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8/9/2019 BEAMS_Unit 7 Linear Inequalities
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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
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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
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Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities
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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.
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Unit 7: Linear Inequalities
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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.
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Unit 7: Linear Inequalities
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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
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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
Arithmetic Form Algebraic Form
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.
Arithmetic Form Algebraic Form
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
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(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 Algebraic Form
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
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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.
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PART B:
POSSIBLE SOLUTIONS FOR A
GIVEN LINEAR INEQUALITY INONE UNKNOWN
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 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.
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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
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Draw a number line to represent the following inequalities:
(a) x > 1
(b) 2x
(c) 2x
(d) 3x
TEST YOURSELF B
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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 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
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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
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(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
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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
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TEACHING AND LEARNING STRATEGIES
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
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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
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(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
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(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
The inequalitysign is reversed.
1 > 3
x3 2 1 0 1
( 1) (1) < (1) (3)
31
x1 0 1 2 3
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Solve the following inequalities:
(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
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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.
The inequalitysign is reversed.
EXAMPLES
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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
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TEACHING AND LEARNING STRATEGIES
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
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PART E:
FURTHER PRACTICE ON COMPUTATIONS
INVOLVING LINEAR INEQUALITIES
Solve the following 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
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3. (a) m + 4 > 4m + 1
(b) mm 614
(c) mm 433
4. (a) 64 x
(b) 12315 m
(c) 54
3 x
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(d) 1835 x
(e) 1031 p
(f) 432
x
(g) 85
3 x
(h) 43
2
p
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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
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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
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1
2 3
4
5
6
7
8
9
10
11 12
ACTIVITY
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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.
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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
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Unit 7: Linear Inequalities
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