Solving equations and inequalities revision notes

Solving equations and inequalities

Specification context

Solving equations and inequalities appears in AQA GCSE Mathematics 8300.

Topic overview

Prepare learners to solve algebraic problems, derive equations from contexts and represent inequality solutions. When revising this area, students should focus on accurate precise mathematical notation and terminology, secure number, algebra, geometry, graphs, probability, statistics, trigonometry, vectors, and mathematical reasoning, and the ability to explain each idea in a way that would score in an exam. The specification expects understanding, not just recognition, so revision should combine definitions, comparisons, worked methods, and answer checks.

Learning objectives

• [Foundation and Higher] Solve linear equations in one unknown algebraically, including equations with brackets.
• [Foundation and Higher] Find approximate solutions to linear equations using a graph.
• [Foundation and Higher] Include equations with the unknown on both sides where tier-appropriate.
• [Foundation and Higher] Solve quadratic equations algebraically by factorising, including rearranged equations where tier-appropriate.
• [Foundation and Higher] Find approximate quadratic solutions using a graph.
• [Higher only] Solve quadratic equations by completing the square and using the quadratic formula.
• [Foundation and Higher] Solve two simultaneous linear equations in two variables algebraically.
• [Foundation and Higher] Find approximate solutions to simultaneous equations using a graph.
• [Higher only] Solve simultaneous equations involving one linear and one quadratic equation.
• [Higher only] Find approximate solutions to equations numerically using iteration.
• [Higher only] Use suffix notation in recursive formulae where required.
• [Foundation and Higher] Translate simple situations or procedures into algebraic expressions or formulae.
• [Foundation and Higher] Derive equations, solve them and interpret the solutions in context.
• [Higher only] Derive and solve two simultaneous equations from a contextual problem.
• [Foundation and Higher] Solve linear inequalities in one variable and represent solution sets on a number line.
• [Higher only] Solve linear inequalities in one or two variables and quadratic inequalities in one variable.
• [Higher only] Represent inequality solutions using set notation and graphs.

Objective-by-objective revision

Linear equations: [Foundation and Higher] Solve linear equations in one unknown algebraically, including equations with brackets.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Linear equations: [Foundation and Higher] Find approximate solutions to linear equations using a graph.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Linear equations: [Foundation and Higher] Include equations with the unknown on both sides where tier-appropriate.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Quadratic equations: [Foundation and Higher] Solve quadratic equations algebraically by factorising, including rearranged equations where tier-appropriate.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Quadratic equations: [Foundation and Higher] Find approximate quadratic solutions using a graph.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Quadratic equations: [Higher only] Solve quadratic equations by completing the square and using the quadratic formula.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Simultaneous equations: [Foundation and Higher] Solve two simultaneous linear equations in two variables algebraically.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Simultaneous equations: [Foundation and Higher] Find approximate solutions to simultaneous equations using a graph.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Simultaneous equations: [Higher only] Solve simultaneous equations involving one linear and one quadratic equation.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Iteration: [Higher only] Find approximate solutions to equations numerically using iteration.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Iteration: [Higher only] Use suffix notation in recursive formulae where required.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Algebraic modelling: [Foundation and Higher] Translate simple situations or procedures into algebraic expressions or formulae.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Algebraic modelling: [Foundation and Higher] Derive equations, solve them and interpret the solutions in context.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Algebraic modelling: [Higher only] Derive and solve two simultaneous equations from a contextual problem.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Inequalities: [Foundation and Higher] Solve linear inequalities in one variable and represent solution sets on a number line.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Inequalities: [Higher only] Solve linear inequalities in one or two variables and quadratic inequalities in one variable.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Inequalities: [Higher only] Represent inequality solutions using set notation and graphs.

To revise this objective well, start by naming the key mathematical idea in clear language. Then explain what it means in the context of Solving equations and inequalities, using accurate precise mathematical notation and terminology rather than short labels. A high-quality answer should show the method, notation, evidence, or reasoning chain that the objective requires. Students often lose marks when they give an answer without linking it back to the exact calculation, reasoning, representation, and interpretation being tested. A stronger response connects the idea to the specification, uses a direct GCSE Mathematics example, and keeps each sentence focused on the wording of the objective rather than repeating broad topic knowledge. A helpful self-check is to ask whether you could answer a new question on this objective without reading from the page. If you can identify the method, justify the working, and check the final answer or conclusion, you are more likely to score in questions that reward accurate mathematical working and conclusion.

Key terms

• linear equation
• unknown
• graph
• quadratic equation
• completing the square
• quadratic formula
• linear
• simultaneous equations
• quadratic
• iteration

Exam focus

Use precise precise mathematical notation and terminology, show each calculation, reasoning, representation, and interpretation step clearly, and check that the answer form matches the question. Read the command word carefully, because a question that asks you to calculate needs a different answer style from one that asks you to explain, compare, or justify.

Common mistakes to avoid

• Avoid a vague answer when the question asks you to [foundation and higher] solve linear equations in one unknown algebraically, including equations with brackets..
• Avoid a vague answer when the question asks you to [foundation and higher] find approximate solutions to linear equations using a graph..
• Avoid a vague answer when the question asks you to [foundation and higher] include equations with the unknown on both sides where tier-appropriate..
• Avoid a vague answer when the question asks you to [foundation and higher] solve quadratic equations algebraically by factorising, including rearranged equations where tier-appropriate..
• Avoid a vague answer when the question asks you to [foundation and higher] find approximate quadratic solutions using a graph..
• Avoid a vague answer when the question asks you to [higher only] solve quadratic equations by completing the square and using the quadratic formula..

Revision strategy

A practical way to revise this topic is to learn the key terms first, then test yourself with flashcards, then move on to MCQs and practice explanations. If you can teach the idea aloud in a logical order and connect it directly to the learning objective, you are much more likely to produce a precise exam answer under time pressure.

How exam questions usually test this topic

Questions on this topic often reward precise use of precise mathematical notation and terminology, clear sequencing, and the ability to connect a named method to the values, diagram, graph, expression, or context in the question. A strong answer names the mathematical idea, applies it carefully, and then ties the final line back to the exact wording of the question.

Final knowledge check

Before moving on, make sure you can define the main terms, explain the important processes in full sentences, compare similar ideas accurately where needed, and recognise common traps in multiple-choice questions. If one part still feels uncertain, return to the matching learning objective and rebuild your explanation from the key vocabulary upward.