AQA GCSE Maths · 3.2.3

Solving equations and inequalities

50 maths questions with worked answers with answers for 3.2.3 Solving equations and inequalities in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Linear equations · A17
Solve x + 8 = 19.
Answer:
Subtract 8 from both sides: x = 19 − 8.
x = 11.
Question 2 · Linear equations · A17
Solve 5x = 35.
Answer:
Divide both sides by 5: x = 35 ÷ 5.
x = 7.
Question 3 · Linear equations · A17
Solve x4 = 6.
Answer:
Multiply both sides by 4: x = 6 × 4.
x = 24.
Question 4 · Linear equations · A17
Solve 3x + 5 = 20.
Answer:
Subtract 5: 3x = 15.
Divide by 3: x = 5.
Question 5 · Linear equations · A17
Solve 2x − 7 = 11.
Answer:
Add 7: 2x = 18.
Divide by 2: x = 9.
Question 6 · Linear equations · A17
Solve 4(x + 2) = 28.
Answer:
Divide by 4: x + 2 = 7.
Subtract 2: x = 5.
Question 7 · Linear equations · A17
Solve 5x − 3 = 2x + 12.
Answer:
Subtract 2x: 3x − 3 = 12.
Add 3 and divide by 3: x = 5.
Question 8 · Linear equations · A17
Solve 3(2x − 1) = 21.
Answer:
Divide by 3: 2x − 1 = 7.
Add 1, then divide by 2: x = 4.
Question 9 · Linear equations · A17
Solve x + 53 = 7.
Answer:
Multiply by 3: x + 5 = 21.
Subtract 5: x = 16.
Question 10 · Linear equations · A17
Solve 7 − 2x = −5.
Answer:
Subtract 7: −2x = −12.
Divide by −2: x = 6.
Question 11 · Linear equations · A17
Solve 0.5x + 4 = 9.
Answer:
Subtract 4: 0.5x = 5.
Divide by 0.5: x = 10.
Question 12 · Linear equations · A17
Solve 2(x − 3) = x + 4.
Answer:
Expand: 2x − 6 = x + 4.
Subtract x and add 6: x = 10.
Question 13 · Linear equations · A17
Solve 4 − 3x = 2x + 14.
Answer:
Add 3x and subtract 14: −10 = 5x.
Divide by 5: x = −2.
Question 14 · Linear equations · A17
Solve x3 + 2 = 5.
Answer:
Subtract 2: x3 = 3.
Multiply by 3: x = 9.
Question 15 · Linear equations · A17
A number doubled then reduced by 4 is 18. Find the number.
Answer:
Let the number be x: 2x − 4 = 18.
Add 4 and divide by 2: x = 11.
Question 16 · Quadratics · A18
Solve x² − 9 = 0.
Answer:
Add 9: x² = 9.
Both 3² and (−3)² equal 9.
x = 3 or x = −3.
Question 17 · Quadratics · A18
Solve x² − 5x + 6 = 0.
Answer:
Factorise: (x − 2)(x − 3) = 0.
Set each factor to zero: x = 2 or x = 3.
Question 18 · Quadratics · A18
Solve x² + x − 12 = 0.
Answer:
Factorise: (x + 4)(x − 3) = 0.
Set each factor to zero: x = −4 or x = 3.
Question 19 · Quadratics · A18
Solve x² = 25.
Answer:
The numbers 5 and −5 both square to 25.
Therefore x = 5 or x = −5.
Question 20 · Quadratics · A18 · Higher only
Solve 2x² − 7x + 3 = 0.
Answer:
Factorise: (2x − 1)(x − 3) = 0.
Set each factor to zero: x = 12 or x = 3.
Question 21 · Quadratics · A18 · Higher only
Solve x² + 4x + 1 = 0 exactly.
Answer:
Complete the square: (x + 2)² − 3 = 0.
(x + 2)² = 3, so x + 2 = ±√3.
Therefore x = −2 ± √3.
Question 22 · Quadratics · A18 · Higher only
Solve x² − 2x − 3 = 0.
Answer:
Factorise: (x − 3)(x + 1) = 0.
Set each factor to zero: x = 3 or x = −1.
Question 23 · Quadratics · A18 · Higher only
Solve (x − 2)² = 7.
Answer:
Take both square roots: x − 2 = ±√7.
Add 2: x = 2 ± √7.
Question 24 · Quadratics · A18 · Higher only
Solve 3x² = 12.
Answer:
Divide by 3: x² = 4.
Taking both square roots gives x = 2 or x = −2.
Question 25 · Simultaneous equations · A19
Solve x + y = 10 and x − y = 2.
Answer:
Add the equations: 2x = 12, so x = 6.
Substitute into x + y = 10: y = 4.
Question 26 · Simultaneous equations · A19
Solve x + y = 9 and 2x + y = 14.
Answer:
Subtract the first equation from the second: x = 5.
Substitute into x + y = 9: y = 4.
Question 27 · Simultaneous equations · A19
Solve 2x + y = 13 and x + y = 8.
Answer:
Subtract x + y = 8 from 2x + y = 13: x = 5.
Then y = 8 − 5 = 3.
Question 28 · Simultaneous equations · A19
Solve 3x + 2y = 16 and x + y = 6.
