Representing inequalities graphically - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Representing inequalities graphically

Equations and graphs

Lesson 2 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and graphs

Lesson 2 of 6

Answer each one, then check.

1

What does x > 3 mean?

2

What does y ≤ 2 mean?

3

What is the line x = 4?

4

What is the line y = 2?

5

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and graphs

Lesson 2 of 6

1

What does x > 3 mean?

x is greater than 3

2

What does y ≤ 2 mean?

y is less than or equal to 2

3

What is the line x = 4?

A vertical line through 4 on the x-axis

4

What is the line y = 2?

A horizontal line through 2 on the y-axis

5

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

1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and graphs

Lesson 2 of 6

1

Draw a boundary line for an inequality.

2

Choose a solid or dashed line.

3

Shade the correct region.

4

Write inequalities for a region and list integer points.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 2 of 6

INEQUALITY REGIONS

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

Solid line for ≤ or ≥. Dashed line for < or >.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Region from Three Inequalities

Equations and graphs

Lesson 2 of 6

The region R satisfies all three inequalities.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solid or Dashed?

Equations and graphs

Lesson 2 of 6

The boundary is part of the region only for ≤ and ≥.

Inequality

Boundary line

Points on the line

y ≥ 2

Solid

Included

y > 2

Dashed

Not included

x + y ≤ 6

Solid

Included

y < x

Dashed

Not included

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Test a Point

Equations and graphs

Lesson 2 of 6

Show which side of the line x + y = 6 satisfies x + y ≤ 6.

1

Test the origin

0 + 0 = 0

2

Compare

0 ≤ 6 is true

3

So shade

The side of the line containing the origin

ANSWER

Shade the region below the line x + y = 6, including the line.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Integer Points in a Region

Equations and graphs

Lesson 2 of 6

List the integer points in x ≥ 1, y ≥ 1, x + y ≤ 4.

1

Try x = 1

y = 1, 2, 3

2

Try x = 2

y = 1, 2

3

Try x = 3

y = 1

ANSWER

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Writing Inequalities

Equations and graphs

Lesson 2 of 6

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.

1

Above y = 1

y ≥ 1

2

Left of x = 4

x ≤ 4

3

Below y = x

y ≤ x

ANSWER

y ≥ 1, x ≤ 4, y ≤ x

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Tips

Equations and graphs

Lesson 2 of 6

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and graphs

Lesson 2 of 6

Inequality

A statement using <, >, ≤ or ≥.

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 2 of 6

YOUR TASK

Sketch the Region

12 minutes

Draw the region satisfying x ≥ 2, y ≥ 1 and x + y ≤ 6. List the integer points that lie in it.

1

Draw each boundary line.

2

Count integer points column by column.

WHAT 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).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and graphs

Lesson 2 of 6

Draw x = a and y = b lines.

Draw sloping boundary lines.

Use solid and dashed lines.

Test a point.

Shade the correct region.

Write inequalities for a region.

List integer points.

Label R clearly.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

≤ and ≥: 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)

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Representing inequalities graphically

Equations and graphs

Lesson 2 of 6

Answer all questions. Show your working. · 25 minutes

Question 1 · 3 marks · Write down

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

Question 2 · 3 marks · Write down

x and y are integers with x ≥ 1, y ≥ 1 and x + y ≤ 4. Write down all the possible pairs (x, y).

Question 3 · 3 marks · Write down

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…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Representing inequalities graphically (cont.)

Equations and graphs

Lesson 2 of 6

Question 4 · 2 marks · Explain

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

Question 5 · 2 marks · Decide

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

Question 6 · 3 marks · Shade

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 3 marks · Write down

Equations and graphs

Lesson 2 of 6

The region R is shown shaded on the grid. Write down the three inequalities that define R. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Equations and graphs

Lesson 2 of 6

3 marks available. Award a mark for each point made.

One correct inequality

B1

Two correct

B1

All three correct

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Write down

Equations and graphs

Lesson 2 of 6

“

x and y are integers with x ≥ 1, y ≥ 1 and x + y ≤ 4. Write down all the possible pairs (x, y).

HOW TO ANSWER IT

Command word: Write down. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Equations and graphs

Lesson 2 of 6

3 marks available. Award a mark for each point made.

At least 3 correct pairs and none wrong

M1

5 correct

A1

All 6 correct

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Write down

Equations and graphs

Lesson 2 of 6

“

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.

HOW TO ANSWER IT

Command word: Write down. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Equations and graphs

Lesson 2 of 6

3 marks available. Award a mark for each point made.

One correct

B1

Two correct

B1

All three

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Explain

Equations and graphs

Lesson 2 of 6

“

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

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Equations and graphs

Lesson 2 of 6

2 marks available. Award a mark for each point made.

Dashed for < or >

M1

Points on the line not included

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Decide

Equations and graphs

Lesson 2 of 6

“

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

HOW TO ANSWER IT

Command word: Decide. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Equations and graphs

Lesson 2 of 6

2 marks available. Award a mark for each point made.

Substitutes

M1

No, with reason

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 3 marks · Shade

Equations and graphs

Lesson 2 of 6

“

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

HOW TO ANSWER IT

Command word: Shade. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Equations and graphs

Lesson 2 of 6

3 marks available. Award a mark for each point made.

x = 2 solid

B1

y = 3 dashed

B1

Correct region

B1

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