Algebra - Solving equations and inequalities — Edexcel GCSE Mathematics

Solving equations and inequalities in Pearson Edexcel GCSE Mathematics involves finding the value(s) of an unknown that make an equation true or satisfy an inequality. Key skills include using inverse operations to isolate the variable, expanding brackets, solving linear and quadratic equations, and representing solutions on a number line.

Solving equations and inequalities is a fundamental algebraic skill that appears across all areas of GCSE Mathematics. You will learn to manipulate expressions, use inverse operations, and apply systematic methods to find unknown values. This topic underpins many real-world problems, from calculating costs to interpreting data.

Mastering these techniques is essential for tackling more advanced topics such as simultaneous equations, quadratic equations, and graphical inequalities. It also develops logical reasoning and problem-solving skills that are valuable in everyday life and further study.

• →Equations require finding exact values that make the statement true, while inequalities describe a range of possible values.
• →Use inverse operations to isolate the variable, performing the same operation on both sides of the equation or inequality.
• →When multiplying or dividing an inequality by a negative number, the inequality sign must be reversed.
• →Always expand brackets and simplify expressions before solving.
• →Represent solutions to inequalities on a number line using open or closed circles and arrows.

• a correct first step that does the same thing to both sides, such as subtracting the smaller x term from each side
• collecting the unknown on one side and the numbers on the other
• a correct final division even if an earlier arithmetic slip made the number wrong
• the final value of the unknown, written as a fraction or decimal when it is not a whole number
• on a graphical question, draw each straight line accurately and read the x coordinate of their intersection
• Rearranging the equation so that one side is zero before factorising.
• Correctly factorising the quadratic expression into two brackets.
• Setting each bracket equal to zero to form separate linear equations.
• Solving the linear equations to find the solution(s) for x, including a repeated root when a bracket is squared.
• When using a graph, accurately reading the x-coordinates of the points where the curve meets the x-axis, including a point of tangency for a repeated root.
• rearranging to a quadratic with zero on one side, including expanding brackets or clearing a fraction first
• substituting the values of a, b and c into the formula with their signs kept, even if the arithmetic afterwards is wrong
• the correct value under the root sign before it is simplified
• a correct completed square form, and a further mark for the solutions that follow from it
• both roots given to the accuracy the question states, whether that is a surd or a rounded decimal
• on a graphical question, plotting the curve and reading the x-coordinates of the intersections with the horizontal axis as approximate solutions
• multiplying one or both equations so that the coefficients of one letter match
• adding or subtracting correctly to eliminate that letter, even if the arithmetic afterwards is wrong
• the value of the first unknown
• substituting back and finding the second unknown, credited even when the first value was wrong
• on a graphical question, draw each line accurately and read both coordinates of their intersection
• eliminating one letter correctly when both equations are linear, for example adding 2x + y = 7 and x − y = 2 to get 3x = 9
• making one letter the subject of the linear equation and substituting it into the quadratic to reach a quadratic in one letter only
• solving that quadratic by factorising or the formula, with the usual method marks available
• finding the matching value of the second letter for each root, and presenting the answers as pairs
• reading approximate solutions from a graph, giving coordinates to the accuracy the grid allows
• a rearrangement that leads to the given iterative formula when you are asked to derive it
• a correct first iteration, with the starting value seen substituted into the formula
• the later iterations, each carried to enough decimal places to support the answer asked for
• the final root rounded exactly as the question states
• substituting two values and stating that the results have opposite signs, when asked to show a root lies between them
• Deriving a correct formula from a described procedure, with all variables clearly defined.
• Forming a correct expression, equation, or pair of simultaneous equations from the information given.
• Solving the derived equation(s) correctly, with marks available for method even if the set-up was flawed.
• Interpreting the solution back in the original context, with correct units and sensible rounding.
• a correct first step that treats both sides alike, as in an equation
• the answer written as an inequality rather than as an equation
• turning the sign round when both sides are divided by a negative number
• a number line showing the right kind of circle at the boundary and an arrow in the right direction
• a complete list of the integers in the range, with none missing and none extra
• the critical values of a quadratic inequality, found by factorising or by using the formula
• choosing the correct stretch of values, supported by a sketch of the curve or by testing a value
• the answer in the notation asked for, whether that is an inequality, set notation or a number line
• each boundary line drawn correctly on a two letter question, dashed or solid to match its inequality
• identifying the region that satisfies every inequality at the same time

