AQA GCSE Mathematics 8300: Solving equations and inequalities – Revision Notes

These revision notes cover subsection 3.2.3, Solving equations and inequalities (A17 to A22), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). Linear equations, factorising quadratics, linear/linear simultaneous equations, forming equations and linear inequalities are for both tiers. Completing the square, the quadratic formula, linear/quadratic simultaneous equations, iteration, quadratic inequalities, set notation and regions are Higher tier only. This content can appear on the non-calculator Paper 1 or the calculator Papers 2 and 3.

For full explanations and worked examples, read the solving equations and inequalities study guide. Then try the practice questions. The Algebra revision notes summarise the rest of Topic 2. The AQA GCSE Mathematics hub, the printable checklist and the free diagnostics help you plan what to revise next.

Key words

• Solution (or root): a value that makes the equation true.
• Simultaneous equations: two equations that must both be true for the same pair of values.
• Iteration: repeating a formula, feeding each answer back in, to approach a solution.
• Strict inequality: < or >; the boundary value is not included.
• Solution set: all the values that satisfy an inequality.

Facts to know

Fact	Detail	Tier
Quadratic formula	x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0, a ≠ 0. Not given in the exam.	Higher tier only
Completed square	x² + bx + c = (x + b/2)² − (b/2)² + c	Higher tier only
Inequality sign rule	Multiply or divide by a negative: reverse the sign	Both
Number line	Open circle for < or >; closed circle for ≤ or ≥	Both
Graph boundary	Dashed line for < or >; solid line for ≤ or ≥	Higher tier only
Set notation	{x : x ≥ 2}; two parts joined with ∪	Higher tier only
Iteration notation	xₙ₊₁ is the next term, xₙ the current one	Higher tier only

A17: Linear equations – method in steps

1. Expand brackets.
2. Move the unknown to the side with the larger coefficient.
3. Move the numbers to the other side.
4. Divide.

Reminder: 2(x − 4) = x + 3 → 2x − 8 = x + 3 → x = 11.

Graphically: the solution of an equation is the x-coordinate where the two sides, drawn as graphs, cross.

A18: Quadratic equations – method in steps

1. Rearrange to … = 0 (needed first in Higher questions such as 3x² = 5x + 2).
2. Try to factorise. If it factorises, set each bracket equal to 0.
3. Higher tier only: if it does not factorise, or the question asks for surds, complete the square. If it asks for decimals, use the formula.

Reminders:

x^2 - 7x + 12 = 0  ->  (x - 3)(x - 4) = 0  ->  x = 3 or x = 4
x^2 + 4x - 1 = 0   ->  (x + 2)^2 - 5 = 0
                   ->  x + 2 = ±√5  ->  x = -2 ± √5     (Higher)

Graphically: roots are where y = ax² + bx + c crosses the x-axis. To solve ax² + bx + c = k, read where the curve meets the line y = k.

A19: Simultaneous equations – method in steps

Linear/linear (both tiers), elimination:

1. Line up x and y terms.
2. Multiply to make one pair of coefficients equal.
3. Same signs: subtract. Different signs: add.
4. Solve, substitute back, check in the other equation.

Reminder: 2x − y = 7 and x + y = 8. Add: 3x = 15, so x = 5, y = 3.

Linear/quadratic (Higher tier only), substitution:

1. Make y (or x) the subject of the linear equation.
2. Substitute into the quadratic.
3. Rearrange to = 0 and solve.
4. Find the matching value of the other letter for each solution.

Reminder (Higher): y = x − 2 and y = x² − 4x + 2.

x^2 - 4x + 2 = x - 2
x^2 - 5x + 4 = 0
(x - 1)(x - 4) = 0
x = 1, y = -1   and   x = 4, y = 2

Graphically: read the coordinates of each intersection point.

A20: Iteration – method in steps (Higher tier only)

1. Show a root lies between a and b: f(a) and f(b) have opposite signs.
2. Rearrange f(x) = 0 into x = g(x) when asked.
3. Use xₙ₊₁ = g(xₙ) from the given starting value. Use the ANS key.
4. Write every iterate to at least 4 d.p.
5. To confirm a root to 2 d.p., check for a sign change across the rounding interval.

Reminder: xₙ₊₁ = √(xₙ + 4) with x₁ = 1 gives x₂ = 2.2361, x₃ = 2.4972, x₄ = 2.5490. The values are approaching a solution of x² − x − 4 = 0. To show that solution is 2.56 to 2 d.p., let h(x) = x² − x − 4: h(2.555) = −0.027 and h(2.565) = 0.014. The sign changes, so the solution lies between 2.555 and 2.565.

