them visually, and how they apply to everyday situations.
What Is an Inequality?
An inequality is a mathematical statement that shows two quantities are not equal, but one is larger or smaller than the other.
Instead of a single solution, inequalities usually have many possible solutions.
Example: x > 5
Inequality Symbols You Must Know
Each inequality symbol has a clear meaning. Understanding them correctly is essential for exams.
| Symbol | Meaning | Example |
| > | Greater than | x > 3 |
| < | Less than | x < 7 |
| ≥ | Greater than or equal to | x ≥ 5 |
| ≤ | Less than or equal to | x ≤ 10 |
Understanding Inequalities Using a Number Line
A number line helps us visually understand which values satisfy an inequality.
Open circle (○) means the value is not included. Closed circle (●) means the value is included.
This visual shows all values greater than 5, but not including 5.
Solving Simple Inequalities
Solving inequalities is similar to solving equations, but instead of one answer, we find a range of answers.
Example: x + 4 > 9
Step 1: Subtract 4 from both sides Step 2: x > 5
Inequalities with Subtraction
When subtracting a number from both sides, the inequality direction remains the same.
Example: x − 3 ≤ 7
Add 3 to both sides → x ≤ 10
Inequalities with Multiplication
Multiplying both sides by a positive number does not change the inequality symbol.
Example: 2x > 8
Divide both sides by 2 → x > 4
The Most Important Rule: Negative Numbers
When you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be reversed.
Example: −2x > 6
Divide both sides by −2 → x < −3
This rule is a very common exam trap.
Compound Inequalities
Sometimes an inequality contains two conditions at the same time. This creates a range of values.
Example: 3 < x ≤ 8
This means x is greater than 3 and less than or equal to 8.
Writing Inequalities from Word Problems
Many real-life situations naturally form inequalities. You must learn to convert words into symbols.
Example: A person must be at least 18 years old.
Let age = x → x ≥ 18
Inequalities in Real Life
Inequalities appear in many everyday decisions.
- Speed limits (speed ≤ 60 km/h)
- Minimum marks required to pass
- Budget limits (expenses ≤ income)
- Age restrictions
Life is full of ranges, not exact numbers.
Inequalities in Competitive Exams
Exams often test:
- Sign reversal mistakes
- Compound inequalities
- Number line interpretation
Careful step-by-step solving avoids most errors.
Common Mistakes to Avoid
Students frequently lose marks due to these errors.
- Forgetting to reverse the symbol
- Incorrect number line marking
- Confusing > with ≥
Attention to symbols is critical.
Practice Questions
Q1. Solve: x + 7 < 15
Q2. Solve: −3x ≥ 9
Q3. Write an inequality for: “Marks must be less than 40”.
Quick Quiz
Q1. When do we reverse the inequality sign?
Q2. Which symbol includes the value itself?
Quick Recap
- Inequalities compare values, not exact answers
- Symbols represent greater or smaller relationships
- Negative multiplication reverses the sign
- Number lines make inequalities easy to understand
- Used in exams, life, and decision-making