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SCHOLAR Study Guide - National 5 Mathematics Course Materials

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National 5 Mathematics

Course Materials

Topic 14: Solving equations andinequations

Authored by:

Margaret Ferguson

Reviewed by:

Jillian Hornby

Previously authored by:

Eddie Mullan

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2016 by Heriot-Watt University SCHOLAR.

Copyright © 2016 SCHOLAR Forum.

Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide.

Distributed by the SCHOLAR Forum.

SCHOLAR Study Guide Course Materials Topic 14: National 5 Mathematics

1. National 5 Mathematics Course Code: C747 75

Acknowledgements

Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the

SCHOLAR programme:

The Scottish Qualifications Authority for permission to use Past Papers assessments.

The Scottish Government for financial support.

The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders. 1

Topic 1

Solving equations and inequations

14.1 Solving linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

14.2 Solving linear inequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

14.3 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

14.4 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Learning objectives

By the end of this topic, you should be able to:

solve linear equations; interpret inequality symbols; solve linear inequations.

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS3

1.1 Solving linear equations

When solving equations we are trying to find the value of the variable.xis avariable, some other letters which are commonly used instead ofxared,n,p,t,yandz. A linear equation takes the formax+b=cwherea,bandcareconstantsand a?=0. You will already know how to solve equations like3x=15andx-5=13so you already know how to solve simple linear equations. There are several methods that you may already have used such as the cover-up, balancing and change side change sign. In this topic we are going to learn how to solve more complex linear equations. To solve an equation we are trying to find the value of the variable which makes the equation true.

Examples

Problem:

Solve5x-4=26

Solution:

Note: We don't have to show the lines of working to remove the-4and find x on its own by÷5. If we substitutex=6into the equation we will see that it makes the equation true.

5x-4=26

5×6-4=26

30-4=26

26 = 26..........................................

Problem:

Solve3y-6=y-10

Solution:

Notice that there areyterms on both sides of the equation. Our aim should always be to collect the letters on one side of the equation and the numbers on the other.

©HERIOT-WATTUNIVERSITY

4TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Problem:

Solve3(a+2)=18

Solution:

Problem:

Solve2b+7(b+2)=4-b

Solution:

Problem:

Solve3d-4(d-1) = 7(4-d)

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS5

Solution:

Problem:

Solve50-(f-5) = 2(f+ 11)

Solution:

Problem:

Solve 2 3 g+4=6

Solution:

Notice that each term in the equation must be multiplied by 3 so that the meaning of the equation is not changed. 8.

Problem:

Solve 1 4 (8p-1) = 5

©HERIOT-WATTUNIVERSITY

6TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Solution:

Notice that the answer is an improper fraction.

You can leave it like this or change it into a mixed numberp=2 3 4

Solving linear equations practice

Go online

Q1:Solve8z+4=-20

Q2:Solve5-2t-2=t

Q3:Solve4(2r+1)=20

Q4:Solve5(6-m)+2=m+2

Q5:Solve2(3k+5)=3k-4(k+1)

Q6:Solve-2(2n+1)=3-(n-1)

Q7:Solve

3 4 j+5=2

Solving linear equations exercise

Go online

Solving linear equations

Q8:Solve:

a)6a-7=23 b)4b+2=b-10 c)5c+7=8c-2

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS7

Solving linear equations with brackets

Q9:Solve:

a)3(d+2)=21 b)2(4-e)=4e-10 c)5(2f-1) = 8(f+2) d)g-3(g-2) = 3(2-g) e)6(h-1)-5=4-(h+1)

Solving linear equations with fractions

Q10:Solve:

a) 1 4 j+3=2 b) 3 5 k-1=2 c) 2 3 m+ 5 3 =7 d) 1 2 (n+ 14) = 6 e) 5 8 p+3=5

1.2 Solving linear inequations

An inequality or inequation defines the relationship between two quantities. We know the equality sign '=' which we have already seen in equations. The four signs that appear >means 'is greater than' ≥means 'is greater than or equal to' ©HERIOT-WATTUNIVERSITY

8TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Examples

Problem:

Solve5x+11>26

Solution:

The solution to this inequality is:xis greater than 3.

