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Exam questions · Maths · Further Algebra

Inequalities and Regions

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Write down [2 marks]

    Write down all the integer values of \(n\) that satisfy \(-3 \leq n < 2\).

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    Model answer

    The integers are \(-3, -2, -1, 0, 1\).

    Mark scheme

    • At least four correct and no more than one wrong — M1
    • \(-3, -2, -1, 0, 1\) — A1
  2. 2 Write down [5 marks]

    The diagram shows a shaded region \(R\). (a) Write down the three inequalities that define the region \(R\). (3 marks) (b) Write down the number of points with integer coordinates inside or on the boundary of \(R\). (2 marks)

    A shaded triangular region R bounded by a vertical line, a horizontal line and a diagonal line.
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    Model answer

    (a) \(x \geq 2\), \(y \geq 1\) and \(x + y \leq 7\). (b) For \(x = 2, 3, 4, 5, 6\) there are \(5, 4, 3, 2, 1\) points, so the total is \(5 + 4 + 3 + 2 + 1 = 15\).

    Mark scheme

    • (a) \(x \geq 2\) — B1
    • (a) \(y \geq 1\) — B1
    • (a) \(x + y \leq 7\) — B1
    • (b) Counts by columns, such as 5, 4, 3, 2, 1 — M1
    • (b) 15 — A1
  3. 3 Solve [2 marks]

    Solve the inequality \(3x - 2 > 10\).

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    Model answer

    Add 2: \(3x > 12\). Divide by 3: \(x > 4\).

    Mark scheme

    • \(3x > 12\) — M1
    • \(x > 4\) — A1
  4. 4 Solve [3 marks]

    Solve the inequality \(x^2 - 2x - 8 < 0\).

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    Model answer

    \(x^2 - 2x - 8 = (x - 4)(x + 2)\), with roots \(-2\) and 4. The curve is below the \(x\)-axis between the roots, so \(-2 < x < 4\).

    Mark scheme

    • \((x - 4)(x + 2)\) or the roots \(-2\) and 4 — M1
    • A sketch or a statement that the quadratic is negative between the roots — M1
    • \(-2 < x < 4\) — A1
  5. 5 Solve [4 marks]

    (a) Solve \(x^2 \geq 16\). (2 marks) (b) Solve \(x^2 - 5x + 6 > 0\). (2 marks)

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    Model answer

    (a) \(x^2 = 16\) gives \(x = \pm 4\), and the curve is above the axis outside the roots, so \(x \leq -4\) or \(x \geq 4\). (b) \((x - 2)(x - 3) > 0\), so \(x < 2\) or \(x > 3\).

    Mark scheme

    • (a) Roots \(\pm 4\) — M1
    • (a) \(x \leq -4\) or \(x \geq 4\) — A1
    • (b) Roots 2 and 3 — M1
    • (b) \(x < 2\) or \(x > 3\) — A1
  6. 6 Work out [3 marks]

    \(n\) is an integer. \(-2 < 2n + 1 \leq 9\). Write down all the possible values of \(n\).

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    Model answer

    Subtract 1: \(-3 < 2n \leq 8\). Divide by 2: \(-1.5 < n \leq 4\). The integers are \(-1, 0, 1, 2, 3, 4\).

    Mark scheme

    • \(-3 < 2n \leq 8\) or \(-1.5 < n \leq 4\) — M1
    • At least four correct integers — M1
    • \(-1, 0, 1, 2, 3, 4\) — A1

Quick check

  1. 1

    What does a dashed boundary line mean on a graph of an inequality?

    1. AThe line is included
    2. BThe inequality has no solutions
    3. CThe region is below the line
    4. DThe line itself is not included
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    D: The line itself is not included

    A dashed line goes with \(<\) or \(>\).

  2. 2

    Which inequality describes the region to the right of the line \(x = 1\), including the line?

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

    To the right means larger \(x\), and the line is included.

  3. 3

    Which inequality describes the region on or below the line \(y = 2x\)?

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

    Below the line means smaller \(y\), and the line is included.

  4. 4

    Which integers satisfy \(-2 < x \leq 3\)?

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

    \(-2\) is not included but 3 is.

  5. 5

    The origin is tested in \(x + y \leq 6\). What does this show?

    1. AThe origin is not in the region
    2. BThe line is dashed
    3. CThe inequality has no solutions
    4. DThe origin is in the region, so shade that side
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    D: The origin is in the region, so shade that side

    \(0 + 0 \leq 6\) is true.

  6. 6

    How many points with whole-number coordinates satisfy \(x \geq 1\), \(y \geq 1\) and \(x + y \leq 6\)?

    1. A\(10\)
    2. B\(21\)
    3. C\(15\)
    4. D\(25\)
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    C: \(15\)

    The columns \(x = 1, 2, 3, 4, 5\) have \(5, 4, 3, 2, 1\) points.

  7. 7

    Which kind of boundary line goes with the inequality \(y > 3\)?

    1. AA solid line
    2. BA dashed line
    3. CA curved line
    4. DNo line is drawn
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    B: A dashed line

    Strict inequalities do not include the boundary.

  8. 8

    Solve \(x^2 - x - 6 < 0\).

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

    The roots are \(-2\) and 3, and the curve is below the axis between them.

  9. 9

    Solve \(x^2 > 9\).

    1. A\(-3 < x < 3\)
    2. B\(x > 3\)
    3. C\(x > 9\)
    4. D\(x < -3\) or \(x > 3\)
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    D: \(x < -3\) or \(x > 3\)

    The curve is above the axis outside the roots \(-3\) and 3.