A21: Forming equations – method in steps

1. Define the letter, with units (“let the width be x cm”).
2. Write an expression for each quantity.
3. Form the equation from the fact given (total, area, perimeter, angle sum).
4. Solve.
5. Interpret: reject impossible values (negative lengths, non-whole numbers of people) and answer the question in words.

Reminder: the angles of a triangle are x°, (2x + 10)° and (3x − 40)°. Then x + 2x + 10 + 3x − 40 = 180, so 6x − 30 = 180 and x = 35. The angles are 35°, 80° and 65°. Check: they add to 180°.

A22: Inequalities – method in steps

Linear (both tiers): solve like an equation; reverse the sign if you multiply or divide by a negative. For a double inequality, do the same thing to all three parts.

Reminder: 7 − 3x ≤ 1 → −3x ≤ −6 → x ≥ 2. Number line: closed circle at 2, arrow to the right.

Quadratic (Higher tier only):

1. Solve the equation = 0 to find the critical values.
2. Sketch the U-shaped parabola through them.
3. “< 0” or “≤ 0”: the part below the axis, between the roots, one inequality.
4. “> 0” or “≥ 0”: the parts above the axis, outside the roots, two inequalities.

Reminder: x² − 4x − 5 < 0 → (x − 5)(x + 1) < 0 → −1 < x < 5.

Regions (Higher tier only): draw each boundary line (dashed for strict, solid for included), test a point not on the line, and identify the region that satisfies every inequality.

Must-know distinctions

• Exact vs approximate: factorising, completing the square and surd answers are exact; graph readings and iteration are approximate.
• Equation vs inequality: an equation has particular solutions; an inequality has a range of solutions.
• Between vs outside for quadratic inequalities: < 0 is between the roots; > 0 is outside.
• Open vs closed circle, and dashed vs solid line: strict boundaries are open or dashed.
• Elimination vs substitution: elimination for two linear equations; substitution whenever one equation is quadratic.

Quick self-test

1. Solve 6 − x = 2x − 9.
2. Solve 5(x + 3) = 2(x − 3).
3. Solve x² + x − 20 = 0.
4. Solve x² − 8x = 0.
5. Solve 3x + y = 11 and x − y = 1.
6. Solve 4 − 3x > 13.
7. Three consecutive integers add to 72. Find them.
8. List the integers n with −3 ≤ 2n − 1 < 7.
9. (Higher) Solve x² + 10x + 7 = 0 by completing the square. Give exact answers.
10. (Higher) Solve x² − 4x − 3 = 0, giving answers to 2 d.p.
11. (Higher) Solve x² − 2x − 8 ≤ 0. Write the answer in set notation.
12. (Higher) Given xₙ₊₁ = 2 + 1/xₙ and x₁ = 2, find x₂ and x₃.

Answers

1. 15 = 3x, so x = 5.
2. 5x + 15 = 2x − 6, so 3x = −21, x = −7.
3. (x + 5)(x − 4) = 0, so x = −5 or x = 4.
4. x(x − 8) = 0, so x = 0 or x = 8.
5. Add: 4x = 12, so x = 3, y = 2.
6. −3x > 9, so x < −3 (sign reversed).
7. n + (n + 1) + (n + 2) = 72, so 3n + 3 = 72 and n = 23: 23, 24, 25.
8. −2 ≤ 2n < 8, so −1 ≤ n < 4: −1, 0, 1, 2, 3.
9. (x + 5)² − 18 = 0, so x + 5 = ±√18 = ±3√2: x = −5 ± 3√2.
10. x = (4 ± √28)/2 = 2 ± √7: x = 4.65 or x = −0.65.
11. (x − 4)(x + 2) ≤ 0, between the roots: {x : −2 ≤ x ≤ 4}.
12. x₂ = 2 + 1/2 = 2.5; x₃ = 2 + 1/2.5 = 2.4.

Where marks are usually lost

• Dividing x² − 8x = 0 by x and giving only x = 8.
• Reversing the inequality sign when subtracting a negative term (only multiplying or dividing by a negative reverses it).
• Writing a quadratic inequality answer such as −2 > x > 4, which no number satisfies; “outside the roots” needs two separate inequalities.
• Pairing the wrong x with the wrong y in a linear/quadratic answer.
• Substituting −b incorrectly when b is negative: for b = −4, −b = +4.
• Rounding iterates to 2 d.p. part-way through, which changes later values.
• Using a closed circle or solid line for a strict inequality.
• Giving a context answer as “x = 5” without saying what 5 measures.
• Rejecting nothing when one root is a negative length.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.2.3 Solving equations and inequalities (A17 to A22), with the Appendix on mathematical formulae.