• 💡Substitute your answer into the original equation before moving on; if the two sides disagree you still have time to find the slip.
• 💡When the equation contains a fraction, multiply every term by the denominator first so you are working with whole numbers.
• 💡On a graph question read to the nearest small square and say that the answer is approximate, because that is what is being asked for.
• 💡After factorising, quickly expand the brackets in the margin to check they match the original quadratic expression.
• 💡Give all real solutions the equation has; do not force two if the factorised form shows a repeated root, and check whether the context rules out a value.
• 💡For graphical solutions, be as precise as possible when reading from the axes, using the scale provided. A small ruler can help align your eye with the grid lines.
• 💡Write down a, b and c with their signs before substituting, and put brackets round any negative value.
• 💡A negative number under the root sign may be a sign slip, but it can also be a valid no-real-roots result, so check the coefficients and then interpret the outcome.
• 💡Completing the square also hands you the turning point of the curve, which a later part of the same question often asks for.
• 💡Number your equations and write what you did to them, such as doubling the second, so the elimination can be followed.
• 💡Check your pair in the equation you did not use for the substitution; both sides should balance.
• 💡When neither letter matches easily, multiply each equation by the other equation's coefficient of the same letter.
• 💡Substitute for the letter already on its own, usually y, as that saves rearranging.
• 💡Set out the final answer as two pairs, for example x = 3 with y = 6 and x = −1 with y = −2, so the pairing is unmistakable.
• 💡Test one pair in both original equations; it costs one line of working and catches sign slips.
• 💡For a graphical solution, draw both graphs accurately and state the coordinates of each intersection, checking they satisfy both equations approximately.
• 💡Keep the full calculator display for every term and round only at the very end.
• 💡Write each term on its own line with its label, so the examiner can see which iteration is which.
• 💡If you are asked to show that a root lies between two numbers, substitute both into the equation and say that the two answers have opposite signs.
• 💡When creating a formula, identify what changes (the variable) and what stays the same (the constant or fixed charge).
• 💡Before writing an equation, state what your letter stands for and its units (e.g., 'let t be the time in hours').
• 💡After solving, reread the last line of the question to ensure you are answering what was actually asked.
• 💡Say your answer aloud, as in x is less than or equal to five, to check the sign is pointing the right way.
• 💡To avoid dividing by a negative number, add the negative term to both sides instead and work from there.
• 💡Label the numbers on your number line; an unlabelled line cannot show which value the circle sits on.
• 💡Sketch the curve, mark the critical values on it and shade the part you want; the sketch settles the direction of the signs for you.
• 💡If you are unsure which stretch to take, test one value from each stretch in the original inequality.
• 💡Set notation needs the letter as well as the condition, as in {x : x ≥ 4}, so do not write the condition on its own.
• 💡Show all steps of your working, especially when expanding brackets or rearranging. Method marks are awarded even if the final answer is wrong.
• 💡For inequalities, always represent the solution on a number line as requested, using the correct type of circle (open for < or >, closed for ≤ or ≥).
• 💡Check your solution by substituting it back into the original equation or inequality to ensure it satisfies the condition.

