BEAMS_Unit 7 Linear Inequalities

Basic Essential Additional Mathematics Skills

UNIT 7 LINEAR INEQUALITIES Unit 1: Negative Numbers

Curriculum Development Division Ministry of Education Malaysia

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

Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils‟ understanding of the concept involved in performing computations on linear inequalities. 2. This module can be used as a guide for teachers to help pupils master the basic skills required 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 required for this topic.

Curriculum Development Division Ministry of Education Malaysia

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

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 inequality by 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.

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART A: LINEAR INEQUALITY OVERALL LESSON NOTES

1.0

Inequality Signs a. The sign “” means „greater than‟. Example: 5 > 3 c. The sign “  ” means „less than or equal to‟. d. The sign “  ” means „greater than or equal to‟.

2.0 Inequality and Number Line

−3

−2

−1

0

1

2

−3 < − 1 −3 is less than − 1

1 − 3 −1 is greater than − 3

3>1

3

x

3 is greater than 1

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

3.0

Properties of Inequalities (a) Addition Involving Inequalities Arithmetic Form 12  8 so 12  4  8  4 29

so 2  6  9  6

Algebraic Form If a > b, then a  c  b  c If a < b, then a  c  b  c

(b) Subtraction Involving Inequalities Arithmetic Form 7 > 3 so 7  5  3  5 2 < 9 so 2  6  9  6

(c)

Algebraic Form If a > b, then a  c  b  c If a < b, then a  c  b  c

Multiplication and Division by Positive Integers

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

5>3

so 5 (7) > 3(7) 12 9 12 > 9 so  3 3 25

so 2(3)  5(3)

8  12 so

8 12  2 2

Algebraic Form

If a > b and c > 0 , then ac > bc a b If a > b and c > 0, then  c c If a  b and c  0 , then ac  bc a b If a  b and c  0 , then  c c

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

(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. Arithmetic Form

8>2 6 7(−3)

16 > 8

so

16 8  4 4 10 15  5 5

10 b and c < 0, then ac < bc If a < b and c < 0, then ac > bc a b If a > b and c < 0, then  c c a b If a < b and c < 0, then  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 a number. Example:

x > 12 4m

(b)

A solution of an inequality is any value of the variable that satisfies the inequality. Examples: (i)

Consider the inequality x  3 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 are greater than 3, meaning that there are infinitely many solutions! But, if the values of x are integers, then x  3 can be written as

x  4, 5, 6, 7, 8,...

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

A number line is normally used to represent all the solutions of an inequality. To draw a number line representing x  3 , 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.

(ii)

The open dot means the value 2 is not included.

x>2

o −2

(iii)

−2

−1

0

x

2

1

3

The solid dot means the value 3 is included.

x3

−1

4

0

1

2

x 3

4

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART B: POSSIBLE SOLUTIONS FOR A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN

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) x  h (d) x  h (ii) representing a linear inequality: (a) x  h (b) x  h (c) x  h (d) x  h on a number line and vice versa.

TEACHING AND LEARNING STRATEGIES 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 number line. 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. ______________________________________________________________________________ Curriculum Development Division Ministry of Education Malaysia

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART B: POSSIBLE SOLUTIONS FOR A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN EXAMPLES

List out all the possible integer values for x in the following inequalities: (You can use the number line to represent the solutions) (1)

x>4 Solution:

−2

−1

0

2

1

3

6

5

4

7

9

8

x

10

The possible integers are: 5, 6, 7, …

(2)

x  3

Solution:

−8

−7

−6

−5

−4

−3

−2

−1

0

1

3

2

x

4

The possible integers are: – 4, − 5, −6, …

(3)

 3  x 1

Solution:

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

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

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x

Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

TEST YOURSELF B

Draw a number line to represent the following inequalities: (a)

x>1

(b)

x2

(c)

x  2

(d)

x3

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART C: COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION ON LINEAR INEQUALITIES

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.

TEACHING AND LEARNING STRATEGIES 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 sign remains the same.

______________________________________________________________________________ Curriculum Development Division Ministry of Education Malaysia

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART C: COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION ON LINEAR INEQUALITIES LESSON NOTES

Operation on Inequalities

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

Examples: (i) 2 < 4 22−3 1>−1

−1

The inequality sign is unchanged.

x 0

1

2

EXAMPLES

(1)

Solve x  5  14 . Solution: x  5  14 x  5  5  14  5 x9

(2)

Subtract 5 from both sides of the inequality. Simplify.