Possible values forxare 4, 11, 12·3, 25

1 2 , 100,... Note: We don't have to show the lines of working to remove the+11and÷5. 2.

Problem:

Solution:

Problem:

Solve3a+8<26

Solution:

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS9

Key point

Consider8>6.

Dividing both sides by -2 gives-4>-3but this is no longer true. We must reverse the inequality sign to make this statement true, giving:-4<-3 Similarly when we multiply on both sides of an inequality by a negative we must reverse the inequality sign.

Examples

Problem:

Solve10-2(d+3)
Solution:
2.
Problem:
Solve 1 /2t-2<5+t
Solution:
3.
Problem:
A gym offers a monthly membership for £32. As a member the cost for each class attended is £1. It is also possible for non-members to drop-in to classes. The drop-in cost for a class is £5.
Letnbe the number of classes attended in a month.
a) Construct an expression for the total cost for a member to attendnclasses. b) Construct an expression for the total cost for a non-member to attendnclasses.
©HERIOT-WATTUNIVERSITY

10TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS
c) Set up an inequation and solve it to find the minimum number of classes that must be attended each month before taking out a membership is the cheaper option.
Explain your answer.

Solution:
a) Members would pay£32 +n×£1=32+n b) Non-members would payn×£5=5n c) We could check to see when non-members pay more than members.
5n>32 +n
4n>32 n>8 It is best to try a value fornto explain your answer.
Whenn=9members pay32 + 9 =£41

Whenn=9non-members pay5×9=£45
When 9 or more classes are attended each month it is cheaper to be a member.
Key point
Remember: when we solve an inequation by multiplying or dividing by a negative we must reverse the inequality sign.
Solving inequations practice

Go online

Q11:Solve3z+4<-20

Q12:Solve10z-23≥3z+26

Q14:Solve2g-7(g+2)≥3(2-g)

Solving inequations exercise

Go online

Solving simple inequations

Q15:Solve:
a)12a+7>31 c)2(3c+7)≥2(2-2c) d)15-(d+3)<2-2d
©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS11

Solving more complex inequations

Q16:Solve:
a)5-2f<-9 b)-6g+8>2g-8 d) 1 2 (6-j)≥22-j e)2(15-k)< 1 2 k
©HERIOT-WATTUNIVERSITY

12TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

1.3 Learning points

Solving linear equations

Show every step as a new line in your solution.
To get rid of a term to the other side of an equation carry out the inverse operation: add becomes subtract; subtract becomes add; multiply becomes divide; divide becomes multiply; If the equation has brackets multiply them out first. If the equation has a fraction, multiply every term in the equation by the denominator (or by the common denominator if there is more than one fraction).
Solving inequations
>means 'is greater than' ≥means 'is greater than or equal to' ©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS13

1.4 End of topic test

End of topic 16 test

Go online

Solving Linear Equations

Q17:Solve:

a)7x+5=3x-7 b)6(x-3) + 8 = 2(3-x) c) 2 5 x+9=7

Solving Inequations

Q18:Solve:

a)8x+4≥5x-8 b)2x+4>5x-14 c)x+5< 1 2 (4x-10)

©HERIOT-WATTUNIVERSITY

14GLOSSARY

Glossary

constants a mathematical constant is a numerical value or symbol which is fixed e.g. 5,

2·075, -1000,π,...

variable a mathematical variable is a symbol for a number we do not know yet, it is usually a letter likexory

©HERIOT-WATTUNIVERSITY

ANSWERS: TOPIC 1415

Answers to questions and activities

14 Solving equations and inequations

Solving linear equations practice (page 6)

Q1:z=

Steps:

8z=-24

Answer:z=-3

Q2:t=

Steps:

-2t+3=t -3t+3=0 -3t=-3

Answer:t=1

Q3:r=

Steps:

8r+4=20

8r=16

Answer:r=2

Q4:m=5

Steps:

30-5m+2=m+2

-5m+32=m+2 -6m+32=2 -6m=-30

Answer:m=5

Q5:k=

Steps:

6k+10=3k-4k-4

6k+10=-k-4

7k+10=-4

7k=-14

Answer:k=-2

Q6:n=

Steps:

16ANSWERS: TOPIC 14

-4n-2=3-n+1 -3n-2=4 -3n=6

Answer:n=-2

Q7:j=

Steps:

3j+20=8

3j=-12

Answer:j=-4

Solving linear equations exercise (page 6)

Q8: a)a=5 b)b=-4 c)c=3 Q9: a)d=5 b)e=3 c)f=10·5 d)g=0 e)h=2 Q10: a)j=-4 b)k=5 c)m=8 d)n=-2 e)p=3.2or 16 5 or3 1 5

Solving inequations practice (page 10)

Q11:z<

Steps:

3z<-24

Answer:z<-8

ANSWERS: TOPIC 1417

Q12:z≥

Steps:

7z-23≥26

7z≥49

Answer:z≥7

Q13:b≥

Steps:

Answer:b≥-7

Steps:

2g-7g-14≥6-3g

-5g-14≥6-3g -2g-14≥6 -2g≥20

Answer:g≥-10

Solving inequations exercise (page 10)

Q15: a)a>2 c)c≥-1 d)d<-10 Q16: a)f>7 b)g<2 c)h≥-1 d)j≥38 e)k>12

18ANSWERS: TOPIC 14

End of topic 16 test (page 13)

Q17: a)x=-3 b)x=2 c)x=-5 Q18: a)x≥-4 b)x<6 c)x>10

quotesdbs_dbs47.pdfusesText_47

[PDF] SCHOLAR Study Guide - National 5 Mathematics Course Materials

SCHOLAR Study Guide

National 5 Mathematics

Course Materials

Topic 14: Solving equations andinequations

Authored by:

Margaret Ferguson

Reviewed by:

Jillian Hornby

Previously authored by:

Eddie Mullan

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

Copyright © 2016 SCHOLAR Forum.

Distributed by the SCHOLAR Forum.

1. National 5 Mathematics Course Code: C747 75

Acknowledgements

SCHOLAR programme:

The Scottish Government for financial support.

Topic 1

Solving equations and inequations

Contents

14.1 Solving linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

14.2 Solving linear inequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

14.3 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

14.4 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Learning objectives

By the end of this topic, you should be able to:

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS3

1.1 Solving linear equations

Examples

Problem:

Solve5x-4=26

Solution:

5x-4=26

5×6-4=26

30-4=26

26 = 26..........................................

Problem:

Solve3y-6=y-10

Solution:

©HERIOT-WATTUNIVERSITY

4TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Problem:

Solve3(a+2)=18

Solution:

Problem:

Solve2b+7(b+2)=4-b

Solution:

Problem:

Solve3d-4(d-1) = 7(4-d)

©HERIOT-WATTUNIVERSITY

TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS5

Solution:

Problem:

Solve50-(f-5) = 2(f+ 11)

Solution:

Problem:

Solution:

Problem:

©HERIOT-WATTUNIVERSITY

6TOPIC 1. SOLVING EQUATIONS AND INEQUATIONS

Solution:

Notice that the answer is an improper fraction.

Solving linear equations practice

Go online

Q1:Solve8z+4=-20

Q2:Solve5-2t-2=t

Q3:Solve4(2r+1)=20

Q4:Solve5(6-m)+2=m+2

Q5:Solve2(3k+5)=3k-4(k+1)

Q6:Solve-2(2n+1)=3-(n-1)

Q7:Solve

Solving linear equations exercise

Go online

Solving linear equations

Q8:Solve: