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Maths · Equations and graphs

Representing inequalities graphically

Shade the region that satisfies one or more inequalities on a graph, use solid and dashed boundaries, and write inequalities for a given region.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    What does \(x > 3\) mean?

    Show answerHide answer

    \(x\) is greater than 3

  2. 2

    What does \(y \le 2\) mean?

    Show answerHide answer

    \(y\) is less than or equal to 2

  3. 3

    What is the line \(x = 4\)?

    Show answerHide answer

    A vertical line through 4 on the x-axis

  4. 4

    What is the line \(y = 2\)?

    Show answerHide answer

    A horizontal line through 2 on the y-axis

  5. 5

    Draw the line \(y = x\): what is its gradient?

    Show answerHide answer

    \(1\)

Learning Objectives

  1. 1Draw a boundary line for an inequality.
  2. 2Choose a solid or dashed line.
  3. 3Shade the correct region.
  4. 4Write inequalities for a region and list integer points.

INEQUALITY REGIONS

An inequality describes a region. Draw the boundary line, then shade the side where the inequality is true.

Solid line for \(\le\) or \(\ge\). Dashed line for \(<\) or \(>\).

Solid or Dashed?

The boundary is part of the region only for \(\le\) and \(\ge\).

  • \(y \ge 2\)

    Boundary line: Solid. Points on the line: Included

  • \(y > 2\)

    Boundary line: Dashed. Points on the line: Not included

  • \(x + y \le 6\)

    Boundary line: Solid. Points on the line: Included

  • \(y < x\)

    Boundary line: Dashed. Points on the line: Not included

Test a Point

Show which side of the line \(x + y = 6\) satisfies \(x + y \le 6\).

Show the solutionHide the solution
  1. 1 Test the origin \(0 + 0 = 0\)
  2. 2 Compare \(0 \le 6\) is true
  3. 3 So shade The side of the line containing the origin

AnswerShade the region below the line \(x + y = 6\), including the line.

Integer Points in a Region

List the integer points in \(x \ge 1\), \(y \ge 1\), \(x + y \le 4\).

Show the solutionHide the solution
  1. 1 Try \(x = 1\) \(y = 1, 2, 3\)
  2. 2 Try \(x = 2\) \(y = 1, 2\)
  3. 3 Try \(x = 3\) \(y = 1\)

Answer\((1,1), (1,2), (1,3), (2,1), (2,2), (3,1)\)

Writing Inequalities

A region is bounded by \(y = 1\), \(x = 4\) and \(y = x\), and lies above \(y = 1\), to the left of \(x = 4\) and below \(y = x\). Write the inequalities.

Show the solutionHide the solution
  1. 1 Above \(y = 1\) \(y \ge 1\)
  2. 2 Left of \(x = 4\) \(x \le 4\)
  3. 3 Below \(y = x\) \(y \le x\)

Answer\(y \ge 1\), \(x \le 4\), \(y \le x\)

Tips

Keep it clear.

  • Label R

    Mark the required region clearly.

  • Check with a point

    Test one point in the region in every inequality.

  • Boundary lines

    Draw all lines even if only part of them bounds the region.

  • Read the instruction

    Some questions ask you to shade the region that is NOT wanted.

Sketch the Region

Draw the region satisfying \(x \ge 2\), \(y \ge 1\) and \(x + y \le 6\). List the integer points that lie in it.

1. Draw each boundary line.

2. Count integer points column by column.

A good answer shows: The region is a triangle with corners \((2, 1)\), \((5, 1)\) and \((2, 4)\). Integer points: \((2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(4,1),(4,2),(5,1)\).

Can I...?

  1. 1Draw x = a and y = b lines.
  2. 2Draw sloping boundary lines.
  3. 3Use solid and dashed lines.
  4. 4Test a point.
  5. 5Shade the correct region.
  6. 6Write inequalities for a region.
  7. 7List integer points.
  8. 8Label R clearly.

Summary & Exam Focus

  • \(\le\) and \(\ge\): solid line. \(<\) and \(>\): dashed.
  • Test a point to decide which side to shade.
  • The region satisfies all inequalities at once.
  • Integer points on a solid boundary are included.

Exam focus

The diagram shows a region R. Write down the three inequalities that define R. (3 marks) (3 marks)

Name each boundary line first, then decide which side is R.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Inequality
A statement using \(<\), \(>\), \(\le\) or \(\ge\).
Boundary
The line separating the region from the rest.
Region
The set of points that satisfy the inequalities.
Integer
A whole number.
Test point
A point used to decide which side to shade.
Solid line
Boundary included.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Write down 3 marks

    The region R is shown shaded on the grid. Write down the three inequalities that define R.

