BEAMS_Unit 7 Linear Inequalities

8/9/2019 BEAMS_Unit 7 Linear Inequalities

1/32

Unit 1:Negative Numbers

UNIT 7

LINEAR INEQUALITIES

B a s i c E s s e n t i a l

A d d i t i o n a l M a t h e m a t i c s S k i l l s

Curriculum Development DivisionMinistry of Education Malaysia


2/32

TABLE OF CONTENTS

Module Overview 1

Part A: Linear Inequalities 2

1.0 Inequality Signs 3

2.0 Inequality and Number Line 3

3.0 Properties of Inequalities 4

4.0 Linear Inequality in One Unknown 5

Part B: Possible Solutions for a Given Linear Inequality in One Unknown 7

Part C: Computations Involving Addition and Subtraction on Linear Inequalities 10

Part D: Computations Involving Division and Multiplication on Linear Inequalities 14

Part D1: Computations Involving Multiplication and Division on

Linear Inequalities 15

Part D2: Perform Computations Involving Multiplication of Linear

Inequalities 19

Part E: Further Practice on Computations Involving Linear Inequalities 21

Activity 27

Answers 29


3/32

Basic Essential Additional Mathematics Skills (BEAMS) Module

Unit 7: Linear Inequalities

1Curriculum Development DivisionMinistry of Education Malaysia

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils understanding of the concept involvedin performing computations on linear inequalities.

2. This module can be used as a guide for teachers to help pupils master the basic skillsrequired to learn this topic.

3. This module consists of six parts and each part deals with a few specific skills.Teachers may use any parts of the module as and when it is required.

4. Overall lesson notes given in Part A stresses on important facts and concepts requiredfor this topic.


4/32



______________________________________________________________________________


PART A:

LINEAR INEQUALITIES

LEARNING OBJECTIVE

Upon completion of Part A, pupils will be able to understand and use the

concept of inequality.

TEACHING AND LEARNING STRATEGIES

Some pupils might face problems in understanding the concept of linear

inequalities in one unknown.

Strategy:

Teacher should ensure that pupils are able to understand the concept of inequalityby emphasising the properties of inequalities. Linear inequalities can also be

taught using number lines as it is an effective way to teach and learn inequalities.


5/32



______________________________________________________________________________


PART A:

LINEAR INEQUALITY

1.0 Inequality Signs

a. The sign means greater than.Example: 5 > 3

c. The sign means less than or equalto.

d. The sign means greater than or equalto.

2.0 Inequality and Number Line

3 < 1

3 is less than 1

and

1 > 3

1 is greater than 3

1 < 31 is lessthan3

and

3 > 1

3 is greaterthan 1

OVERALL LESSON NOTES

1 2 3x

0 1 2 3


6/32



______________________________________________________________________________


3.0 Properties of Inequalities

(a) Addition Involving Inequalities

Arithmetic Form Algebraic Form

812 so 48412

92 so 6962

Ifa > b, then cbca

Ifa < b, then cbca

(b) Subtraction Involving Inequalities


7 > 3 so 5357

2 < 9 so 6962

Ifa > b, then cbca

Ifa < b, then cbca

(c) Multiplication and Division by Positive Integers

When multiply or divide each side of an inequality by the same positive number, therelationship between the sides of the inequality sign remains the same.


5 > 3 so 5 (7) > 3(7)

12 > 9 so12 9

3 3

Ifa > b and c > 0 , then ac > bc

Ifa > b and c > 0, thena b

c c

52 so )3(5)3(2

128 so2

12

2

8

If ba and 0c , then bcac

If ba and 0c , thenc

b

c

a


7/32



______________________________________________________________________________


(d) Multiplication and Division by Negative Integers

When multiply or divide both sides of an inequality by the same negative number, the

relationship between the sides of the inequality sign is reversed.


8 > 2 so 8(5) < 2(5)

6 < 7 so 6(3) > 7(3)

16 > 8 so 16 8

4 4

10 b and c < 0, then ac < bc

If a < b and c < 0, then ac > bc

Ifa > b and c < 0, thena b

c c

Ifa < b and c < 0, thena b

c c

Note: Highlight that an inequality expresses a relationship. To maintain the same

relationship or balance, pupils must perform equal operations on both sides of

the inequality.

4.0 Linear Inequality in One Unknown

(a) A linear inequality in one unknown is a relationship between an unknown and anumber.

Example: x > 12

m4

(b) A solution of an inequality is any value of the variable that satisfies the inequality.

Examples:

(i) Consider the inequality 3x The solution to this inequality includes every number that is greater than 3.

What numbers are greater than 3? 4 is greater than 3. And so are 5, 6, 7, 8, and

so on. What about 5.5? What about 5.99? And 5.000001? All these numbers aregreater than 3, meaning that there are infinitely many solutions!

But, if the values of x are integers, then 3x can be written as,...8,7,6,5,4x


8/32



______________________________________________________________________________


A number line is normally used to represent all the solutions of an inequality.

(ii) x > 2

(iii) 3x The solid dot

means the value

3 is included.

The open dotmeans the value2 is not

included.

3 2 1 10 2x

4

o

0 1 2x

1 2 3 4

To draw a number line representing 3x , place an

open dot on the number 3. An open dot indicates that

the number is not part of the solution set. Then, to

show that all numbers to the right of 3 are included in

the solution, draw an arrow to the right of 3.


9/32



______________________________________________________________________________


PART B:

POSSIBLE SOLUTIONS FOR A

GIVEN LINEAR INEQUALITY INONE UNKNOWN


Some pupils might have difficulties in finding the possible solution for a given

linear inequality in one unknown and representing a linear inequality on a numberline.

Strategy:

Teacher should emphasise the importance of using a number line in order to solve

linear inequalities and should ensure that pupils are able to draw correctly the

arrow that represents the linear inequalities.

LEARNING OBJECTIVES

Upon completion of Part B, pupils will be able to solve linear

inequalities in one unknown by:

(i) determining the possible solution for a given linear inequality in one

unknown:

(a) x h

(b) x h

(c) hx

(d) x h

(ii) representing a linear inequality:

(a) x h

(b) x h

(c) hx (d) x h

on a number line and vice versa.


10/32



______________________________________________________________________________


PART B:

POSSIBLE SOLUTIONS FOR

A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN

List out all the possible integer values forx in the following inequalities: (You can use the

number line to represent the solutions)

(1) x > 4

Solution:

The possible integers are: 5, 6, 7,

(2) 3x

Solution:

The possible integers are: 4, 5, 6,

(3) 13 x

Solution:

The possible integers are: 2, 1, 0, and 1.

258x

1 0 217 46 3 3 4

EXAMPLES

412

5 6 871 20 3 9 10

258x

1 0 217 46 3 3 4


11/32



______________________________________________________________________________


Draw a number line to represent the following inequalities:

(a) x > 1

(b) 2x

(c) 2x

(d) 3x

TEST YOURSELF B


12/32



______________________________________________________________________________



Some pupils might have difficulties when dealing with problems involving

addition and subtraction on linear inequalities.

Strategy:

Teacher should emphasise the following rule:

1) When a number is added or subtracted from both sides of the inequality,the inequality signremains the same.

LEARNING OBJECTIVES

Upon completion of Part C, pupils will be able perform computations

involving addition and subtraction on inequalities by stating a new

inequality for a given inequality when a number is:

(a) added to; and

(b) subtracted from

both sides of the inequalities.

PART C:

COMPUTATIONS INVOLVINGADDITION AND SUBTRACTION ON

LINEAR INEQUALITIES


13/32



______________________________________________________________________________


PART C:

COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION

ON LINEAR INEQUALITIES

Operation on Inequalities

1) When a number is added or subtracted from both sides of the inequality, the inequalitysignremains the same.

Examples:

(i) 2 < 4

Adding 1 to both sides of the inequality:

The inequalitysign is

unchanged.

LESSON NOTES

1x

2 3 4

2 < 4

4x

2 3 5

2 + 1 < 4 + 1

3 < 5


14/32



______________________________________________________________________________


(ii) 4 > 2

Subtracting 3 from both sides of the inequality:

(1) Solve 145 x .

