Edexcel GCSE Maths · A17–A22

Solving equations and inequalities

50 maths questions with worked answers with answers for A17–A22 Solving equations and inequalities in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

All 50 questions and answers

Question 1 · Linear equations · A17
Solve 5x + 3 = 28.
Answer:
5x = 25.
x = 5.
Question 2 · Linear equations · A17
Solve 4(x − 2) = 20.
Answer:
x − 2 = 5.
x = 7.
Question 3 · Linear equations · A17
Solve 7x − 4 = 3x + 16.
Answer:
4x = 20.
x = 5.
Question 4 · Linear equations · A17
Solve x3 + 2 = 9.
Answer:
x3 = 7.
x = 21.
Question 5 · Linear equations · A17
Solve 2x + 15 = 3.
Answer:
2x + 1 = 15.
2x = 14, so x = 7.
Question 6 · Linear equations · A17 · Higher only
Solve x + 12 + x − 23 = 4.
Answer:
Multiply by 6: 3(x + 1) + 2(x − 2) = 24.
5x − 1 = 24, so x = 5.
Question 7 · Forming equations · A17
The angles of a triangle are x, 2x and 3x. Find x.
Answer:
x + 2x + 3x = 180.
6x = 180, so x = 30°.
Question 8 · Forming equations · A17
A rectangle has length 2x + 3 and width x. Its perimeter is 36. Find x.
Answer:
2(2x + 3) + 2x = 36, so 6x + 6 = 36.
x = 5.
Question 9 · Quadratics · A18
Solve x² + 5x + 6 = 0.
Answer:
(x + 2)(x + 3) = 0.
x = −2 or x = −3.
Question 10 · Quadratics · A18
Solve x² − 4x = 0.
Answer:
x(x − 4) = 0.
x = 0 or x = 4.
Question 11 · Quadratics · A18
Solve x² − 7x + 10 = 0.
Answer:
(x − 2)(x − 5) = 0.
x = 2 or x = 5.
Question 12 · Quadratics · A18 · Higher only
Solve 2x² + x − 6 = 0.
Answer:
(2x − 3)(x + 2) = 0.
x = 1.5 or x = −2.
Question 13 · Quadratic formula · A18 · Higher only
Solve x² + 4x − 3 = 0, giving answers to 2 d.p.
Answer:
x = −4 ± √(16 + 12)2 = −4 ± √282.
x = 0.65 or x = −4.65.
Question 14 · Quadratic formula · A18 · Higher only
Solve 3x² − 5x − 1 = 0 to 2 d.p.
Answer:
x = 5 ± √(25 + 12)6 = 5 ± √376.
x = 1.85 or x = −0.18.
Question 15 · Completing the square · A18 · Higher only
Solve x² + 6x − 4 = 0 by completing the square. Leave your answer in surd form.
Answer:
(x + 3)² − 13 = 0, so (x + 3)² = 13.
x = −3 ± √13.
Question 16 · Discriminant · A18 · Higher only
How many real solutions does x² + 2x + 5 = 0 have?
Answer:
b² − 4ac = 4 − 20 = −16.
It is negative, so there are no real solutions.
Question 17 · Forming quadratics · A18 · Higher only
A rectangle has width x and length x + 3. Its area is 40. Find x.
Answer:
x(x + 3) = 40, so x² + 3x − 40 = 0.
(x + 8)(x − 5) = 0; length cannot be negative, so x = 5.
Question 18 · Simultaneous equations · A19
Solve x + y = 10 and x − y = 4.
Answer:
Add: 2x = 14, so x = 7.
y = 3.
Question 19 · Simultaneous equations · A19
Solve 3x + 2y = 16 and x + 2y = 8.
Answer:
Subtract: 2x = 8, so x = 4.
4 + 2y = 8, so y = 2.
Question 20 · Simultaneous equations · A19
Solve 2x + 3y = 13 and 5x − y = 7.
Answer:
Multiply the second by 3: 15x − 3y = 21. Add: 17x = 34, so x = 2.
10 − y = 7, so y = 3.
Question 21 · Simultaneous equations · A19
2 apples and 3 pears cost £1.30. 2 apples and 1 pear cost 70p. Find the cost of a pear.
Answer:
Subtract: 2 pears cost 60p.
A pear costs 30p.
Question 22 · Simultaneous equations · A19 · Higher only
Solve y = x + 1 and x² + y² = 13.
Answer:
x² + (x + 1)² = 13, so 2x² + 2x − 12 = 0, x² + x − 6 = 0.
(x + 3)(x − 2) = 0: x = 2, y = 3 or x = −3, y = −2.
Question 23 · Simultaneous equations · A19 · Higher only
Solve y = x² and y = 2x + 3.
Answer:
x² = 2x + 3, so x² − 2x − 3 = 0, (x − 3)(x + 1) = 0.
x = 3, y = 9 or x = −1, y = 1.
Question 24 · Iteration · A20 · Higher only
Use xₙ₊₁ = √(2xₙ + 5) with x₀ = 3 to find x₁ and x₂ to 3 d.p.
Answer:
x₁ = √11 = 3.317.
x₂ = √(2 × 3.3166 + 5) = √11.633 = 3.411.
Question 25 · Iteration · A20 · Higher only
