Linear Inequalities: Solving and Representing — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers inequalities containing a letter to the first power. By the end of this guide you should be able to read the four inequality symbols, solve a linear inequality, and represent the solution on a number line.

You should also be able to handle double-ended inequalities, list the integers satisfying one, deal with the case where you multiply or divide by a negative number, and represent a region on a graph using inequalities.

The organising idea is that an inequality is solved exactly like an equation, with one exception: multiplying or dividing by a negative number reverses the sign. Everything else transfers directly — do the same to both sides, undo operations in reverse order, collect the letters on one side. The one exception is not arbitrary either: 3 is less than 5, but multiply both by −1 and −3 is greater than −5, because negating flips the number line end to end. Knowing there is exactly one difference, and why it exists, turns this into a topic you already largely know.

Key terms and definitions

Inequality — a statement that two quantities are not necessarily equal, using <, >, ≤ or ≥.

Strict inequality — < or >, which exclude the boundary value.

Non-strict inequality — ≤ or ≥, which include it.

Solution set — all the values satisfying the inequality.

Integer — a whole number, positive, negative or zero.

Number line — a diagram showing the solution set, using an open or filled circle.

Region — the area of a graph satisfying one or more inequalities.

Core concepts

The four symbols

x < 3 means x is less than 3, not including 3 itself.

x > 3 means x is greater than 3, again excluding 3.

x ≤ 3 means x is less than or equal to 3, so 3 is allowed.

x ≥ 3 means x is greater than or equal to 3.

The open end of the symbol always faces the larger quantity, which is a reliable way to read one that has been written in an unfamiliar order. In 5 > x, the open end faces the 5, so 5 is the larger — which is the same statement as x < 5.

Being able to flip an inequality round like that matters, because answers are conventionally written with the letter first.

Solving like an equation

Use inverse operations, doing the same to both sides, exactly as with an equation.

For 3x + 4 > 19: subtract 4 from both sides to get 3x > 15, then divide by 3 to get x > 5.

The answer is a range, not a single value, and it should be written as an inequality. Writing x = 5 is the commonest way of throwing away a correct solution at the last moment.

Brackets and fractions are handled as they would be in an equation: expand, or multiply through by the denominator.

The one exception: negatives

Multiplying or dividing both sides by a negative number reverses the inequality sign.

For −2x > 6: dividing both sides by −2 gives x < −3, with the sign flipped.

The reason is worth seeing once. Start with the true statement 3 < 5. Multiply both sides by −1 and you get −3 and −5; but −3 is greater than −5, so the sign must turn round for the statement to stay true.

Many students prefer to avoid the situation entirely, and that is a legitimate strategy: for −2x > 6, add 2x to both sides and subtract 6 instead, giving −6 > 2x, so 2x < −6 and x < −3. Same answer, no flipping.

Adding or subtracting a negative number never affects the sign — only multiplying or dividing does.

The unknown on both sides

Collect the letter terms on the side that leaves a positive coefficient, and the exception above never arises.

For 5x − 2 < 3x + 8: subtracting 3x gives 2x − 2 < 8, then adding 2 gives 2x < 10 and x < 5.

Collecting on the other side would give −2x > −10, which still gives x < 5 after dividing by −2 and flipping — but with an extra opportunity for error.

Number lines

A number line shows the solution set with a circle at the boundary and an arrow along the values that satisfy it.

An open (unfilled) circle marks a boundary that is not included, used for < and >.

A filled circle marks a boundary that is included, used for ≤ and ≥.

So x > 2 has an open circle at 2 with an arrow pointing right, while x ≤ −1 has a filled circle at −1 with an arrow pointing left.

The circle type is worth a mark on its own, and mixing the two up is the most frequent error in the representing half of this topic.

Double-ended inequalities

An inequality such as −3 < x ≤ 4 sets bounds at both ends, and it is shown as a line segment between two circles — open at −3, filled at 4.

To solve one, apply each operation to all three parts.

For 1 < 2x + 5 ≤ 11: subtract 5 from every part to get −4 < 2x ≤ 6, then divide every part by 2 to get −2 < x ≤ 3.

Working on only two of the three parts is the usual slip, and it produces an inequality that no longer says anything true.

