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Inequalities

FoundationHigherAQAEdexcelOCR

Get to grips with inequalities using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on solving and representing inequalities, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. Flip the inequality sign when you multiply or divide by a negative.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An inequality compares two expressions that are not necessarily equal, using the symbols for less than, greater than, and their or-equal-to versions.

Solving one works almost exactly like solving an equation: undo the operations, doing the same to both sides. The answer is a range of values rather than a single one.

The one crucial difference is that multiplying or dividing by a negative number reverses the inequality sign. If \(-2x > 6\), dividing by \(-2\) gives \(x < -3\), with the sign flipped. Forgetting this produces an answer that is exactly the wrong way round.

Revision notes

Solving like an equation

Undo the operations in reverse order, applying each to both sides.

For \(3x + 4 \le 19\): subtract \(4\) to get \(3x \le 15\), then divide by \(3\), giving \(x \le 5\).

Dividing by a negative

Multiplying or dividing both sides by a negative reverses the direction of the sign.

From \(-4x < 12\), dividing by \(-4\) gives \(x > -3\). An alternative that avoids the issue entirely is to add \(4x\) to both sides first and keep everything positive.

Number lines and integer solutions

On a number line, an open circle means the value is excluded and a filled circle means it is included.

For \(-2 < x \le 3\), the integer solutions are \(-1, 0, 1, 2, 3\). The \(-2\) is excluded because the inequality is strict.

Key points

  • An inequality gives a range of values.
  • Solve using the same steps as an equation.
  • Multiplying or dividing by a negative reverses the sign.
  • An open circle excludes a value; a filled circle includes it.
  • \(\le\) and \(\ge\) include the endpoint.
  • List integer solutions carefully at both ends.

Worked examples

Example 1

Solve \(5x - 3 > 12\).

Working

\[5x > 15\]add 3 to both sides
\[x > 3\]divide both sides by 5

Example 2

Solve \(-3x \ge 9\).

Working

\[-3x \ge 9\]divide both sides by −3
\[x \le -3\]the sign reverses because you divided by a negative

Example 3

List the integer values satisfying \(-1 \le x < 4\).

Working

\[\text{Include } -1 \text{, exclude } 4\]the left is inclusive, the right is strict
\[-1, 0, 1, 2, 3\]list every integer in the range

Common mistakes

  • Forgetting to reverse the sign after dividing by a negative.

    −2x > 6 gives x < −3. The direction flips.

  • Including an endpoint that is excluded.

    For x < 4 the value 4 is not a solution, so it is not listed.

  • Writing the answer as an equation.

    The answer is x > 3, not x = 3. An inequality has a range of solutions.

  • Mixing up open and filled circles.

    Filled means included, matching ≤ or ≥.

Exam tips

  • Avoid dividing by a negative by moving terms instead.
  • Say aloud whether each endpoint is included.
  • Draw a number line for double inequalities.
  • Check a value from within your range satisfies the original inequality.

Key terms

Inequality
A statement that one expression is greater or less than another.
Strict inequality
One using < or >, excluding the endpoint.
Integer
A whole number, positive, negative or zero.
Number line
A line used to represent a range of values.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.