Show that x³ − 4x − 1 = 0 has a root between 2 and 3.
Answer:
When x = 2: 8 − 8 − 1 = −1. When x = 3: 27 − 12 − 1 = 14.
The sign changes, so there is a root between 2 and 3.
Question 26 · Iteration · A20 · Higher only
Show that x² − 3x − 1 = 0 can be rearranged to x = 3 + 1x.
Answer:
x² = 3x + 1.
Divide by x: x = 3 + 1x.
Question 27 · Forming equations · A21
I think of a number, double it and add 9. The result is 25. Find the number.
Answer:
2n + 9 = 25.
2n = 16, so n = 8.
Question 28 · Forming equations · A21
Tom is 4 years older than Sara. Their ages add up to 30. How old is Sara?
Answer:
s + (s + 4) = 30.
2s = 26, so Sara is 13.
Question 29 · Interpreting · A21
A solution to a quadratic for the length of a side is x = −6. Why is it rejected?
Answer:
A length cannot be negative.
Only the positive solution makes sense.
Question 30 · Inequalities · A22
Solve 3x − 5 > 10.
Answer:
3x > 15.
x > 5.
Question 31 · Inequalities · A22
Solve 2(x + 3) ≤ 14.
Answer:
x + 3 ≤ 7.
x ≤ 4.
Question 32 · Inequalities · A22
List the integers n such that −2 < n ≤ 3.
Answer:
−2 is not included; 3 is included.
−1, 0, 1, 2, 3.
Question 33 · Inequalities · A22
Solve −3 ≤ 2x + 1 < 7.
Answer:
Subtract 1: −4 ≤ 2x < 6.
−2 ≤ x < 3.
Question 34 · Inequalities · A22
Solve 5 − 2x > 11.
Answer:
−2x > 6.
Dividing by a negative reverses the sign: x < −3.
Question 35 · Number lines · A22
On a number line, what does an open circle at 4 mean?
Answer:
4 is not included.
It shows < or >.
Question 36 · Quadratic inequalities · A22 · Higher only
Solve x² − 9 < 0.
Answer:
The roots are x = ±3; the graph is below the axis between them.
−3 < x < 3.
Question 37 · Quadratic inequalities · A22 · Higher only
Solve x² − 5x + 4 ≥ 0.
Answer:
(x − 1)(x − 4) ≥ 0. The graph is above the axis outside the roots.
x ≤ 1 or x ≥ 4.
Question 38 · Regions · A22 · Higher only
Which side of y = 2x + 1 is the region y > 2x + 1?
Answer:
Test (0, 0): 0 > 1 is false.
The region above the line, not containing the origin.
Question 39 · Regions · A22 · Higher only
When shading a region, when is the boundary line dashed?
Answer:
For strict inequalities.
Use a dashed line for < or >, and a solid line for ≤ or ≥.
Question 40 · Linear equations · A17
Solve 9 − 2x = 3.
Answer:
−2x = −6.
x = 3.
Question 41 · Linear equations · A17
Solve 3(2x + 1) = 2(x + 9).
Answer:
6x + 3 = 2x + 18.
4x = 15, so x = 3.75.
Question 42 · Simultaneous equations · A19
Solve y = 2x − 1 and y = x + 3.
Answer:
2x − 1 = x + 3.
x = 4, y = 7.
Question 43 · Quadratics · A18
Solve x² = 64.
Answer:
Take square roots, remembering both signs.
x = 8 or x = −8.
Question 44 · Quadratics · A18 · Higher only
Solve (x − 2)² = 25.
Answer:
x − 2 = ±5.
x = 7 or x = −3.
Question 45 · Inequalities · A22
Write down the largest integer x that satisfies 4x < 23.
Answer:
x < 5.75.
x = 5.
Question 46 · Forming equations · A21 · Higher only
A square has side x + 2 and area 49. Find x.
Answer:
(x + 2)² = 49, so x + 2 = 7 (side must be positive).
x = 5.
Question 47 · Linear equations · A17
Solve 0.5x + 3 = 8.
Answer:
0.5x = 5.
x = 10.
Question 48 · Simultaneous equations · A19 · Higher only
Solve 4x + 3y = 1 and 3x − 2y = 5.
Answer:
×2 and ×3: 8x + 6y = 2 and 9x − 6y = 15. Add: 17x = 17, x = 1.
4 + 3y = 1, so y = −1.
Question 49 · Iteration · A20 · Higher only
Why is iteration used to solve some equations?
Answer:
Some equations cannot be solved exactly by algebra.
Iteration gives an approximate solution to any accuracy required.
Question 50 · Inequalities · A22
Solve 4x + 1 ≥ 2x − 9.
Answer:
2x ≥ −10.
x ≥ −5.

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