    A shaded triangle R bounded by y equals 1, x equals 4 and y equals x.
    Show answerHide answer

    Model answer

    \(y \ge 1\), \(x \le 4\), \(y \le x\)

    Mark scheme

    • One correct inequality — B1
    • Two correct — B1
    • All three correct — B1
  2. Question 2 Write down 3 marks

    \(x\) and \(y\) are integers with \(x \ge 1\), \(y \ge 1\) and \(x + y \le 4\). Write down all the possible pairs \((x, y)\).

    Show answerHide answer

    Model answer

    \((1,1), (1,2), (1,3), (2,1), (2,2), (3,1)\)

    Mark scheme

    • At least 3 correct pairs and none wrong — M1
    • 5 correct — A1
    • All 6 correct — A1
  3. Question 3 Write down 3 marks

    A region on a grid is bounded by the lines \(x = 1\), \(y = 1\) and \(x + y = 6\), and lies above \(y = 1\), to the right of \(x = 1\) and below \(x + y = 6\). Write down the three inequalities that define the region.

    Show answerHide answer

    Model answer

    \(x \ge 1\), \(y \ge 1\), \(x + y \le 6\)

    Mark scheme

    • One correct — B1
    • Two correct — B1
    • All three — B1
  4. Question 4 Explain 2 marks

    Explain when you would draw a dashed line rather than a solid line for the boundary of an inequality.

    Show answerHide answer

    Model answer

    Use a dashed line for \(<\) or \(>\), because points on the line are not included in the region. Use a solid line for \(\le\) or \(\ge\).

    Mark scheme

    • Dashed for \(<\) or \(>\) — M1
    • Points on the line not included — C1
  5. Question 5 Decide 2 marks

    Is the point \((2, 3)\) in the region \(y > 2x - 1\)? Give a reason.

    Show answerHide answer

    Model answer

    No. When \(x = 2\), \(2x - 1 = 3\), and \(3 > 3\) is false, so the point is on the boundary, which is not included.

    Mark scheme

    • Substitutes — M1
    • No, with reason — A1
  6. Question 6 Shade 3 marks

    On a grid, show by shading the region defined by \(x \ge 2\), \(y < 3\) and \(y \ge 0\). Describe the shape you have shaded.

    Show answerHide answer

    Model answer

    A rectangular strip: the region to the right of the line \(x = 2\) (solid), below the dashed line \(y = 3\) and above the x-axis (solid line \(y = 0\)).

    Mark scheme

    • \(x = 2\) solid — B1
    • \(y = 3\) dashed — B1
    • Correct region — B1

Quick check

  1. A dashed line is used for...

    1. A\(\le\) and \(\ge\)
    2. B\(<\) and \(>\)
    3. C\(=\) only
    4. DEvery inequality
    Show answerHide answer

    B: \(<\) and \(>\)

    Strict inequalities: \(<\) and \(>\).

  2. The line \(x = 3\) is...

    1. AHorizontal
    2. BThrough the origin
    3. CVertical
    4. DSloping
    Show answerHide answer

    C: Vertical

    A vertical line through 3 on the x-axis.

  3. To decide which side to shade, you can...

    1. ATest a point
    2. BGuess
    3. CShade both sides
    4. DMeasure the gradient
    Show answerHide answer

    A: Test a point

    Test a point, such as the origin.

  4. \(x + y \le 6\) is true at \((1, 1)\) because...

    1. A\(1 + 1 = 6\)
    2. B\(1 + 1 > 6\)
    3. CThe point is on the line
    4. D\(2 \le 6\)
    Show answerHide answer

    D: \(2 \le 6\)

    \(1 + 1 = 2\) and \(2 \le 6\).

  5. The integer points in \(x \ge 1\), \(y \ge 1\), \(x + y \le 3\) are...

    1. A\((1,1)\) only
    2. B\((1,1),(1,2),(2,1)\)
    3. C\((1,1),(2,2)\)
    4. D\((0,0),(1,1)\)
    Show answerHide answer

    B: \((1,1),(1,2),(2,1)\)

    \((1,1)\), \((1,2)\), \((2,1)\).

  6. A region satisfies all its inequalities...

    1. AOne at a time
    2. BOnly the first
    3. CAt the same time
    4. DNever
    Show answerHide answer

    C: At the same time

    At the same time.

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