Answer:
Double x + y = 6 to get 2x + 2y = 12.
Subtract from 3x + 2y = 16: x = 4.
Then y = 6 − 4 = 2.
Question 29 · Simultaneous equations · A19
Two pens and one book cost £7. One pen and one book cost £5. Find each price.
Answer:
Let p be the pen price and b the book price.
2p + b = 7 and p + b = 5; subtract to get p = 2.
Then b = 3: pen £2, book £3.
Question 30 · Simultaneous equations · A19 · Higher only
Solve y = x² and y = 4.
Answer:
Set x² = 4, giving x = 2 or x = −2.
In both cases y = 4.
Solutions: (2, 4) and (−2, 4).
Question 31 · Simultaneous equations · A19 · Higher only
Solve y = x² and y = x + 2.
Answer:
Set x² = x + 2 and rearrange: x² − x − 2 = 0.
Factorise: (x − 2)(x + 1) = 0.
For x = 2, y = 4; for x = −1, y = 1.
Solutions: (2, 4) and (−1, 1).
Question 32 · Inequalities · A22
Solve x + 4 < 9.
Answer:
Subtract 4 from both sides.
x < 9 − 4, so x < 5.
Question 33 · Inequalities · A22
Solve 3x ≥ 12.
Answer:
Divide both sides by positive 3.
x ≥ 12 ÷ 3, so x ≥ 4.
Question 34 · Inequalities · A22
Solve 2x − 5 ≤ 7.
Answer:
Add 5: 2x ≤ 12.
Divide by 2: x ≤ 6.
Question 35 · Inequalities · A22
Solve −2x > 8.
Answer:
Divide by −2 and reverse the inequality sign.
x < −4.
Question 36 · Inequalities · A22
List the integers satisfying −2 < x ≤ 2.
Answer:
The lower bound excludes −2 and the upper bound includes 2.
The integers are −1, 0, 1, 2.
Question 37 · Inequalities · A22
Solve 3 < x + 2 ≤ 8.
Answer:
Subtract 2 from all three parts.
3 − 2 < x ≤ 8 − 2, so 1 < x ≤ 6.
Question 38 · Inequalities · A22 · Higher only
Solve x² − 9 < 0.
Answer:
Rearrange to x² < 9.
The values whose squares are less than 9 lie between −3 and 3.
−3 < x < 3.
Question 39 · Inequalities · A22 · Higher only
Solve x² − 4 ≥ 0.
Answer:
Rearrange to x² ≥ 4.
The distance of x from 0 must be at least 2.
x ≤ −2 or x ≥ 2.
Question 40 · Inequalities · A22 · Higher only
Does (2, 3) satisfy y > 2x − 2?
Answer:
Substitute x = 2 and y = 3.
3 > 2 × 2 − 2 gives 3 > 2, which is true.
Question 41 · Iteration · A20 · Higher only
An iteration is xₙ₊₁ = xₙ + 6xₙ2. If x₀ = 2, find x₁.
Answer:
Substitute x₀ = 2: 6 ÷ 2 = 3.
x₁ = (2 + 3) ÷ 2 = 2.5.
Question 42 · Iteration · A20 · Higher only
An iteration is xₙ₊₁ = xₙ + 10xₙ2. If x₀ = 2, find x₁.
Answer:
Substitute x₀ = 2: 10 ÷ 2 = 5.
x₁ = (2 + 5) ÷ 2 = 3.5.
Question 43 · Iteration · A20 · Higher only
If xₙ₊₁ = xₙ + 52 and x₀ = 1, find x₂.
Answer:
First x₁ = (1 + 5) ÷ 2 = 3.
Then x₂ = (3 + 5) ÷ 2 = 4.
Question 44 · Modelling · A21
Three identical tickets cost £27. Write and solve an equation for price p.
Answer:
Let p be the price of one ticket: 3p = 27.
Divide by 3: p = £9.
Question 45 · Modelling · A21
A rectangle has width x and length x + 3. Its perimeter is 22. Find x.
Answer:
Perimeter: 2x + 2(x + 3) = 22.
Expand and simplify: 4x + 6 = 22.
4x = 16, so x = 4.
Question 46 · Modelling · A21
A number plus three times itself is 28. Find the number.
Answer:
Let x be the number: x + 3x = 28.
4x = 28, so x = 7.
Question 47 · Modelling · A21
A taxi costs £4 plus £3 per mile. The fare is £19. How many miles?
Answer:
Let m be the miles: 4 + 3m = 19.
3m = 15, so m = 5 miles.
Question 48 · Error check · A17
A student solves 2x + 5 = 13 as x = 9. Correct the solution.
Answer:
Subtract 5: 2x = 8.
Divide by 2: x = 4.
Question 49 · Error check · A18
Why is x = 5 not the only solution of x² = 25?
Answer:
Both 5² = 25 and (−5)² = 25.
The solutions are x = 5 and x = −5.
Question 50 · Error check · A22
A student divides −3x < 12 by −3 and writes x < −4. Correct it.
Answer:
Divide by −3 and reverse the inequality.
x > −4.

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Solving equations and inequalities

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