Representing inequalities on a number line is a fundamental GCSE skill tested on both Foundation and Higher papers. It turns an algebraic statement into a visual diagram, making it easy to see which values are included in the solution set.
What Are Inequalities on a Number Line?
A number line diagram shows which values satisfy an inequality. Two key conventions are used:
- An open circle (○) means the value is not included (used with < or >).
- A closed circle (●) means the value is included (used with ≤ or ≥).
A solid line or arrow extends from the circle in the direction of the values that satisfy the inequality.
Key Formulas
Step-by-Step Method
- Solve the inequality algebraically (if needed) to find the boundary value(s).
- Draw a number line and mark the boundary value(s).
- Choose the correct circle: open for < or >, closed for ≤ or ≥.
- Draw an arrow or solid line in the direction of the solution set.
- For double inequalities (e.g., 2 < x ≤ 5), mark both boundary values and shade the region between them.
Worked Example 1 — Foundation Level
Question: Represent x > 3 on a number line.
Working:
Draw a number line. Mark the value 3.
Place an open circle at 3 (because 3 is not included — it is strictly greater than).
Draw an arrow going to the right from the open circle, indicating all values greater than 3.
Answer: Open circle at 3, arrow pointing right.
Worked Example 2 — Higher Level
Question: Solve 2x + 1 ≤ 9 and represent the solution on a number line.
Working:
2x + 1 ≤ 9
2x ≤ 8
x ≤ 4
Draw a number line. Place a closed circle at 4 (because 4 is included — less than or equal to).
Draw an arrow going to the left from the closed circle.
Answer: x ≤ 4. Closed circle at 4, arrow pointing left.
Worked Example 3 — Exam Style
Question: Solve −3 < 2x − 1 ≤ 7 and represent the solution on a number line. List all integer values that satisfy the inequality.
Working:
Solve the double inequality by adding 1 to all three parts:
−3 + 1 < 2x ≤ 7 + 1
−2 < 2x ≤ 8
Divide all three parts by 2:
−1 < x ≤ 4
On the number line: open circle at −1 (not included), closed circle at 4 (included), solid line between them.
Integer values: 0, 1, 2, 3, 4.
Answer: −1 < x ≤ 4. Integers: 0, 1, 2, 3, 4.
Common Mistakes
- Using the wrong type of circle. An open circle means the value is excluded (< or >), and a closed circle means it is included (≤ or ≥). Mixing these up loses marks immediately.
- Drawing the arrow in the wrong direction. For x > 3, the arrow goes right (towards larger values). For x < 3, it goes left. Read the inequality carefully before drawing.
- Forgetting to list all integers in a double inequality. When asked for integer values, include the boundary if it is a closed circle. In −1 < x ≤ 4, the value −1 is not included but 4 is.
Exam Tips
- When listing integers, be careful at the boundaries. An open circle means you do not include that integer; a closed circle means you do.
- For double inequalities, solve all three parts together — add, subtract, multiply or divide across the entire inequality in one step.
- If you divide or multiply by a negative number, remember to reverse the inequality signs.
Practice Questions
Q1 (Foundation): Represent x ≥ −2 on a number line.
Q2 (Foundation): Solve 3x − 4 > 8 and show the solution on a number line.
Q3 (Higher): Solve 1 ≤ 3x + 4 < 16 and list all integer values of x that satisfy the inequality.
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Related Topics
Summary
- An open circle (○) means the boundary value is not included (< or >).
- A closed circle (●) means the boundary value is included (≤ or ≥).
- For single inequalities, draw an arrow from the circle in the direction of the solution set.
- For double inequalities, mark both boundaries with the correct circles and shade between them.
- When listing integers, include boundary values only if the circle is closed.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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