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Exam questions · Maths · Further Algebra

Inequalities and Regions

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

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

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    Model answer

    Dividing by 2 gives \(0.5 \leq n < 4.5\), so \(n = 1, 2, 3, 4\).

    Mark scheme

    • \(0.5 \leq n < 4.5\) or three correct values — M1
    • 1, 2, 3, 4 — A1
  2. 2 Write down [5 marks]

    The diagram shows a shaded region \(R\). (a) Write down the three inequalities that define \(R\). [3 marks] (b) Show that the point \((4, 5)\) is inside \(R\) or on its boundary. [2 marks]

    A shaded triangular region R bounded by a horizontal line, a steep line through the origin and a falling diagonal line.
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    Model answer

    (a) \(y \geq 2\), \(y \leq 2x\) and \(x + y \leq 9\). (b) \(5 \geq 2\), \(5 \leq 2 \times 4 = 8\) and \(4 + 5 = 9 \leq 9\), so the point satisfies all three inequalities, and is on the boundary \(x + y = 9\).

    Mark scheme

    • (a) \(y \geq 2\) — B1
    • (a) \(y \leq 2x\) — B1
    • (a) \(x + y \leq 9\) — B1
    • (b) Substitutes \((4, 5)\) into at least two inequalities — M1
    • (b) All three satisfied, with a conclusion — A1
  3. 3 Solve [3 marks]

    Solve \(-2 < 3x + 1 \leq 10\).

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    Model answer

    Subtract 1: \(-3 < 3x \leq 9\). Divide by 3: \(-1 < x \leq 3\).

    Mark scheme

    • \(-3 < 3x \leq 9\) — M1
    • \(-1 < x\) — A1
    • \(x \leq 3\) — A1
  4. 4 Write down [2 marks]

    A region is above the line \(y = 2\), below the line \(y = 5\) and to the right of the line \(x = 1\), and includes all three lines. Write down the three inequalities that define the region.

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    Model answer

    The inequalities are \(y \geq 2\), \(y \leq 5\) and \(x \geq 1\).

    Mark scheme

    • Two correct inequalities — M1
    • \(y \geq 2\), \(y \leq 5\) and \(x \geq 1\) — A1
  5. 5 Solve [3 marks]

    Solve \(x^2 - 7x + 10 < 0\).

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    Model answer

    \((x - 2)(x - 5) < 0\), with roots 2 and 5. The curve is below the axis between the roots, so \(2 < x < 5\).

    Mark scheme

    • \((x - 2)(x - 5)\) — M1
    • A sketch or a clear statement that the curve is below the axis between the roots — M1
    • \(2 < x < 5\) — A1
  6. 6 Solve [3 marks]

    Solve \(x^2 - x - 12 \leq 0\).

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    Model answer

    \((x - 4)(x + 3) \leq 0\), with roots \(-3\) and 4. The curve is below or on the axis between the roots, so \(-3 \leq x \leq 4\).

    Mark scheme

    • \((x - 4)(x + 3)\) — M1
    • A sketch or a clear statement that the curve is below the axis between the roots — M1
    • \(-3 \leq x \leq 4\) — 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.