Solving equations and inequalities — AQA GCSE Mathematics

Solving equations and inequalities is a fundamental algebra topic in GCSE Mathematics. It involves finding the value(s) of an unknown variable that satisfy a given equation or inequality. Equations use the equals sign (=) and typically have one or more specific solutions, while inequalities use symbols like <, >, ≤, ≥ and have a range of solutions. Mastery of this topic is essential for higher-level algebra, problem-solving, and many real-world applications.

This topic builds on basic arithmetic and algebraic manipulation, extending to linear, quadratic, and simultaneous equations. It appears across all exam boards, including AQA GCSE, and is a prerequisite for topics such as graphs, sequences, and calculus. A strong understanding of inverse operations, balancing, and the rules for inequalities is crucial for success in both foundation and higher tier exams.

• →Equations must be balanced: whatever operation you perform on one side, you must do to the other.
• →Use inverse operations to isolate the variable: for example, if 3x + 2 = 11, subtract 2 then divide by 3.
• →Inequalities are solved similarly to equations, but the inequality sign reverses when multiplying or dividing by a negative number.
• →For quadratic equations, factorise, complete the square, or use the quadratic formula to find solutions.
• →Simultaneous equations can be solved by elimination or substitution, finding values that satisfy both equations.

• Use inverse operations that preserve equality to isolate the unknown, performing the same operation on both sides.
• Check the algebraic solution by substituting it back into the original equation.
• For a graphical method, plot each side of the equation as a separate straight-line graph on the same axes.
• Read the x-coordinate of the point of intersection to obtain the approximate solution.
• Recognise that the graphical solution may be approximate because it depends on reading accuracy.
• Collect the unknown terms on one side of the equation by adding or subtracting the same term from both sides.
• Collect the constant terms on the other side by adding or subtracting the same number from both sides.
• Divide or multiply both sides by the coefficient of the unknown to isolate it.
• Check the solution by substituting it back into the original equation and confirming both sides are equal.
• For a graphical approach, plot each side as a straight line and read the x-coordinate of the intersection point.
• Rearranges a quadratic equation into the standard form ax² + bx + c = 0 before factorising.
• Factorises a quadratic expression correctly, including cases where a common factor is taken out first.
• Uses the zero product property: if (x + p)(x + q) = 0 then x = -p or x = -q.
• Solves quadratic equations where the coefficient of x² is greater than 1 by factorising.
• Reads approximate solutions from the x-intercepts of the graph of y = ax² + bx + c.
• Checks algebraic solutions by substituting back into the original equation.
• Rearranges a quadratic equation into the form ax² + bx + c = 0 before completing the square or using the quadratic formula.
• Completes the square correctly, including when the coefficient of x² is not 1, by factoring out that coefficient first.
• Solves the equation obtained after completing the square, remembering both the positive and negative square roots.
• Substitutes the correct values of a, b and c into the quadratic formula, including negative signs.
• Simplifies the resulting expression, leaving surd answers in exact form where required.
• Correctly eliminate one variable by adding or subtracting the equations, or by substitution, to form a single equation in one variable.
• Solve the resulting linear equation accurately to find the value of one variable.
• Substitute the found value back into one of the original equations to find the other variable.
• State the solution as a coordinate pair (x, y) and verify by substitution into both original equations.
• For graphical solutions, plot both lines accurately and read the coordinates of the intersection point, giving an approximate answer.
• Rearrange the linear equation to express one variable in terms of the other.
• Substitute this expression into the quadratic equation to obtain a quadratic equation in one variable.
• Solve the quadratic equation correctly, finding both solutions where they exist.
• Substitute each value back into the linear equation to find the corresponding value of the other variable.
• State all solution pairs clearly as coordinates and verify by substitution into both original equations.
• Rearranges the given equation correctly into the iterative form x = g(x), for example x = ∛(20 − x) from x³ + x = 20.
• Uses a sensible starting value x₀ and substitutes it correctly into the iteration formula to obtain x₁.
• Repeats the iteration accurately, showing successive values, until two consecutive values agree to the required degree of accuracy.
• States the approximate solution to the required accuracy, for example x ≈ 2.59 to 2 decimal places, and shows sufficient iterations to justify it.
• Recognises when an iteration diverges or oscillates and adjusts the rearrangement or starting value accordingly.
• Defines a variable clearly, for example let w be the width in centimetres, and uses it consistently.
• Translates the situation correctly into an algebraic expression or formula, such as 4w + 6 for the perimeter.
• Forms a correct equation or a correct pair of simultaneous equations from the given information.
• Solves the equation or simultaneous equations accurately using an appropriate method, showing sufficient working.
• Interprets the solution in the context of the problem, stating the values of the unknowns with correct units and checking against the original conditions.
• Rearranges the inequality correctly by adding or subtracting the same quantity from both sides.
• Divides or multiplies both sides by the coefficient of the variable, reversing the inequality sign when the multiplier or divisor is negative.
• States the solution set correctly, for example x < 3 or x ≤ 2.
• Draws a number line with an open circle for < or > and a filled circle for ≤ or ≥.
• Indicates the correct direction of the solution set with an arrow or line extending from the circle.
• Solves a linear inequality in one or two variables by rearranging correctly and, where necessary, reversing the inequality sign when multiplying or dividing by a negative number.
• Finds the critical values of a quadratic inequality by solving the associated quadratic equation.
• Tests intervals between and beyond the critical values to determine where the quadratic inequality is satisfied.
• Represents the solution set on a number line with correct open or filled circles and direction.
• Uses set notation correctly, for example {x : 2 < x < 3} or {x : x ≤ −1} ∪ {x : x ≥ 4}.
• Draws the boundary line correctly on a graph, using a dashed line for strict inequalities and a solid line for inclusive inequalities, and shades the correct region.

