Representing Solution Sets Revision Notes | GCSE AQA Higher Maths

1. Why Representing Solution Sets Matters

Solving an inequality gives you a range of values, not a single answer. The challenge is communicating that range clearly — and in GCSE maths, there are three precise ways to do it: a number line, set notation, and a graph. Each method carries its own conventions, and examiners award marks specifically for using them correctly.

Think about a situation where a ride requires passengers to be at least 120 cm tall. The solution isn't one height — it's every height from 120 cm upwards. Representing that range unambiguously is exactly what this topic is about. Get the notation right, and you collect every available mark.

2. The Number Line: Open and Closed Circles

A number line is the most visual way to show a solution set. The key decision you make every time is whether the boundary value is included or excluded — and you signal that with a circle.

An open circle says: 'we get infinitely close to this value, but we never actually reach it.' A closed circle says the opposite — the boundary is part of the answer.

The third example above shows a combined inequality: $-1 < x \leq 4$. Here you have two boundaries with different circle types — open at $-1$ because the inequality is strict ($<$), and closed at $4$ because it is non-strict ($\leq$). This mixed case appears frequently in exam questions.

3. Set Notation: Writing the Solution Formally

Once you can draw the solution set, you also need to write it. Set notation is the formal mathematical language for this, and the examiner expects you to read and write it fluently.

The structure $\{ x: \text{condition} \}$ is always the same: curly brackets, the variable, a colon, then the inequality. The colon is read aloud as 'such that', so $\{ x: x > 3 \}$ reads as 'the set of all $x$ such that $x$ is greater than 3.'

4. Graphical Representation: Dashed and Solid Lines

When an inequality involves two variables — typically $x$ and $y$ — you represent the solution set on a coordinate grid. The boundary becomes a line, and the solution set becomes a shaded region. The same open/closed logic from the number line reappears here, but in a new form.

The dashed line is the graphical equivalent of an open circle: it marks the boundary but signals that points on the line itself are not part of the solution. A solid line is the graphical equivalent of a closed circle: points on the line are included in the solution set.

After drawing the boundary line, you shade the correct region. To decide which side to shade, substitute a test point — $(0, 0)$ works well unless it lies on the boundary line itself. If the test point satisfies the inequality, shade the side containing it; if not, shade the other side.

Worked example: Representing $y \leq 2x - 1$ on a Graph

Represent the inequality $y \leq 2x - 1$ on a coordinate grid.

1. Boundary line: $y = 2x - 1$ — Replace the inequality symbol with $=$ to find the boundary. [M1 — identifying the correct boundary line]
2. Symbol is $\leq$, so draw a SOLID line. — Non-strict inequality ($\leq$) means the boundary is included — solid line. If it were $<$, we would draw a dashed line.
3. When $x = 0$: $y = 2(0) - 1 = -1$, giving point $(0, -1)$. When $x = 2$: $y = 2(2) - 1 = 3$, giving point $(2, 3)$. Draw a solid line through these points. — Two points are enough to draw a straight line accurately. [M1 — correct line drawn]
4. Test $(0, 0)$: Is $0 \leq 2(0) - 1$? Is $0 \leq -1$? No. — $(0, 0)$ does NOT satisfy the inequality, so we shade the opposite side — below and to the right of the line.
5. Shade the region below the solid line $y = 2x - 1$. — Every point in the shaded region satisfies $y \leq 2x - 1$. [A1 — correct region shaded]

Answer: A solid line through $(0, -1)$ and $(2, 3)$, with the region below the line shaded.

The graph above shows the boundary line $y = 2x - 1$. In an exam, this line would be drawn solid (because the inequality is $\leq$), and the region below it — where $y$ values are less than or equal to $2x - 1$ — would be shaded. Always label the line with its equation and indicate the shaded region clearly.

5. Worked Examples Across All Three Representations

The real skill is moving fluently between all three representations. An exam question might give you a number line and ask for set notation, or give you set notation and ask you to shade a graph. Let's practise both directions.

Worked example: From Set Notation to Number Line: $\{ x: -3 \leq x < 5 \}$

Represent $\{ x: -3 \leq x < 5 \}$ on a number line.

1. $-3 \leq x < 5$ means $x$ is at least $-3$ and strictly less than $5$. — Identify both boundaries and their inequality types before drawing anything.
2. At $-3$: symbol is $\leq$, so CLOSED circle. At $5$: symbol is $<$, so OPEN circle. — [M1 — correct circle types at both boundaries]
3. Draw a number line. Place a closed circle at $-3$ and an open circle at $5$. Shade (or draw a thick line) between them. — The shaded segment represents every value from $-3$ (included) up to but not including $5$. [A1 — correct diagram]

Answer: Number line with a closed circle at $-3$, an open circle at $5$, and the region between them shaded.

Worked example: From a Graph Description to Set Notation: $x + y > 4$

A graph shows a dashed line through $(0, 4)$ and $(4, 0)$, with the region above-right shaded. Write the inequality in set notation.

1. The line passes through $(0, 4)$ and $(4, 0)$. Gradient $= \frac{0 - 4}{4 - 0} = \frac{-4}{4} = -1$. Equation: $y = -x + 4$, i.e. $x + y = 4$. — Always find the equation of the boundary line first.
2. The boundary is DASHED, so the inequality is strict: $>$ or $<$. — Dashed line = boundary not included = strict inequality ($<$ or $>$). [M1]
3. Test $(5, 5)$ (a point in the shaded region): $5 + 5 = 10 > 4$. True. — The shaded region satisfies $x + y > 4$, confirming the direction of the inequality. [M1]
4. $\{ (x, y): x + y > 4 \}$ — For two-variable inequalities, the set contains coordinate pairs $(x, y)$. [A1]

Answer: $\{ (x, y): x + y > 4 \}$

6. Deliberate Practice

These problems cover all three representation methods. Work through each one before checking the solution — the process of attempting it first is what builds the skill.

7. Exam Technique and Common Mistakes

This topic is one where marks are lost not through misunderstanding the maths, but through careless notation. The following mistakes appear repeatedly in mark schemes — knowing them in advance puts you ahead.

Notice how the open/closed circle rule from the number line and the dashed/solid line rule from the graph are the same idea in different settings. If you remember one, you remember both: strict inequality means the boundary is excluded, non-strict means it is included.

One final point on exam language: if a question says 'represent on a number line', you must draw a number line with correctly placed circles and a shaded segment or arrow — a written inequality alone will not earn the marks. Similarly, 'use set notation' requires the $\{ x: \ldots \}$ format, not a diagram. Read the command carefully and give the representation that is asked for.