Stretch objectives

Before you start this chapter, mark how confi dent you feel about each of the statements below:

I can solve linear equations involving fractions.

I can solve quadratic equations by factorising.

I can solve two inequalities and compare them to fi nd values that satisfy both inequalities.

Check-in questions

• Complete these questions to assess how much you remember about each topic. Then mark your work using the answers at the end of the lesson.

• If you score well on all sections, you can go straight to the Revision Checklist and Exam-style Questions at the end of the lesson. If you don’t score well, go to the lesson section indicated and work through the examples and practice questions there.

1 Solve the equation 3 13

x − = 4 + 2x Go to 7.1

2 Solve these quadratic equations. Go to 7.2

a x2 - 7x = 0 b x2 + 8x + 15 = 0 c x2 - 5x + 6 = 0

3 a Solve the inequality 4 + x > 7x - 8

b Solve the inequality 3 54x + 5. Represent the solutions

on a copy of the number line. Go to 7.2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

7 Stretch lesson: Equations and inequalities

7.1 Linear equations with fractionsWhen equations involve fractions, multiply both sides by the denominator to eliminate the fraction part of the equation.

Solve: x + 43

= 10

x + 4 = 30

x = 26Multiply both sides by 3.

Subtract 4 from both sides.

Example

1Q

A

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017

Solve: x + 23

+ x − 12

= 156

2(x + 2) + 3(x - 1) = 15

2x + 4 + 3x - 3 = 15

5x + 1 = 15

5x = 15 - 1 x = 14

5 or 2 45

Example

3Q

A 6 is the lowest common multiple of 2, 3 and 6, so multiply both sides of the equation by 6.

Expand the brackets.

Solve.

You can write your answer as an improper fraction, a mixed number or an exact decimal.

Exam tips Make sure that you write down each step in the solution.

Practice questions 1 Solve these equations.

a x + =65 2 b x − =3

2 5 c x + =164 6

2 Solve these equations.

a 92 3− =x b 15 2

3 3− =x c 29 35 7− =x

3 Solve these equations.

a x x+x x+x x+x x+x x − =3x x3x x2

14 5 b x x+x x+x x+x x+x x − =6x x6x x

52

10115

c 2 12

34

154

x x2 1x x2 12 1+2 12 1x x2 1+2 1x x2 1 +x x+x x + =

4 Tzun is asked to solve 2 68 162 6x2 62 6+2 6 = .

This is his working: 28 = 16 – 6x

28 = 10x

2x = 10 − 8

2x = 2

x = 1

Identify where Tzun went wrong and work out the correct value for x.

Solve: 3 2

5( )3 2( )3 2 1( )1x( )x −( )−

= 6

3(2x - 1) = 6 × 5

6x - 3 = 30

6x = 33

x = 336

x = 5.5

First, multiply both sides by 5.

Example

2Q

A

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017

7.2 Quadratic equationsA quadratic equation written in the form ax2 + bx + c = 0 can be solved by factorising into two brackets (x ± ?)(x ± ?) = 0. (See Chapter 6 for more on factorisation.)

Since the equation equals zero, at least one of the brackets must equal zero.

To solve the equation x2 - x - 6 = 0:

• Factorise into two brackets. (x + 2)(x - 3) = 0

• Either (x + 2) = 0 or (x - 3) = 0

So x = -2 or x = 3

Solve: x2 - 7x + 10 = 0

(x - 2)(x - 5) = 0

Either (x - 2) = 0 or (x - 5) = 0

So x = 2 or x = 5

Example

4Q

A

Solve: x2 - 6x - 16 = 0

(x − 8)(x + 2) = 0

Either (x − 8) = 0 or (x + 2) = 0

So x = 8 or x = −2

Example

5Q

A

Exam tips Check that the equation is written in the form ax2 + bx + c = 0 before you factorise.