• 💡Show each algebraic step clearly so that method marks can be awarded even if the final answer is wrong.
• 💡When using a graph, draw the lines accurately and label the intersection point before reading the x-value.
• 💡Substitute your solution back into the original equation to confirm it works, especially in multiple-choice questions where options may be close.
• 💡Write down each step of your rearrangement so that method marks can be awarded even if the final answer is incorrect.
• 💡After solving, substitute your value back into the original equation to check both sides are equal.
• 💡In multiple-choice questions, substitute each option into the equation to see which one works, especially if you are unsure of your algebraic method.
• 💡Always rearrange the equation to the form ax² + bx + c = 0 before factorising, and show this step clearly.
• 💡When using a graph, state that you are reading the x-axis intercepts and give your answers to a suitable degree of accuracy, such as one decimal place.
• 💡Check each solution by substituting it back into the original equation; this catches sign errors quickly.
• 💡If a question asks for solutions in surd form, leave the answer with a square root and simplify it fully; do not round to a decimal.
• 💡When using the quadratic formula, write down the values of a, b and c before substituting to reduce sign errors.
• 💡After completing the square, show the step where you take the square root of both sides, including the ± sign, to secure method marks.
• 💡Show your working clearly, especially the elimination or substitution steps, as method marks may be available.
• 💡When using a graph, draw the lines with a ruler and label them; read the intersection carefully and state that it is approximate.
• 💡Always check your solution by substituting back into both original equations; this can catch arithmetic errors.
• 💡Write down the substitution step clearly so that the examiner can follow your method.
• 💡If the quadratic does not factorise easily, use the quadratic formula and leave answers in surd form if required, unless told to round.
• 💡Always check each solution pair in both original equations to ensure they satisfy both.
• 💡Always write down your rearrangement and starting value before iterating, so the method is clear.
• 💡Show at least three or four iterations in a table, keeping full calculator display values until the final rounding.
• 💡Check convergence by comparing consecutive values; if they are not settling, reconsider your rearrangement.
• 💡Write a clear 'let' statement defining each variable before forming any equation.
• 💡Show each stage of forming and solving the equation so method marks can be awarded even if an arithmetic slip occurs.
• 💡Substitute your solution back into the original word problem to check it satisfies all conditions.
• 💡Finish with a sentence interpreting the answer in context, including units.
• 💡Always check whether the coefficient of the variable is negative before dividing, because this determines whether the inequality sign reverses.
• 💡Substitute a test value from your solution set back into the original inequality to confirm the direction is correct.
• 💡Label the number line clearly and use the correct circle type: open for strict inequalities and filled for inclusive inequalities.
• 💡For quadratic inequalities, always find the critical values first and then test a value in each interval to confirm the solution set.
• 💡When drawing a graph, label the boundary line clearly and use the correct line style for strict or inclusive inequalities.
• 💡Write the solution set in more than one form where possible, such as on a number line and in set notation, to show full understanding.
• 💡Check whether the inequality is strict or inclusive before deciding on open or filled circles and dashed or solid lines.
• 💡Always show your working, even for simple equations. Method marks are awarded for correct steps, so you can still gain marks even if the final answer is wrong.
• 💡For inequalities, represent the solution on a number line as requested. Use an open circle for < or > and a closed circle for ≤ or ≥.
• 💡Check your solution by substituting it back into the original equation or inequality. This helps catch errors and confirms your answer is correct.

