AM015 20212022

AM015

MATHEMATICS

Session 2022/2023

Lecture Notes
&

Tutorial Questions
KMPP

The only way to learn Mathematics is to do Mathematics

Table 4 - Course Information

1 Name of Course : Mathematics 1
Course Code : AM015

2. Synopsis : This course is to equip students with knowledge of Mathematics and the ability to solve problems and propose solutions using
mathematical terms, notation, principles and methods to pursue education in first-year degree in the field of social science
3. Name(s) of and business.
academic staff :
1. Ahmad Afif bin Mohd Nawawi
2. Refer to Appendix 1

4. Semester and Year offered : Sem 1 Year 1

5. Credit Value : 5

6. Prerequisite/co-requisite: (if any)

7. Course Learning Outcomes (CLO) : At the end of the course the students will be able to:
C(eLxOa1mple) - explain thCeobmapsircephreinncdipfulensdoafmimenmtaulnciosantcieopnt(sCa2n,PdLtOh1e)ories in Algebra, Calculus and Financial Mathematics (C2, PLO 1, MQF LOC i)

CLO2 Solve problems using appropriate methods and tools in Algebra, Calculus and Financial Mathematics (C3, PLO 2, MQF LOC ii)

CLO3 Compare familiar and uncomplicated numerical and graphical data in Algebra. (C4, P3, PLO 7, MQF LOC iii e)
8. Mapping of the Course Learning Outcomes to the Programme Learning Outcomes, Teaching Methods and Assessment :

Course Learning Programme Programme Learning Learning Outcome Clusters Learning Teaching Assessment
Outcomes (CLO) Educational Outcomes (PLO) (LOC) Taxonomy Methods
Objectives (PEO)
Level
PEO1 PEO2 PEO3 PLO1 PLO2 PLO7 PLO8 LOD1 LOD2 LOD3 LOD4 LOD5

CLO 1 √ √ √ C1, C2 Summative
Lecture Assessment Test,

Examination

CLO 2 √√ √ C1, C2, C3 Tutorial Assignment
Examination

CLO 3 √ √ √ C1,C2,C3,C4, Tutorial Assignment
P1, P2,P3

Indicate the relevancy between the CLO and PLO by ticking “/“ the appropriate relevant box.

(This description must be read together with Standards 2.1.2 , 2.2.1 and 2.2.2 in Area 2 - pages 16 & 18)

9. Transferable Skills (if applicable) 1 Critical Thinking and Problem Solving
(Skills learned in the course of study which can 2 Numeracy Skills
be useful and utilized in other settings)

1 Distribution of Student Learning Time (SLT) Teaching and Learning Activities
0.
CLO* Guided Learning Guided Learning Independent Learning SLT
Course Content Outline (F2F) (NF2F) eg : e- (NF2F)
1,2 Learning 4
Topic 1 : 1.1 Real Numbers 1,2 L TO 6
Topic 1 : 1.2 Indices 1,2 10
Topic 1 : 1.3 Surds 1,2 110 0 2 10
Topic 1 : 1.4 Logarithms 1,2 10
Topic 2 : 2.1 Inequalities 1,2 120 0 3 10
Topic 2 : 2.2 Absolute Values 1,2,3 4
Topic 3 : 3.1 Matrices 1,2,3 230 0 5 6
Topic 3 : 3.2 Determinant of Matrices 1,2,3 8
Topic 3 : 3.3 Inverse of a Matrix 1,2,3 230 0 5 6
Topic 3 : 3.4 Systems of Linear Equations with Three 1,2,3 6
VToapriiacb4le:s4.1 Systems of Linear Inequalities 1,2,3 230 0 5 10
Topic 4 : 4.2 Solving Problem Involving System of 1,2 10
TLoinpeiacr5In:e5q.u1aFluitniecstions 1,2 230 0 5 10
Topic 5 : 5.2 Composite Functions 1,2 14
Topic 5 : 5.3 Inverse Functions 110 0 2

120 0 3

220 0 4

120 0 3

120 0 3

230 0 5

230 0 5

230 0 5

340 0 7

Topic 5 : 5.4 Polynomials 1,2 2 3 0 0 5 10
Topic 6 : 6.1 Arithmetics Sequences and Series 1,2 1 2 0 0 3 6
Topic 6 : 6.2 Geometric Sequences and Series 1,2 2 3 0 0 5 10
Topic 7 : 7.1 Simple Interest 1,2 1 1 0 0 2 4
Topic 7 : 7.2 Simple Discount 1,2 1 2 0 0 3 6
Topic 7 : 7.3 Compound Interest 1,2 2 3 0 0 5 10
Topic 7 : 7.4 Annuity 1,2 2 3 0 0 5 10

Total 180

Continuous Assessment Percentage F2F NF2F SLT
(%)
4 4
1 Assignment 1 : Topic 3 & 4 (Numeracy Skills) 20 0 4 4
8
2 Assignment 2 : Topic 5 20 0

Total

Final Assessment Percentage F2F NF2F SLT
Summative Assessment Test (UPS) : (%) 1.5 4.5 6
1 Topic 1-5
20

Final Assessment Percentage F2F NF2F SLT
(%) 8

2 Final Examination (Topic 1,2,6 &7) 40 2 6 202

**Please tick (√) if this course is Latihan Industri/Clinical

Placement/Practicum/WBL using 2-weeks, 1 credit formula

L=Lecture, T=Tutorial, P=Practical, O=Others, F2F=Face to Face, NF2F=Non Face to GRAND TOTAL SLT
Face
11 Identify special requirement to deliver the course (e.g: software, nursery, computer lab, simulation room, etc) :

12 References (include 1. Lial, M.L., Greenwell, R.N, & Ritchey, N.P. (2016).Finite Mathematics and Calculus with Applications (10th. Edition).
required and further Pearson New International Edition.

readings, and should be the 2. Haeussler E.F.,Paul R.S., Wood R.J. (2018). Introductory Mathematical Analysis: For Business, Economics, and the

most current) Life and Social Science (14th Edition)

3. Croft, A. & Davidson, R. (2020). Foundation Maths (7th. Edition). Pearson.

4. Barnett, R Ziegler, M, J. Stocker & Byleen,(2019) College Mathematics for Business, Economics, Life Science &

Social Science (14th Edition) Pearson Education.

13 Other additional
information :

Appendix 1 1 Ahmad Afif bin Mohd Nawawi AM015
AM015
2 Salawathy Dharwina binti Md AM015
Yusof

3 Farah Alwani binti Zolkifli

4 Maglambikai A/P Saundarajan AM015
Pillai

5 Nur Hidayah Hanim binti AM015
Ahamad
AM015
6 Lim Joo Sim AM015
7 Nursyamsaziela binti Morad

MATHEMATICS UNIT
PENANG MATRICULATION COLLEGE
MINISTRY OF EDUCATION MALAYSIA 13200 PONGSU SERIBU,

KEPALA BATAS, PULAU PINANG.

SEMESTER TEACHING PLAN (RPS)
MATHEMATICS AM015

SEMESTER 1 SESSION 2022/2023

WEEK DATE TOPIC/SUBTOPIC LECTURE TUTOR NOTES
1. NUMBER SYSTEM AND
1 01/08/2022 1.1 (a), (b) 1.1 (a), (b)
- EQUATIONS 1.2 (a), (b), (c) 1.2 (a), (b), (c)
(6L+9T hours)
05/08/2022
1.1 Real Numbers
1.2 Indices

08/08/2022

2 - 1.3 Surds 1.3 (a), (b), (c), (d) 1.3 (a), (b), (c),
(d)
12/08/2022

15/08/2022

3 - 1.4 Logarithms 1.4 (a), (b), (c), (d) 1.4 (a), (b), (c),
(d)
19/08/2022

2. INEQUALITIES AND

22/08/2022 ABSOLUTE VALUES
(4L+6T hours)
4 - 2.1 (a), (b), (c), (d), 2.1 (a), (b), (c),
26/08/2022 (e) (d), (e)

2.1 Inequalities

29/08/2022 Independence
Day
5 - 2.2 Absolute value 2.2 (a), (b), (c) 2.2 (a), (b), (c)
(31 Aug 2022)
02/09/2022

MID SEMESTER BREAK (05/09/2022- 09/09/2022)

3. MATRICES SYSTEMS OF

12/09/2022 LINEAR EQUATIONS

- (5L+7T hours) Malaysia Day
16/09/2022
6 3.1 (a), (b), (c),(d) (16 Sept 2022)

3.1 Matrices 3.1 (a), (b), (c), (d)

3.2 Determinant of Matrices 3.2 (a), (b), (c) 3.2 (a), (b), (c)

19/09/2022

7 - 3.3 Inverse of a Matrix 3.3 (a), (b) 3.3 (a), (b)

23/09/2022

3.4 Systems of Linear 3.4 (a), (b) 3.4 (a), (b)
Equations with Three
Variables

26/09/2022 4. LINEAR
-
8 30/09/2022 PROGRAMMING Assignment 1

(3L+5T hours) Submission of
individual
4.1 System of Linear 4.1 (a), (b), (c) 4.1 (a), (b), (c) Assignment
Inequalities Deepavali
(24-26 Oct
03/10/2022 4.2 Solving Problems 2022)
Individual
9 - involving system of Linear 4.2 (a), (b), (c), (d) 4.2 (a), (b), (c), assignment
07/10/2022 Inequalities (d) moderation
Assignment
5. FUNCTIONS AND Group

10/10/2022 GRAPHS Submission of
(9L+13T hours) Group
10 - 5.1 (a), (b), (c) 5.1 (a), (b), (c)
14/10/2022 Assignment
Group
5.1 Functions
Assignment
17/10/2022 moderation

11 - 5.1 Functions 5.1 (c), (d) 5.1 (c), (d)
21/10/2022 5.2 Composite functions 5.2 (a), (b), (c) 5.2 (a), (b), (c)

24/10/2022

12 - 5.3 Inverse functions 5.3 (a), (b), (c), (d) 5.3 (a), (b), (c),
28/10/2022 (d)

31/10/2022

13 - 5.4 Inverse functions 5.4 (a), (b), (c) 5.4 (a), (b), (c)
5.5 (a), (b), (c)
04/11/2022 6.1 (a), (b)

5.5 Polynomials 5.5 (a), (b), (c) 6.2 (a), (b)
7.1 (a), (b), (c)
07/11/2022 6. SEQUENCES AND

14 - SERIES
11/11/2022 (3L+5T hours)
6.1 (a), (b)

6.1 Arithmetic Sequences
and series

14/11/2022

15 - 6.2 Geometric Sequences 6.2 (a), (b)
18/11/2022 and series

21/11/2022 7. MATHEMATICS OF

16 - FINANCE 7.1 (a), (b), (c)
25/11/2022 (6L+9T hours)

7.1 Simple interest

17 28/11/2022 7.2 Simple Discount 7.2 (a), (b), (c) 7.2 (a), (b), (c) Mesyuarat
- 7.3 Compound interest 7.3 (a), (b), (c) 7.3 (a), (b), (c) Pengesahan

02/12/2022 7.4 (a), (b) 7.4 (a), (b) Markah

05/12/2022

18 - 7.4 Annuity
09/12/2022

10/12/2022 REVISION WEEK
- FINAL EXAM

14/12/2022

15/12/2022
–

22/12/2022

7/7/2022

TOPIC 1 1.1 REAL NUMBERS

NUMBER SYSTEMS AND
EQUATIONS

LEARNING OUTCOMES Types of Real Numbers

At the end of this lesson, students are able Types Symbol Description & Example
N
to:   W     1. Natural Positive numbers that are
(a) Review natural numbers (  ), whole Numbers used for counting.
{1, 2, 3, …}
rnautmionbaelrsnu( Wmb)e,irnste(ge)rs(  ), prime numbers,
and irrational numbers(  ). Prime Natural number that are
Numbers greater than one that can
(b) Represent the relationship of number sets be divided by itself and
in a real number system. one only.
{ 2, 3, 5, 7, …}

