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Exam questions · Maths · Algebra

Solving Linear Equations and Inequalities

  • 7 exam questions
  • 21 marks
  • 10 quick checks
  1. 1 Solve [2 marks]

    Solve \(5x + 7 = 32\).

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    Model answer

    Subtract 7 from both sides to get \(5x = 25\), then divide by 5 to get \(x = 5\).

    Mark scheme

    • \(5x = 25\) or \(\dfrac{32 - 7}{5}\) — M1
    • \(x = 5\) — A1
  2. 2 Solve [3 marks]

    Solve \(9 - 2x = 3x - 11\).

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    Model answer

    Add \(2x\) to both sides to get \(9 = 5x - 11\). Add 11 to get \(20 = 5x\). So \(x = 4\). Check: \(9 - 8 = 1\) and \(12 - 11 = 1\).

    Mark scheme

    • \(9 = 5x - 11\) or \(20 - 2x = 3x\) — M1
    • \(5x = 20\) — M1
    • \(x = 4\) — A1
  3. 3 Solve [3 marks]

    Solve \(2(3x + 4) = 5(x + 2) + 3\).

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    Model answer

    Expand both sides: \(6x + 8 = 5x + 10 + 3\), which is \(6x + 8 = 5x + 13\). Subtract \(5x\): \(x + 8 = 13\). So \(x = 5\).

    Mark scheme

    • Expands correctly: \(6x + 8\) and \(5x + 10\) — M1
    • \(6x + 8 = 5x + 13\) or \(x + 8 = 13\) — M1
    • \(x = 5\) — A1
  4. 4 Solve [3 marks]

    Solve \(\dfrac{3x + 1}{2} = x + 3\).

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    Model answer

    Multiply both sides by 2 to get \(3x + 1 = 2(x + 3)\), which is \(3x + 1 = 2x + 6\). Subtract \(2x\) and 1: \(x = 5\).

    Mark scheme

    • \(3x + 1 = 2(x + 3)\) or equivalent — M1
    • \(3x + 1 = 2x + 6\) — M1
    • \(x = 5\) — A1
  5. 5 Work out [4 marks]

    Ben is \(x\) years old. His sister is 3 years older than Ben. Their mother is 4 times as old as Ben. The sum of their three ages is 69 years. Work out Ben's age and his mother's age.

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    Model answer

    Ben is \(x\), his sister is \(x + 3\) and his mother is \(4x\). So \(x + (x + 3) + 4x = 69\), which gives \(6x + 3 = 69\), so \(6x = 66\) and \(x = 11\). Ben is 11 and his mother is \(4 \times 11 = 44\).

    Mark scheme

    • \(x + 3\) and \(4x\) as expressions for the sister and mother — M1
    • \(x + (x + 3) + 4x = 69\) or equivalent — M1
    • Ben is 11 — A1
    • Mother is 44 — A1
  6. 6 Solve [3 marks]

    Solve \(2 - 3x \leq 14\).

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    Model answer

    Subtract 2 from both sides to get \(-3x \leq 12\). Divide both sides by \(-3\) and reverse the inequality sign, giving \(x \geq -4\).

    Mark scheme

    • \(-3x \leq 12\) — M1
    • Divides by \(-3\) and reverses the sign — M1
    • \(x \geq -4\) — A1
  7. 7 Solve [3 marks]

    (a) Solve \(4x + 3 > 15\). [2 marks] (b) Write down the smallest integer that satisfies \(4x + 3 > 15\). [1 mark]

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    Model answer

    (a) Subtract 3 to get \(4x > 12\), then divide by 4 to get \(x > 3\). (b) 3 is not included because the sign is \(>\), so the smallest integer is 4.

    Mark scheme

    • (a) \(4x > 12\) — M1
    • (a) \(x > 3\) — A1
    • (b) 4 — B1

Quick check

  1. 1

    Three consecutive integers add up to 72. What is the smallest of them?

    1. A24
    2. B23
    3. C25
    4. D21
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    B: 23

    Let the numbers be \(n\), \(n + 1\) and \(n + 2\). Then \(3n + 3 = 72\), so \(n = 23\).

  2. 2

    Which inequality means "at most 20"?

    1. A\(x > 20\)
    2. B\(x \geq 20\)
    3. C\(x < 20\)
    4. D\(x \leq 20\)
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    D: \(x \leq 20\)

    "At most 20" means 20 or less, which is \(x \leq 20\).

  3. 3

    Solve \(2x + 7 = 19\).

    1. A\(x = 13\)
    2. B\(x = 12\)
    3. C\(x = 6\)
    4. D\(x = 5\)
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    C: \(x = 6\)

    Subtract 7 to get \(2x = 12\), then divide by 2.

  4. 4

    Solve \(5x - 3 = 2x + 9\).

    1. A\(x = 6\)
    2. B\(x = 12\)
    3. C\(x = 2\)
    4. D\(x = 4\)
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    D: \(x = 4\)

    Subtract \(2x\) to get \(3x - 3 = 9\), add 3 to get \(3x = 12\), and divide by 3.

  5. 5

    Solve \(3(x + 2) = 18\).

    1. A\(x = 16\)
    2. B\(x = 4\)
    3. C\(x = 5\)
    4. D\(x = 8\)
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    B: \(x = 4\)

    Expand to get \(3x + 6 = 18\), so \(3x = 12\) and \(x = 4\).

  6. 6

    Solve \(\dfrac{x}{5} - 2 = 3\).

    1. A\(x = 5\)
    2. B\(x = 15\)
    3. C\(x = 1\)
    4. D\(x = 25\)
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    D: \(x = 25\)

    Add 2 to get \(\dfrac{x}{5} = 5\), then multiply both sides by 5.

  7. 7

    Solve \(-2x > 6\).

    1. A\(x < 3\)
    2. B\(x > 3\)
    3. C\(x > -3\)
    4. D\(x < -3\)
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    D: \(x < -3\)

    Dividing by \(-2\) reverses the inequality sign, so \(x < -3\).

  8. 8

    Which list shows all the integers that satisfy \(-2 \leq x < 2\)?

    1. A\(-2, -1, 0, 1\)
    2. B\(-2, -1, 0, 1, 2\)
    3. C\(-1, 0, 1\)
    4. D\(-1, 0, 1, 2\)
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    A: \(-2, -1, 0, 1\)

    \(-2\) is included because of \(\leq\), but 2 is not included because of \(<\).

  9. 9

    Which inequality is shown by an open circle at 5 with an arrow pointing to the right?

    1. A\(x > 5\)
    2. B\(x < 5\)
    3. C\(x \geq 5\)
    4. D\(x \leq 5\)
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    A: \(x > 5\)

    An open circle means 5 is not included, and the arrow to the right means larger numbers, so \(x > 5\).

  10. 10

    I think of a number, double it and subtract 3. The answer is 11. Which equation represents this?

    1. A\(x - 6 = 11\)
    2. B\(2x - 3 = 11\)
    3. C\(2(x - 3) = 11\)
    4. D\(2x + 3 = 11\)
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    B: \(2x - 3 = 11\)

    Doubling gives \(2x\) and subtracting 3 gives \(2x - 3\), which equals 11.