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Exam questions · Maths · Further Algebra

Inequalities and Regions

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

    Write down all the integers that satisfy \(2 < x \leq 6\).

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    Model answer

    The integers are 3, 4, 5, 6.

    Mark scheme

    • At least three correct and no more than one wrong — M1
    • 3, 4, 5, 6 — 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) How many points with integer coordinates are inside \(R\) or on its boundary? [2 marks]

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

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

    Mark scheme

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

    Solve \(5x - 3 \leq 17\).

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    Model answer

    \(5x \leq 20\), so \(x \leq 4\).

    Mark scheme

    • \(5x \leq 20\) — M1
    • \(x \leq 4\) — A1
  4. 4 Decide [2 marks]

    Here are two points: \((1, 3)\) and \((2, 6)\). For each point, say whether it satisfies the inequality \(y \leq 2x + 1\). You must show your working.

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    Model answer

    For \((1, 3)\): \(2 \times 1 + 1 = 3\) and \(3 \leq 3\), so it does. For \((2, 6)\): \(2 \times 2 + 1 = 5\) and \(6 \leq 5\) is false, so it does not.

    Mark scheme

    • \((1, 3)\) satisfies it, with \(2 \times 1 + 1 = 3\) — B1
    • \((2, 6)\) does not satisfy it, with \(2 \times 2 + 1 = 5\) — B1
  5. 5 Solve [3 marks]

    Solve \(x^2 + x - 12 > 0\).

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    Model answer

    \(x^2 + x - 12 = (x + 4)(x - 3)\), with roots \(-4\) and 3. The curve is above the \(x\)-axis outside the roots, so \(x < -4\) or \(x > 3\).

    Mark scheme

    • \((x + 4)(x - 3)\) or the roots \(-4\) and 3 — M1
    • A sketch or a clear statement of which regions are above the axis — M1
    • \(x < -4\) or \(x > 3\) — A1
  6. 6 Solve [3 marks]

    (a) Solve \(x^2 < 25\). [1 mark] (b) Solve \(x^2 - 3x - 10 \leq 0\). [2 marks]

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    Model answer

    (a) \(-5 < x < 5\). (b) \((x - 5)(x + 2) \leq 0\), so \(-2 \leq x \leq 5\).

    Mark scheme

    • (a) \(-5 < x < 5\) — B1
    • (b) Roots \(-2\) and 5 — M1
    • (b) \(-2 \leq x \leq 5\) — 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.