Solution:

9

51455

145

x

x

x

(2) Solve 3 2.p

Solution:

3 2

3 3 2 3

5

p

p

p

Subtract 5 from both sides

of the inequality.

Simplify.

Add 3 to both sides of the

inequality.

Simplify.

The inequality

sign is

unchanged.

EXAMPLES

x1 0 1 2

1x

2 3 4

4 > 2

4 3 > 2 3

1 > 1


15/32



______________________________________________________________________________


Solve the following inequalities:

(1) 24 m (2) 3.4 2.6x

(3) 613 x (4) 65.4 d

(5) 1723 m (6) 78 54y

(7) 9 5d (8) 2 1p

(9)1

32

m

(10) 3 8x

TEST YOURSELF C


16/32



______________________________________________________________________________



The computations involving division and multiplication on inequalities can beconfusing and difficult for pupils to grasp.

Strategy:

Teacher should emphasise the following rules:

1) When both sides of the inequality is multiplied or divided by a positivenumber, the inequalitysignremains the same.

2) When both sides of the inequality is multiplied or divided by a negativenumber, the inequalitysign is reversed.

LEARNING OBJECTIVES

Upon completion of Part D, pupils will be able perform computationsinvolving division and multiplication on inequalities by stating a new

inequality for a given inequality when both sides of the inequalities are

divided or multiplied by a number.

PART D:

COMPUTATIONS INVOLVINGDIVISION AND MULTIPLICATION

ON LINEAR INEQUALITIES


17/32



______________________________________________________________________________


PART D1:

COMPUTATIONS INVOLVING

MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES

1. When both sides of the inequality is multiplied or divided by a positive number, the

inequalitysignremains the same.

Examples:

(i) 2 < 4

Multiplying both sides of the inequality by 3:

LESSON NOTES

The inequality

sign is

unchanged.

1x

2 3 4

2 < 4

2 3 < 4 3

6 < 12

x6 8 10 12 14


18/32



______________________________________________________________________________


(ii) 4 < 2

Dividing both sides of the inequality by 2:

2. When both sides of the inequality is multiplied or divided by a negative number, the

inequalitysign is reversed.

Examples:

(i) 4 < 6

Dividing both sides of the inequality by 1:

The inequality

sign is reversed.

x6 5 4 3

3x

4 5 6

The inequality

sign is

unchanged.

4x

2 0 2

4 < 6

4 (1) > 6

(1)

4 < 2

4 2 < 2 2

2 < 1

2 1 0 1 2

x


19/32



______________________________________________________________________________


(ii) 1 > 3

Multiply both sides of the inequality by 1:

Solve the inequality 3 12q .

Solution:

(i) 3 12q

312

33

q

4q

Divide each side of the

inequality by 3.

Simplify.

The inequalitysign is reversed.

EXAMPLES


1 > 3

x3 2 1 0 1

( 1) (1) < (1) (3)

31

x1 0 1 2 3


20/32



______________________________________________________________________________



(1) 7 49p (2) 6 18x

(3)5c > 15

(4)200 < 40p

(5) 243 d (6) 82 x

(7) x312 (8) y525

(9) 162 m (10) 276 b

TEST YOURSELF D1


21/32



______________________________________________________________________________


PART D2:

PERFORM COMPUTATIONS INVOLVING

MULTIPLICATION OF LINEAR INEQUALITIES

Solve the inequality 32

x .

Solution:

32

x .

3)2()2

(2 x

6x

Multiply both sides of the

inequality by 2.

Simplify.


EXAMPLES


22/32



______________________________________________________________________________


1. Solve the following inequalities:

(1) 38

d (2) 8

2

n

(3)5

10y

(4) 67

b

(5) 0 128

x (6) 8 0

6

x

TEST YOURSELF D2


23/32



______________________________________________________________________________



Pupils might face problems when dealing with problems involving linear

inequalities.

Strategy:

Teacher should ensure that pupils are given further practice in order to enhance

their skills in solving problems involving linear inequalities.