• Applying an operation to only one side of the equation; correct this by always performing the same operation to both sides.
• Dividing only the unknown term rather than every term on both sides. If you divide first, divide each complete side by the same non-zero number; subtracting the constant first is often simpler.
• Reading the y-coordinate of the intersection point instead of the x-coordinate; correct this by identifying that the solution is the x-value where the two graphs meet.
• Assuming a graph always gives an exact answer; correct this by stating that graphical solutions are approximate and should be checked algebraically where possible.
• Moving a term without changing its sign; correct this by remembering that adding or subtracting a term from both sides reverses its sign on the side it is moved from.
• Collecting unknowns on the side that makes the coefficient negative; correct this by choosing to collect on the side that keeps the coefficient positive where possible, or by carefully handling negative coefficients.
• Forgetting to divide the constant term by the coefficient; correct this by dividing every term by the coefficient of the unknown.
• Reading the y-coordinate of the intersection point on a graph instead of the x-coordinate; correct this by identifying that the solution is the x-value where the two lines cross.
• Forgetting to rearrange to = 0 before factorising; for example, solving x² + 5x = -6 by writing (x + 2)(x + 3) = -6 and then incorrectly setting each bracket to -6. Correction: first add 6 to both sides to get x² + 5x + 6 = 0, then factorise and set each bracket to 0.
• Sign errors when finding the roots from brackets; for example, from (x + 2)(x + 3) = 0 writing x = 2 or x = 3. Correction: solve x + 2 = 0 to get x = -2 and x + 3 = 0 to get x = -3.
• Reading the y-intercept instead of the x-intercepts from a graph. Correction: approximate solutions are the x-values where the curve crosses the x-axis, not where it crosses the y-axis.
• Forgetting to rearrange to = 0 before using the quadratic formula; for example, using the formula on x² + 3x = 4 without first writing x² + 3x - 4 = 0. Correction: always rearrange to ax² + bx + c = 0 and identify a, b and c from that form.
• Sign errors when substituting negative values into the quadratic formula; for example, taking b = -3 when b is actually 3, or mishandling -b. Correction: write down a, b and c clearly, then substitute carefully, using brackets for negative numbers.
• When completing the square, forgetting to subtract (b/2)² after adding it, or losing the constant term. Correction: check the expansion of your completed-square form to confirm it matches the original expression.
• Forgetting to multiply both sides of an equation when using elimination, leading to an incorrect elimination step. Correction: always apply the same operation to the entire equation.
• Substituting the found value into the wrong equation or making an arithmetic error. Correction: substitute into the simpler equation and check in both.
• Reading the intersection point from a graph inaccurately, for example misreading the scale. Correction: use a ruler and check that the point lies on both lines.
• Giving the solution as two separate numbers without indicating which is x and which is y. Correction: always write the answer as a coordinate pair (x, y).
• Substituting into the quadratic without rearranging the linear equation correctly, leading to an equation that is not quadratic. Correction: always rearrange first and check the substitution.
• Losing one of the solutions by dividing by a variable or by not considering both roots. Correction: solve the quadratic fully, using factorisation or the formula, and list both roots.
• Forgetting to find the corresponding values of the other variable for each root. Correction: always substitute each root back into the linear equation.
• Making sign errors when expanding brackets after substitution. Correction: expand carefully and check each term.
• Substituting into the original equation instead of the rearranged iterative formula; the correction is to apply xₙ₊₁ = g(xₙ) at each step.
• Rounding intermediate values too early, which prevents convergence; the correction is to keep full calculator accuracy and round only the final answer.
