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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
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Teacher resources

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Student handouts

The same files the students see, to print or hand out.

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.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 3 marks Easier

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.

Mark scheme — 3 marks available

  • One correct inequality — B1
  • Two correct — B1
  • All three correct — B1

Model answer

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

2. Exam question Write down 3 marks Easier

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

Mark scheme — 3 marks available

  • At least 3 correct pairs and none wrong — M1
  • 5 correct — A1
  • All 6 correct — A1

Model answer

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

3. Exam question Write down 3 marks Easier

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.

Mark scheme — 3 marks available

  • One correct — B1
  • Two correct — B1
  • All three — B1

Model answer

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

4. Exam question Explain 2 marks Easier

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

Mark scheme — 2 marks available

  • Dashed for \(<\) or \(>\) — M1
  • Points on the line not included — C1

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 \(\ge\).

5. Exam question Decide 2 marks Easier

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

Mark scheme — 2 marks available

  • Substitutes — M1
  • No, with reason — A1

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.

6. Exam question Shade 3 marks Easier

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.

Mark scheme — 3 marks available

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

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 \(y = 0\)).

7. Multiple choice 1 mark Easier

A dashed line is used for...

  1. A \(\le\) and \(\ge\)
  2. B \(<\) and \(>\) Correct
  3. C \(=\) only
  4. D Every inequality

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

8. Multiple choice 1 mark Core

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

  1. A Horizontal
  2. B Through the origin
  3. C Vertical Correct
  4. D Sloping

Why: A vertical line through 3 on the x-axis.

9. Multiple choice 1 mark Core

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

  1. A Test a point Correct
  2. B Guess
  3. C Shade both sides
  4. D Measure the gradient

Why: Test a point, such as the origin.

10. Multiple choice 1 mark Core

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

  1. A \(1 + 1 = 6\)
  2. B \(1 + 1 > 6\)
  3. C The point is on the line
  4. D \(2 \le 6\) Correct

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

11. Multiple choice 1 mark Core

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

  1. A \((1,1)\) only
  2. B \((1,1),(1,2),(2,1)\) Correct
  3. C \((1,1),(2,2)\)
  4. D \((0,0),(1,1)\)

Why: \((1,1)\), \((1,2)\), \((2,1)\).

12. Multiple choice 1 mark Stretch

A region satisfies all its inequalities...

  1. A One at a time
  2. B Only the first
  3. C At the same time Correct
  4. D Never

Why: At the same time.