Practice questions 1 Factorise these quadratic expressions.

a x2 + 6x + 8 b x2 + 12x + 20 c x2 + 7x + 12 d x2 + 12x + 36

2 Use factorisation to solve these quadratic equations.

a x2 + 7x + 10 = 0 b x2 + 13x + 36 = 0

c x2 + 13x + 30 = 0 d x2 + 12x + 35 = 0

3 Solve these quadratic equations.

a x2 - x - 2 = 0 b x2 - 5x + 6 = 0

c x2 - 2x - 8 = 0 d x2 - 8x + 16 = 0

4 Solve these.

a x2 + 4x = −3 b x2 - x - 3 = 3 c x2 + 8x + 3 = −9

5 The area of the square is 64 cm². Find the value of x.

(x + 3) cm

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017

Solve: 3 24

x − < 4

3x - 2 < 16

3x < 16 + 2

3x < 18

x < 183

x < 6

Multiply both sides by 4.

Add 2 to both sides.

Divide both sides by 3.

Example

6Q

A

Solve: -7 < 3x - 1 11

-6 < 3x 12

-2 < x 4

The integer values that satisfy this inequality are -1, 0, 1, 2, 3 and 4.

Add 1 to each part of the inequality.

Divide each part of the inequality by 3.

Example

7Q

A

Solve: 2 < 2 53

x − < 5

6 < 2x - 5 < 15

11 < 2x < 20

5.5 < x < 10

The integer values that satisfy this inequality are 6, 7, 8 and 9.

Multiply each part of the inequality by 3.

Add 5 to each part of the inequality.

Divide each part of the inequality by 2.

Example

8Q

A

7.3 Further inequalitiesInequalities involving fractionsFollow the same process for dealing with inequalities involving fractions as you did with equations - multiply through to remove the denominator.

Two inequalitiesWhen there are two inequalities, make sure that you do the same thing to all parts of the inequality.

Practice questions 1 Solve these inequalities.

a 2 15 3x + > b x − <7

4 2 5. c 5 3

9x –

3 d 8 6

10x –

0.3

2 Solve these inequalities.

a 5 2x + 1 < 11 b −8 3x + 1 < 13 c 4 4x < 10 d −10 4x + 2 < 2

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017

Exam-style questions 1 Which integers satisfy 2 < 2x + 5 15?

2 Write down the largest integer which satisfies 2 5

4 2x– –< .

3 Solve: 5 2 40 53x x+( ) = −

4 Write an inequality for the integers that satisfy both of these inequalities.

−5 x 3 −2 < 2x + 2 8

5 Solve: x2 - 8x + 15 = 0

6 This rectangle has area 44 cm2. Find the length of the longest side.

(x – 4) cm

(x + 3) cm

7 Solve: 2x2 + 8x + 6 = 0

8 Solve: x2 - 7x + 6 = −6

9 The area x of a field is given as x2 + x - 12 = 0. Solve to find the value of x.

REVISION CHECKLIST ● Some quadratic equations can be solved by factorisation.

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017

Chapter 7 Stretch lesson: AnswersCheck-in questions

1 x = −4 132 a x = 0 or x = 7

b x = −5 or x = −3

c x = 2 or x = 3

3 a x < 2

b –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

7.1 Linear equations with fractions

1 a x = 4 b x = 13 c x = 8

2 a x = 3 b x = 3 c x = −2

3 a x = 5 b x = 4 c x = 2

4 Tzun doesn’t eliminate the denominator first. He also subtracts 8 instead of multiplying by 8 in the third line.

The correct working is: 2x + 6 = 16 × 8

2x + 6 = 128

2x = 128 – 6

2x = 122

x = 122 ÷ 2

x = 61

7.2 Quadratic equations

1 a (x + 4)(x + 2) b (x + 10)(x + 2) c (x + 3)(x + 4) d (x + 6)(x + 6)

2 a x = –2 or x = –5 b x = –9 or x = –4 c x = –10 or x= –3 d x = –7 or x = –5

3 a x = 2 or x = –1 b x = 2 or x = 3 c x = 4 or x = –2 d x = 4

4 a x = –3 or x = –1 b x = 3 or x = –2 c x = –2 or x = –6

5 x = 5

7.3 Further inequalities

1 a x > 7 b x < 17 c x 6 d x 98

2 a 2 x < 5 b −3 x < 4 c 1 x < 2.5 d −3 x < 0

Exam-style questions

1 –1, 0, 1, 2, 3, 4 and 5

2 x = –2

3 x = 12

4 −2 < x 3

5 x = 3 or x = 5

6 11 cm

7 x = –1 or x = –3

8 x = 3 or x = 4

9 x = 3

Edexcel GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017