• Assuming any rearrangement will converge; the correction is to test the iteration and, if values diverge, choose a different rearrangement or starting value.
• Stopping after one or two iterations without evidence of convergence; the correction is to continue until successive values agree to the required accuracy.
• Failing to define the variable or using different letters for the same unknown; the correction is to state clearly what each letter represents and use it throughout.
• Forming an equation from only part of the information, such as using the perimeter formula without the length relationship; the correction is to include every given condition.
• Solving correctly but not returning to the context, leaving the answer as a bare number; the correction is to state what the value means, for example the width is 5 cm.
• Making sign errors when solving simultaneous equations by elimination; the correction is to check by substituting both values back into both original equations.
• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number; the correction is to reverse the sign whenever the multiplier or divisor is negative.
• Using a filled circle for a strict inequality such as x < 3; the correction is to use an open circle for strict inequalities and a filled circle only for ≤ or ≥.
• Drawing the arrow in the wrong direction after solving, for example showing x > 3 when the solution is x < 3; the correction is to check the final inequality before drawing the arrow.
• Treating the inequality as an equation and giving a single value rather than a solution set; the correction is to state the range of values and show it on the number line.
• Assuming a quadratic inequality is satisfied between the critical values without testing; the correction is to test a value in each interval to determine the correct region.
• Using a solid boundary line for a strict inequality on a graph; the correction is to use a dashed line for < and > and a solid line for ≤ and ≥.
• Writing set notation incorrectly, such as omitting the variable or using the wrong inequality signs; the correction is to write the set clearly, for example {x : x > 2}.
• Forgetting to reverse the inequality sign when dividing by a negative number in a two-variable inequality; the correction is to reverse the sign whenever the multiplier or divisor is negative.
• Students often forget to reverse the inequality sign when multiplying or dividing by a negative number. For example, solving -2x > 6 should give x < -3, not x > -3.
• When solving equations with fractions, students may multiply only one term by the denominator instead of every term. Always multiply the entire equation by the common denominator.
• Students sometimes assume that all equations have integer solutions. Solutions can be fractions, decimals, or surds, and should be left in exact form unless told otherwise.

1. 1Day 1-2: Revise solving linear equations, including those with brackets and fractions. Practice at least 10 equations of increasing difficulty.
2. 2Day 3-4: Learn and practice solving linear inequalities, focusing on the rule for reversing the sign. Represent solutions on number lines.
3. 3Day 5-6: Tackle quadratic equations by factorising, completing the square, and using the formula. Practice identifying the most efficient method.
4. 4Day 7-8: Master simultaneous equations using elimination and substitution. Solve problems involving both linear and quadratic equations.
5. 5Day 9-10: Complete mixed exam-style questions under timed conditions. Review mistakes and revisit weak areas.

• 📋Solve a linear equation: often presented as a straightforward equation like 4x - 7 = 21. Show each step clearly to secure method marks.
• 📋Solve an inequality and represent on a number line: e.g., solve 3 - 2x < 9 and draw the solution. Remember to reverse the sign when dividing by a negative.
• 📋Solve simultaneous equations: typically two linear equations, but higher tier may include one linear and one quadratic. Use elimination or substitution and check your answer.
• 📋Form and solve an equation from a word problem: e.g., 'The perimeter of a rectangle is 30 cm. Its length is 3 cm more than its width. Find the dimensions.' Define variables, set up an equation, solve, and interpret the solution.