Solve p  3  2. Solution: p3 2 p  3 3  2  3 p5

Add 3 to both sides of the inequality. Simplify.

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

TEST YOURSELF C Solve the following inequalities:

(1)

m  4  2

(2)

x  3.4  2.6

(3)

x  13  6

(4)

4.5  d  6

(5)

23  m  17

(6)

y  78  54

(7)

9  d 5

(8)

p  2  1

(9)

m

(10)

3 x 8

1 3 2

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART D: COMPUTATIONS INVOLVING DIVISION AND MULTIPLICATION ON LINEAR INEQUALITIES

LEARNING OBJECTIVES Upon completion of Part D, pupils will be able perform computations involving 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.

TEACHING AND LEARNING STRATEGIES The computations involving division and multiplication on inequalities can be confusing 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 positive number, the inequality sign remains the same. 2) When both sides of the inequality is multiplied or divided by a negative number, the inequality sign is reversed. 3)

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

PART D1: COMPUTATIONS INVOLVING MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES

LESSON NOTES

1. When both sides of the inequality is multiplied or divided by a positive number, the inequality sign remains the same. Examples: (i)

2 4m + 1

(b)

14  m  6  m

(c)

3  3m  4  m

(a)

4  x  6

(b)

15  3m  12

(c)

3

x 5 4

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

(d)

5x  3  18

(e)

1  3 p  10

(f)

x 3 4 2

(g) 3 

(h)

x 8 5

p2 4 3

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

EXAMPLES

What is the smallest integer for x if 5x  3  18 ?

A number line can be used to obtain the answer.

Solution: 5x  3  18

5x  18  3

x3

5x  15 x 3

O 0

1

2

3

4

5

6

x = 4, 5, 6,… Therefore, the smallest integer for x is 4.

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x

Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

TEST YOURSELF E2

3x  1  14, what is the smallest integer for x?

1.

If

2.

What is the greatest integer for m if m  7  4m  1 ?

3.

4.

5.

If

x  3  4 , find the greatest integer value of x. 2

If

p2  4 , what is the greatest integer for p? 3

What is the smallest integer for m if

3 m  9? 2

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

ACTIVITY

1

2

3

4

5

6

7

8

9

10 11

12

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

HORIZONTAL: 4.

1  3 is an ___________.

5.

An inequality can be represented on a number __________.

7.

2  6 is read as 2 is __________ than 6.

9.

Given 2x  1  9 , x  5 is a _____________ of the inequality.

11.

 3x  12 x  4

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

VERTICAL: 1.

x  1 2 x  2

The inequality sign remains unchanged when multiplied by a ___________ integer. 2.

6 x  24 equals to x  4 when both sides are _____________ by 6.

3.

x  5 equals to 3x  15 when both sides are _____________ by 3.

6.

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

8.

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

10.

3x  6 is an example of ____________ inequality.

12.

5  3 is read as 5 is _____________ than 3.

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

ANSWERS

TEST YOURSELF B: (a)

x −3

−2

−1

0

2

1

3

(b)

x −3

−2

−1

0

1

0

1

2

3

(c)

x −3

−2

−1

2

3 x

−3

(d)

−2

−1

0

1

2

3

TEST YOURSELF C: (1) m  6

(2) x  6

(8) p  3

(4) d  1.5 (5) m  6 5 (9) m  (10) x  5 2

(2) x  3

(3) c  3

(4) p  5

(5)

(7) x  4

(8) y  5

(9) m  8

(10) b 

(3) y  50

(4) b  42

(5) x  96

(6) y  24 (7) d  4

(3) x  19

TEST YOURSELF D1: (1)

p7

(6) x  4

d  8

9 2

TEST YOURSELF D2: (1) d  24

(2) n  16

(6) x  48

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Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities

TEST YOURSELF E1: 1. (a) m  5

(b) x  8

(c ) m  1 9 (b) m  21 2. (a) m  4 (c ) x  2 1 3. (a ) m  1 (b) m  4 (c) m  2 4. (a) x  10 (b) m  1 (c) x  8 (d) x  3 (e) p  3 (f) x  2 (g) x  25 (h) p  10

TEST YOURSELF E2: (1) x  6

(2) m  1

(3) x  13

(4) p  9

(5) m  14

ACTIVITY: 1. positive 2. divided 3. multiplied 4. inequality 5. line 6. Equivalent 7. less 8. solid 9. solution 10. linear 11. negative 12. greater

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