7 Stretch lesson: Equations and inequalities - Collins images/Post16maths... · Equations and inequalities 7.1 Linear equations with fractions When equations involve fractions, multiply

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