Listing integers

Questions frequently ask for the integers satisfying an inequality, which means solving it first and then listing carefully.

For −2 < x ≤ 3, the integers are −1, 0, 1, 2 and 3.

Two details decide the marks. The boundary at −2 is excluded because the symbol is strict, so −1 is the smallest. The boundary at 3 is included because the symbol is not strict, so 3 appears in the list.

Do not forget zero, which belongs in the list whenever it lies in the range and is very easily skipped.

Regions on a graph

An inequality in two letters describes a region of the plane rather than a stretch of the number line.

Draw the boundary line first, by treating the inequality as an equation — so for y ≥ 2x + 1, draw y = 2x + 1.

Use a solid line for ≤ or ≥, since the boundary is included, and a dashed line for < or >, since it is not. This mirrors the filled and open circles on a number line.

Then decide which side to shade by testing a point, usually the origin. For y ≥ 2x + 1, substituting (0, 0) gives 0 ≥ 1, which is false, so the origin is not in the region and the other side is shaded.

Where several inequalities apply, draw every boundary and shade only the region satisfying all of them. Label it clearly, and state whether you are shading the required region or the unwanted one — mark schemes accept either, provided you say which.

Worked examples

Example 1: Solving and representing

Solve 4x − 3 ≤ 17 and show the solution on a number line.

Add 3 to both sides: 4x ≤ 20.

Divide both sides by 4: x ≤ 5.

On the number line, put a filled circle at 5, because the symbol includes the boundary, with an arrow pointing left towards smaller values.

Writing x = 5 rather than x ≤ 5 would lose the mark despite the working being right.

Example 2: Dividing by a negative

Solve 8 − 3x < 23.

Subtract 8 from both sides: −3x < 15.

Divide both sides by −3, reversing the sign: x > −5.

The avoiding route gives the same result: adding 3x to both sides gives 8 < 23 + 3x, subtracting 23 gives −15 < 3x, so 3x > −15 and x > −5. ✓

Checking a value confirms it: x = 0 gives 8 < 23, which is true, and 0 does satisfy x > −5.

Example 3: A double-ended inequality

Solve −7 ≤ 3x + 2 < 14 and list the integer solutions.

Subtract 2 from all three parts: −9 ≤ 3x < 12.

Divide all three parts by 3: −3 ≤ x < 4.

The integers are −3, −2, −1, 0, 1, 2 and 3.

Note the ends: −3 is included because the symbol is ≤, and 4 is excluded because the symbol is strict. Zero is in the list.

Common mistakes and how to avoid them

Giving the answer as an equation. The solution is a range, so it keeps an inequality sign.

Forgetting to reverse the sign after dividing by a negative. Or avoid the situation by collecting the letters on the positive side.

Using the wrong circle on a number line. Open for < and >, filled for ≤ and ≥.

Operating on only two parts of a double-ended inequality. Every operation applies to all three.

Missing zero when listing integers. It counts whenever it lies in the range.

Including an excluded boundary in a list. Check whether each end is strict.

Using a solid line for a strict inequality on a graph. Dashed for < and >, solid for ≤ and ≥.

Exam technique for "Linear Inequalities"

Write each step on its own line, treating it as an equation, and note where a negative division forces a reversal.

Collect the letter terms on whichever side leaves a positive coefficient. It costs nothing and removes the topic's one genuine trap.

Keep the inequality sign in the final answer, and check it points the way the working requires.

For a number line, decide the circle type from the symbol before drawing anything.

When listing integers, write the solved inequality above the list and check both ends against their symbols.

For a graph, draw every boundary with the correct line style, test the origin to pick the side, and state clearly which region you have shaded.

Quick revision summary

An inequality is solved exactly like an equation, with one exception: multiplying or dividing by a negative number reverses the sign.

The answer is a range, so it keeps its inequality sign — x ≤ 5, never x = 5.

Collect the letters on the side that leaves a positive coefficient, and the reversal never comes up.

On a number line: an open circle for < and >, a filled circle for ≤ and ≥, with an arrow along the satisfying values.

For a double-ended inequality, apply every operation to all three parts.

When listing integers, check whether each boundary is included, and remember zero.

On a graph, draw the boundary as a dashed line for < and > and a solid line for ≤ and ≥, test a point such as the origin to choose the side, and label the region you shade.