Solving equations and inequalities — AQA GCSE Mathematics

Solving equations and inequalities involves finding the unknown values that make a mathematical statement true, using inverse operations to isolate the variable. For inequalities, the solution is a range of values, and the inequality sign reverses when multiplying or dividing by a negative number.

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.

• a correct first step that does the same thing to both sides, such as taking the same number from each side
• collecting the terms in x on one side and the number terms on the other
• dividing by the coefficient of x, and this method mark can stand even if the final number is wrong
• reading the solution from a graph as the value of x where the line meets the required height, within the tolerance the grid allows
• expanding any brackets correctly before terms are collected
• a correct step that removes the unknown from one side, such as taking 3x from both sides
• gathering the number terms on the other side to leave a one-step equation
• the correct final value, which may be a fraction, with the earlier method marks standing even if that last step slips
• rearranging so that one side is zero before any factorising is attempted
• a correct factorisation, which an examiner can verify by expanding it back
• setting each factor equal to zero to produce two linear equations
• stating every root the factorisation gives, including a negative one, and not only the value that looks tidy
• rearranging to a three-term quadratic with zero on one side and the terms in the usual order
• a correct completed square form such as (x + 3)² − 7
• substituting a, b and c into the formula with their correct signs, which stands even if the arithmetic after it is wrong
• each solution, written to the accuracy asked for or left in surd form when that is what the question wants
• a correct method to match the coefficients of one letter, such as multiplying an equation through
• adding or subtracting correctly so that one letter is eliminated
• the first value found, and a further mark for substituting it back to obtain the second
• both values given as a pair that satisfies both original equations
• recognising that parallel lines give no solution and coincident lines give infinitely many solutions
• Rearranging the linear equation to make one letter the subject.
• Substituting that expression into the quadratic, squaring the bracket as a whole.
• Reaching a three-term quadratic with zero on one side.
• Solving it, and pairing each value of x with its own value of y.
• Recognising that equating can be used when both equations are already expressed as y = …, by setting the right-hand sides equal.
• a correct rearrangement into the iterative form when the question asks you to derive it
• substituting the given starting value into the right-hand side to produce the first iterate
• a further correct iterate obtained from the unrounded previous value
• the final answer written to the degree of accuracy the question states
• using a sign change in f(x) between two values to justify that a root lies between them
• defining the unknown clearly, including the unit it is measured in
• an equation, or a pair of equations, that matches the information given
• solving what you formed, which correct method can earn marks even when the equation itself was set up wrongly
• the answer given in context, with its unit and the quantity it refers to
• a correct first step that treats the inequality as an equation would be treated
• turning the inequality sign round when both sides are multiplied or divided by a negative number
• the answer written as an inequality rather than as a single value
• a number line carrying the correct type of circle at each critical value and the arrow or segment in the right place
• Correctly solving a linear inequality, including reversing the sign when multiplying or dividing by a negative number.
• Finding the correct critical values for a quadratic inequality by solving the corresponding equation.
• Using a sketch or sign analysis to identify the correct region(s) for a quadratic inequality solution, expressed using correct notation.
• Representing a one-variable solution accurately on a number line with the correct circle type (open/closed) and direction.
• Writing the solution in set notation, for example {x : x ≥ −3} or {x : x < −2} ∪ {x : x > 3}.
• For two-variable inequalities, drawing the correct boundary line (solid/dashed) and shading the correct region.

• 💡put your answer back into the original equation; if both sides give the same number you are safe to move on
• 💡clear any fraction by multiplying every term by the denominator before you start collecting terms
• 💡leave a non-whole answer as a fraction in its lowest terms unless the question asks for a decimal
• 💡on a graph question, draw a faint line across from the given height and down to the horizontal axis so your reading is easy to see
• 💡move the unknowns to the side where the coefficient comes out positive; fewer negatives means fewer slips
• 💡substitute your value into each side separately, since both sides must give the same number
• 💡keep one operation per line of working so every balancing step is visible
• 💡expand your brackets back in your head before solving, since a wrong factorisation wastes every mark after it
• 💡check for a common factor or a difference of two squares before hunting through pairs of numbers
• 💡when a graph question asks for approximate solutions, give the values the grid supports rather than rounding to whole numbers
• 💡write down a, b and c with their signs before touching the formula, then substitute in one go
• 💡bracket the whole numerator and bracket 2a on the calculator, or the division will be applied to the wrong part
• 💡if b² − 4ac comes out negative, recheck your signs before concluding there is no real solution
• 💡number your two equations and note what you multiplied each by, so the elimination step can be followed
• 💡test your pair in the equation you did not use for the substitution as a quick check
• 💡on a graph question, read the crossing point to the nearest small division and give both coordinates
• 💡Substitute the letter that is already on its own; it keeps the algebra shortest.
• 💡Set your answers out as pairs, for example x = 2 with y = 5, so the pairing cannot be misread.
• 💡Check one pair in the quadratic equation, since that is where a sign error tends to hide.
• 💡type the formula once with the answer key inside it, then press equals repeatedly for each new value
• 💡write down every iterate, not only the last one, because the intermediate values carry method marks
• 💡once two successive values agree to the accuracy asked for, you have your answer and can stop
• 💡Write one short sentence defining your letter before you form any equation.
• 💡Read your equation back against the words of the question; if it is wrong, correct method may still earn follow-through marks.
• 💡Finish with a sentence that answers the question asked, unit included.
• 💡test one value from your range in the original inequality; if it fails, the sign is the wrong way round
• 💡if negative coefficients trip you up, move the terms in x to the other side so the coefficient stays positive
• 💡check the scale on a printed number line before marking it, since the divisions may not be single units
• 💡Always sketch the parabola for a quadratic inequality. Its shape makes it clear whether the solution is between or outside the critical values.
• 💡When representing a solution on a number line, use an open circle (o) for < and >, and a filled circle (●) for ≤ and ≥, to show if the endpoint is included.
• 💡For two-variable inequalities, test a convenient point like (0,0) in the original inequality to quickly determine which side of the line to shade.
• 💡Write set notation as {x : condition}; for two separate ranges use the union symbol ∪, as in {x : x < −2} ∪ {x : x > 3}.
• 💡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.

• adding the 4 to the right instead of subtracting it from both sides, so 5x + 4 = 19 turns into 5x = 23 rather than 5x = 15
• expanding 2(x + 1) = 10 as 2x + 1 = 10, so only the first term inside the bracket is multiplied
• stopping at 5x = 15 and offering that as the answer instead of x = 3
• losing a minus sign as a term crosses the equals sign
• taking a term off one side only, so 7x − 5 = 3x + 11 becomes 4x − 5 = 3x + 11
• changing the sign of a term that has not crossed the equals sign
• expanding 4(x − 2) as 4x − 2 instead of 4x − 8
• assuming an answer must be a whole number and rounding, when a fraction such as 7/2 is exact
• factorising while a term is still on the right, so the product is not zero and the reasoning collapses
• solving x² + 5x + 6 = 0 and writing the roots as 2 and 3 rather than −2 and −3
• dividing x² − 4x = 0 through by x, which throws away the root x = 0
• reading where the curve crosses the vertical axis instead of the horizontal one
• reading b as positive when the equation has a negative term in x, so −5x is entered with b as 5
• leaving out the ± and giving only one of the two solutions
• halving the coefficient of x but forgetting to subtract the square, so x² + 6x + 2 is written as (x + 3)² + 2
• dividing only the square root part by 2a instead of the whole of the numerator
• multiplying only part of an equation, so 2x + 3y = 12 becomes 4x + 6y = 12
• adding when the matched terms have the same sign, which doubles that letter instead of removing it
• finding x and stopping, so the second unknown is never worked out
• sign slips while subtracting, so subtracting a negative term is treated as subtracting a positive one
• assuming every pair of linear equations has exactly one solution, ignoring parallel or coincident lines
• Squaring term by term, so (x + 3)² is written as x² + 9; the correct expansion is x² + 6x + 9.
• Solving the quadratic and stopping, leaving the second variable unfound; substitute each x back into the linear equation.
• Pairing the answers the wrong way round, which gives points on neither graph; keep each x with its own y.
• Substituting back into the quadratic when the linear equation gives the partner value with far less work.
• Calling the method 'elimination' when equating y = x² and y = x + 3; that is equating, and it requires both equations in y = … form.
• retyping a rounded version of the previous answer, so the digits drift away from the true root
• quoting the final value to more decimal places than the rounded working can support
• carrying out one iteration when the question asks for a sequence of values
• losing track of the order, so the value labelled x₃ is really the second iterate
• assuming iteration is only for equations that cannot be solved exactly, when it can also approximate roots of equations that have exact solutions
• writing an expression where an equation is needed, so there is nothing to solve
• using one letter for two different quantities, such as both ticket prices
• solving correctly and leaving the answer as a bare value with no statement of what it measures
• mixing units, for example pounds in one equation and pence in the other
• leaving the sign unchanged after dividing by a negative, so −2x > 6 is given as x > −3
• replacing the inequality sign with an equals sign and offering one value as the answer
• using a filled circle where the value is excluded, or an open circle where it is included
• drawing the arrow the wrong way, for example pointing right when the solution is x < 5
• Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number.
• Stating the critical values as the final answer for a quadratic inequality, instead of the range(s) they define (e.g. writing x = −2, 3 instead of x < −2 or x > 3).
• Using a solid line for a strict inequality (< or >) or a dashed line for an inclusive one (≤ or ≥) when drawing a boundary on a graph.
• Writing set notation incorrectly, such as {x : x ≥ −3} as {x = −3} or omitting the union symbol between the two parts of a quadratic solution.
• 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.