Sheet № 98 · Foundation + Higher · AQA · Edexcel · OCR
Inequalities on a Number Line –
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.
§Key definitions
Question:
Represent x > 3 on a number line.
Answer:
Open circle at 3, arrow pointing right.
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.
§Formulas to memorise
x > a — open circle at a, arrow pointing right
x ≤ a — closed circle at a, arrow pointing left
a < x ≤ b — open circle at a, closed circle at b, solid line between them
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
Represent x > 3 on a number line.
Working:
⚠ 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.