Solving linear inequalities - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving linear inequalities

Equations and inequalities

Lesson 1 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and inequalities

Lesson 1 of 7

Answer each one, then check.

1

Solve 2x + 3 = 11.

2

Solve 5x = −20.

3

Which is bigger, −3 or −5?

4

What is the symbol for "less than or equal to"?

5

Solve x/3 = 6.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 1 of 7

1

Solve 2x + 3 = 11.

x = 4

2

Solve 5x = −20.

x = −4

3

Which is bigger, −3 or −5?

−3

4

What is the symbol for "less than or equal to"?

≤

5

Solve x/3 = 6.

x = 18

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and inequalities

Lesson 1 of 7

1

Use the inequality symbols <, >, ≤ and ≥.

2

Show inequalities on a number line.

3

Solve linear inequalities, including reversing the sign.

4

Write down integer solutions and solve double inequalities.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Inequality Symbols

Equations and inequalities

Lesson 1 of 7

The open end of the symbol points at the bigger number.

Symbol

Meaning

Circle on a number line

x > 3

x is greater than 3

Open circle at 3

x ≥ 3

x is greater than or equal to 3

Filled circle at 3

x < 3

x is less than 3

Open circle at 3

x ≤ 3

x is less than or equal to 3

Filled circle at 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Inequalities on a Number Line

Equations and inequalities

Lesson 1 of 7

Open circle: not included. Filled circle: included.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving an Inequality

Equations and inequalities

Lesson 1 of 7

Solve 3x + 2 ≤ 14.

1

Subtract 2 from both sides

3x ≤ 12

2

Divide both sides by 3

x ≤ 4

ANSWER

x ≤ 4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 1 of 7

THE ONE NEW RULE

When you multiply or divide an inequality by a negative number, reverse the inequality sign.

Otherwise solve exactly as you would an equation.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Negative Coefficient

Equations and inequalities

Lesson 1 of 7

Solve 5 − 2x > 11.

1

Subtract 5 from both sides

−2x > 6

2

Divide by −2 and reverse the sign

x < −3

3

Check with a value, x = −4

5 − 2(−4) = 13 > 11

ANSWER

x < −3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Double Inequality

Equations and inequalities

Lesson 1 of 7

Solve 5 < 2x − 1 ≤ 9.

1

Add 1 to all three parts

6 < 2x ≤ 10

2

Divide all three parts by 2

3 < x ≤ 5

ANSWER

3 < x ≤ 5

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Integer Solutions

Equations and inequalities

Lesson 1 of 7

List the integers that satisfy −2 < x ≤ 3.

1

−2 itself is not included

Start at −1

2

3 is included

End at 3

ANSWER

−1, 0, 1, 2, 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

Inequalities on a Graph

Shading regions.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Shading a Region

Equations and inequalities

Lesson 1 of 7

Solid line for ≤ or ≥, dashed line for < or >.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Showing an Inequality on a Graph

Equations and inequalities

Lesson 1 of 7

Draw the boundary, then decide which side to shade.

1

Draw the line

Replace the inequality sign with = and draw it

2

Solid or dashed

Solid for ≤ and ≥; dashed for < and >

3

Test a point

For example (0, 0), if it is not on the line

4

Shade the side that works

Shade the wanted region, or the unwanted region, as the question asks

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

A Region from Three Inequalities

Equations and inequalities

Lesson 1 of 7

A region R satisfies y ≥ 1, y ≤ x and x + y ≤ 6. Find the vertices of R.

1

y = 1 meets y = x

(1, 1)

2

y = 1 meets x + y = 6

(5, 1)

3

y = x meets x + y = 6

x + x = 6, so (3, 3)

ANSWER

R is a triangle with vertices (1, 1), (5, 1) and (3, 3).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

A Quadratic Inequality

Equations and inequalities

Lesson 1 of 7

Solve x² − x − 6 < 0.

1

Factorise

(x − 3)(x + 2) < 0

2

The roots are

x = 3 and x = −2

3

The graph is U-shaped, below the axis between the roots

−2 < x < 3

ANSWER

−2 < x < 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and inequalities

Lesson 1 of 7

Inequality

A statement that compares two values using <, >, ≤ or ≥.

Integer

A whole number, positive, negative or zero.

Solution set

All the values that make an inequality true.

Boundary line

The line that separates a region on a graph.

Region

An area of a graph that satisfies one or more inequalities.

Strict inequality

One with < or >, not including the boundary.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 1 of 7

YOUR TASK

True or False?

10 minutes

Decide whether each statement is true or false, and correct it if it is false. (a) −3 > −2 (b) x ≥ 5 includes 5 (c) if −x < 4 then x < −4 (d) the integers satisfying 1 ≤ x < 4 are 1, 2, 3, 4.

1

Use a number line.

2

Test values.

WHAT A GOOD ANSWER SHOWS

(a) False: −3 < −2. (b) True. (c) False: reversing the sign gives x > −4. (d) False: 4 is not included, so the integers are 1, 2, 3.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and inequalities

Lesson 1 of 7

Use the four inequality symbols.

Draw an inequality on a number line.

Solve a linear inequality.

Reverse the sign when multiplying by a negative.

Solve a double inequality.

List integer solutions.

Shade a region on a graph (Higher).

Solve a quadratic inequality (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Open circle for < and >; filled circle for ≤ and ≥.

Solve like an equation, but reverse the sign when dividing by a negative.

Double inequality: do the same to all three parts.

Higher: solid boundary for ≤, ≥; dashed for <, >.

EXAM FOCUS

Solve 5 − 2x > 11. (3 marks)

Dividing by a negative number flips the sign. Check your answer with a test value.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Solving Linear Inequalities

Equations and inequalities

Lesson 1 of 7

Answer all questions. Show your working. · 30 minutes

Question 1 · 2 marks · Non-calculator

Solve 3x + 2 ≤ 14.

Question 2 · 3 marks · Non-calculator

Solve 5 − 2x > 11.

Question 3 · 2 marks · Non-calculator

The number line shows an inequality. Write down the inequality.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Solving Linear Inequalities (cont.)

Equations and inequalities

Lesson 1 of 7

Question 4 · 2 marks · Non-calculator

n is an integer. Write down all the possible values of n when −2 < n ≤ 3.

Question 5 · 3 marks · Non-calculator

Solve 5 < 2x − 1 ≤ 9.

Question 6 · 4 marks · Non-calculator

On the grid, the region R satisfies y ≥ 1, y ≤ x and x + y ≤ 6. Mark the region R with an R.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

“

Solve 3x + 2 ≤ 14.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

3x ≤ 12

M1

x ≤ 4

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

“

Solve 5 − 2x > 11.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

−2x > 6

M1

Dividing by −2

M1

x < −3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 2 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

The number line shows an inequality. Write down the inequality. (2 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

x ≤ 3 or x > −2

B1

−2 < x ≤ 3

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

“

n is an integer. Write down all the possible values of n when −2 < n ≤ 3.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

At least four correct values

M1

All correct

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

“

Solve 5 < 2x − 1 ≤ 9.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

Adding 1 to all parts

M1

Dividing all parts by 2

M1

3 < x ≤ 5

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 4 marks · Non-calculator

Equations and inequalities

Lesson 1 of 7

On the grid, the region R satisfies y ≥ 1, y ≤ x and x + y ≤ 6. Mark the region R with an R. (4 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Equations and inequalities

Lesson 1 of 7

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

One line drawn correctly

M1

All three lines drawn correctly

M1

The correct region

M1

R labelled

A1

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