LEARNING OBJECTIVES

Upon completion of Part E, pupils will be able perform computationsinvolving linear inequalities.

PART E:

FURTHER PRACTICE ONCOMPUTATIONS INVOLVING

LINEAR INEQUALITIES


24/32



______________________________________________________________________________


PART E:

FURTHER PRACTICE ON COMPUTATIONS

INVOLVING LINEAR INEQUALITIES


1. (a) 05 m

(b) 62 x

(c) 3 + m > 4

2. (a) 3m < 12

(b) 2m > 42

(c)4x > 18

TEST YOURSELF E1


25/32



______________________________________________________________________________


3. (a) m + 4 > 4m + 1

(b) mm 614

(c) mm 433

4. (a) 64 x

(b) 12315 m

(c) 54

3 x


26/32



______________________________________________________________________________


(d) 1835 x

(e) 1031 p

(f) 432

x

(g) 85

3 x

(h) 43

2

p


27/32



______________________________________________________________________________


What is the smallest integer forx if 1835 x ?

Solution:

1835 x

3185 x

155 x O

3x

x= 4, 5, 6,

Therefore, the smallest integer forx is 4.

3x

A number line can

be used to obtain the

answer.

210 3 4 5 6

EXAMPLES


28/32



______________________________________________________________________________


1. If ,1413 x what is the smallest integer forx?

2. What is the greatest integer for m if 147 mm ?

3.If 43

2

x, find the greatest integer value ofx.

4.If 4

3

2

p, what is the greatest integer forp?

5.What is the smallest integer for m if 9

2

3

m?

TEST YOURSELF E2


29/32



______________________________________________________________________________


1

2 3

4

5

6

7

8

9

10

11 12

ACTIVITY


30/32



______________________________________________________________________________


HORIZONTAL:

4. 31 is an ___________.

5. An inequality can be represented on a number __________.

7. 62 is read as 2 is __________ than 6.

9. Given 912 x , 5x is a _____________ of the inequality.

11. 123 x

4x

The inequality sign is reversed when divided by a ____________ integer.

VERTICAL:

1.

2

12

x

x

The inequality sign remains unchanged when multiplied by a ___________ integer.

2. 246 x equals to 4x when both sides are _____________ by 6.

3. 5x equals to 153 x when both sides are _____________ by 3.

6. ___________ inequalities are inequalities with the same solution(s).

8. 2x is represented by a ____________ dot on a number line.

10. 63 x is an example of ____________ inequality.

12. 35 is read as 5 is _____________ than 3.


31/32



______________________________________________________________________________


TEST YOURSELF B:

(a)

(b)

(c)

(d)

TEST YOURSELF C:

(1) 6m (2) 6x (3) 19x (4) 5.1d (5) 6m

(6) 24y (7) 4d (8) 3p (9)25m (10) 5x

TEST YOURSELF D1:

(1) 7p (2) 3x (3) 3c (4) 5p (5) 8d

(6) 4x (7) 4x (8) 5y (9) 8m (10)2

9b

TEST YOURSELF D2:

(1) 24d (2) 16n (3) 50y (4) 42b (5) 96x 48(6) x

0 2 3x

1 2 3 1

0 2 3x

1 2 3 1

0 2 3 x1 2 3 1

0 2 3

x

1 2 3 1

ANSWERS


32/32



TEST YOURSELF E1:

1. 5)( ma 8)( xb 1)( mc

2. 4)( ma 21)( mb

2

9)( xc

3.1

( ) 1 ( ) 4 (c)2

a m b m m

4. ( ) 10 (b) 1 (c) 8 (d) 3 (e) 3 (f) 2 (g) 25 (h) 10a x m x x p x x p

TEST YOURSELF E2:

(1) 6x (2) 1m (3) 13x (4) 9p (5) 14m

ACTIVITY:

1. positive2. divided3. multiplied4. inequality5. line6. Equivalent7. less8. solid9. solution10.linear11.negative12.greater

Documents

BEAMS_Unit 7 Linear Inequalities