• moving a term across the equals sign without changing its sign, turning x + 6 = 10 into x = 10 + 6
• dividing only one side by the coefficient, so 2x = 14 is written as x = 14
• expanding a bracket onto the first term only, so 3(x + 4) becomes 3x + 4
• reading the y coordinate of the crossing point when the equation was solved for x
• Incorrectly stating the solutions from the brackets, e.g. for (x + 5)(x − 3) = 0, writing x = 5 and x = −3. The signs must be reversed, so the correct solutions are x = −5 and x = 3.
• Forgetting to rearrange the equation to equal zero first, e.g. trying to factorise x² + 2x = 15 directly.
• Stopping after factorising and giving the brackets as the final answer, instead of solving for x.
• When solving an equation like x² − 4x = 0 by factorising to x(x − 4) = 0, forgetting the x = 0 solution from the single 'x' factor.
• Assuming every quadratic has two distinct solutions. Correction: a repeated root such as (x − 3)² = 0 gives one solution, and some quadratics have no real roots.
• forgetting that the formula begins with −b, so a positive coefficient is substituted without its sign changing
• taking a to be the coefficient of x rather than the coefficient of x²
• halving the coefficient of x when completing the square but not subtracting the square of that half, so x² + 6x is written as (x + 3)² instead of (x + 3)² − 9
• dividing only the first part of the formula by 2a and leaving the root undivided
• assuming a negative value under the root sign must be a slip, when it can legitimately mean the quadratic has no real roots
• subtracting when the matched terms have opposite signs, so 3y and −3y should have been added
• eliminating the letter correctly but forgetting to combine the numbers on the right as well
• finding one letter and stopping, when the question needs both values
• substituting the value back into an equation you created rather than one you were given, which repeats any earlier slip
• expanding a squared bracket wrongly, writing (x + 4)² as x² + 16 instead of x² + 8x + 16
• finding the two x values and stopping, without the y value that goes with each one
• pairing the answers the wrong way round, so one solution's x is quoted with the other solution's y
• trying to eliminate a letter by subtracting the two equations when one contains a squared term, which cannot remove it
• rounding each term before feeding it back in, so the sequence settles on a value that is not the root
• typing the starting value in again for every term instead of using the previous answer
• labelling the terms from the wrong place, so the value called x₃ is really the second iteration
• giving the root to a different accuracy from the one the question asked for
• Confusing the fixed and variable parts when creating a formula, for example swapping a standing charge with a per-unit cost.
• Mixing units within an equation, such as pounds on one side and pence on the other, without conversion.
• Solving correctly but giving the value of the variable (e.g. x) when the question asked for a related quantity (e.g. 2x + 5).
• writing the answer as x = 5 when the solution is a whole range of values
• leaving the sign unchanged after dividing by a negative number, so −2x > 6 is given as x > −3
• filling in the circle for a strict inequality, or leaving it open when the boundary value is included
• including the boundary in a list of integers when the inequality is strict, so −2 is listed for −2 < n
• offering the critical values as the answer, so −2 and 3 are written down instead of the range between them
• taking the values outside the critical values when an upward opening quadratic is below zero, or the values between them when it is above zero
• writing a double inequality the wrong way round, such as 3 < x < −2, which no value can satisfy
• drawing a solid boundary line for a strict inequality on a region question
• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always check by substituting a test value.
• Incorrectly expanding brackets, especially when a negative sign is outside. Multiply every term inside the bracket by the factor outside.
• Thinking that an equation and an inequality are solved in exactly the same way. Inequalities have a range of solutions and require sign reversal when multiplying/dividing by negatives.

1. 1Day 1-2: Revise basic equation solving (one-step and two-step) and check understanding with practice questions.
2. 2Day 3-4: Learn to solve equations with brackets and unknowns on both sides. Practice expanding and simplifying.
3. 3Day 5-6: Study inequalities: solving linear inequalities, reversing the sign, and representing solutions on number lines.
4. 4Day 7-8: Mixed practice: solve a variety of equations and inequalities, including word problems.
5. 5Day 9-10: Complete past paper questions on this topic, mark your work, and review any mistakes.

• 📋Solve a linear equation with brackets and unknowns on both sides. Advice: Expand carefully, collect terms, and check your answer.
• 📋Solve a linear inequality and show the solution on a number line. Advice: Remember to reverse the inequality sign if multiplying/dividing by a negative.
• 📋Form and solve an equation from a word problem. Advice: Define the variable, translate the problem into an equation, solve, and interpret the answer in context.
• 📋Solve a quadratic equation by factorising. Advice: Rearrange to =0, factorise, and set each factor to zero to find both solutions.