Learning Objectives
- Explain what an inequality is.
- Understand the meaning of the four inequality symbols.
- Solve simple inequalities.
- Represent solutions on a number line.
- Know when to reverse the inequality sign.
Introduction
This lesson develops Inequalities from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🎬 Show a range on a number line
An open circle excludes the endpoint; the ray shows all possible values.
🤔 Think About This...
Imagine your school says:
Does that mean you must score exactly 50?
No. You could score:
- 50%
- 60%
- 75%
- 98%
All these scores satisfy the condition.
📖 What is an Inequality?
An inequality compares two quantities that may not be equal.
| Symbol | Meaning |
|---|---|
| < | Less than |
| > | Greater than |
| ≤ | Less than or equal to |
| ≥ | Greater than or equal to |
🧠 The StudyNest Thinking Method
👉 What values make this statement true?
Remember: An equation normally has one answer. An inequality usually has many.
📝 Worked Examples
Example 1
Solve:
Subtract 4 from both sides.
Example 2
Solve:
Add 3 to both sides.
Example 3
Solve:
Divide both sides by 5.
⚠ A Very Important Rule
Example:
Divide both sides by −2.
⚠ A Very Important Rule
Example:
Divide both sides by −2.
🔄 Why Does the Sign Reverse?
Let us begin with a true statement:
Now multiply both sides by −1:
On a number line, −3 is greater than −5.
👀 Visualise It: Number Lines
A number line helps us see which values satisfy an inequality.
x < 4
←────────○════════════
4
The open circle means 4 is NOT included. The arrow points towards all numbers less than 4.
x ≤ 4
←────────●════════════
4
The filled circle means 4 IS included. The solution contains 4 and every number smaller than 4.
x > 2
════════════○────────→
2
The open circle means 2 is not included. The arrow points towards numbers greater than 2.
x ≥ 2
════════════●────────→
2
The filled circle means 2 is included. The arrow points towards all numbers greater than or equal to 2.
- Use an open circle (○) for < and > because the endpoint is not included.
- Use a filled circle (●) for ≤ and ≥ because the endpoint is included.
- Draw the arrow to the left for values less than a number.
- Draw the arrow to the right for values greater than a number.
✏ Practice
Without solving, decide how each inequality would look on a number line.
- x < 6
- x ≥ -2
- x ≤ 5
- x > 0
📝 More Worked Examples
Example 4
Solve:
Subtract 2 from both sides.
Divide both sides by 3.
Example 5
Solve:
Subtract 7 from both sides.
Divide both sides by −2.
Because we divided by a negative number, reverse the sign.
Example 6
Solve and list the integer solutions:
x must be greater than 1, but it may equal 5.
🌍 Real-Life Example
A school bus can carry no more than 30 students.
Let s represent the number of students on the bus.
The bus may carry 30 students or fewer, but it cannot carry more than 30.
🗣 Translating Words into Inequalities
| Words | Symbol |
|---|---|
| Less than | < |
| Greater than | > |
| At most / no more than | ≤ |
| At least / not less than | ≥ |
⚠ Common Mistakes
Treating an inequality exactly like an equation and forgetting that it can have many solutions.
Forgetting to reverse the sign after multiplying or dividing by a negative number.
Using an open circle for ≤ or ≥. These symbols include the endpoint, so use a filled circle.
Reading “at most” as greater than. At most means the value cannot go above the stated number.
Practice Questions
Solve the following inequalities.
- x + 7 < 15
- x − 4 ≥ 9
- 3x ≤ 18
- 2x + 5 > 13
- −4x < 20
- 10 − 2x ≥ 4
- List the integer solutions of 2 < x ≤ 6.
- Write an inequality for: “A student must be at least 16 years old.”
✅ Show Answers
- x < 8
- x ≥ 13
- x ≤ 6
- x > 4
- x > −5
- x ≤ 3
- 3, 4, 5 and 6
- a ≥ 16
🎯 Exam Tip
🧠 Did You Know?
⭐ What You Can Do Now
- ✅ I understand the four inequality symbols.
- ✅ I can solve simple inequalities.
- ✅ I know when to reverse the inequality sign.
- ✅ I can interpret solutions on a number line.
- ✅ I can translate words such as “at least” and “at most.”
🎉 Well Done!
You can now solve inequalities and describe a whole range of possible answers.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
📈 Your Progress
Lesson 13 of 22
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.