Types Symbol Description & Example Types Symbol Description & Example
W 3. Integer Z Whole number including
2. Whole Natural number including their negatives
numbers zero { …,-2,-1,0,1,2,3,…}
{0, 1, 2, 3, …}
{1,2,3,… }
Positive 
Integers  { …,-3,-2,-1 }

Negative
Integers

1

7/7/2022

Types Symbol Description & Types Symbol Descriptions & Examples
Example
Note: 2k , k   4. Rational  A rational number can be
Even Numbers A number is even numbers written as a quotient.
if it is a multiple of
two    a : a, b  , b  0 The decimal representations
{…,-4,-2,0,2,4,…}  b  are terminating or repeating.


Terminating:

Odd Numbers 2k 1,k  A number is an 1  0.5, 3  0.75
odd if it is not a 2 4
multiple of two Repeating:
{…,-3,-1,1,3,5…}
7  2.333...
3

0.45  0.454545...

Types Symbol Description & Examples Number Types Symbol Description & Example

5. Irrational  For an irrational number the 6. Real  The union of
numbers decimal representation is numbers rational numbers and
irrational numbers form
non-repeating. the real numbers system

3  1.732050808...  
e  2.71828182845...

  3.14159...

The relationship of number sets in a real number system Real Number,

 Rational Number,

Irrational Number,

  W Integer, 

Negative Integer, 
This diagram show that   W     Whole Number, W
and      Zero Natural Number,

Positive Integer, 

2

7/7/2022

Example 1 Example 2

For the set of {-5, -4, -3, -1, 0, 1, 2, 3, 8}, Given
identify the set of
S  9, 7,, 22,e,0,4,0.16, 2, 1 ,5.1212...
(a) natural numbers  75 3
(b) whole numbers
(c) prime numbers Find the set of:
(d) even numbers
(e) negative integers (a)  (b) W (c) 
(f) odd numbers
(d)  (e)  (f) 

Solution CONCLUSION

1.2 INDICES 

  W


W  16
and     

LEARNING OUTCOMES

At the end of the lesson, students are able
to :
(a) Define an index.
(b) State and apply the rules of indices.
(c) Solve equations involving indices.

18

3

7/7/2022

Definition 1 Definition 2
For all aR and nZ (positive integer), For all a R and n  Z ,

aa a........ a  an an  a  a  a ... a Positive
index
n factors of a a0 1 (a  0)
Zero
a = base a to the an  1 index
n = exponent or power or index power of an
an = an exponential expression Negative
n index

Definition 3 Indices

If a is a real number, m and n are integers for (a) a0 1, a  0

awhich n is real then ; (b) an  1
an
Ratio index

   am 1

an (c) an  n a
na m
 nm m

(d) an  n am  (n a)m

(a) a0 1, a  0 (b) an  1
an
EX 1 30  11
EX 43  1
22 43
1
2 1
64

4

1 7/7/2022

(c) an  n a m

1 (d) an  n am  (n a)m

EX 53  3 5 2

EX 73  3 72  ( 3 7)2

Rules of Indices Rules of Indices

1. a m  a n  a mn 1. a m  a n  a mn

2. am  a mn 56 56 11
an  3  33  3Example

 3 . a m n  a m n 2. am  a mn
an
4 . a b n  a n b n Example
25
5.  a n  an , b0 23  2 53  22
 b  bn

 3 . a m n  a m n 4 . a b n  a n b n

Example 11 Example   1 0 5    1 5 1 0 5

362  (62 )2  100000

6

5

7/7/2022

5.  a n  an , b0 Example 1
 b  bn
Without using calculator, find :

Example  1 2 2   7 2 9a) 3 b)  1 3
 5  5 2
  2
   25 

 72
52

 49
25

Solution:

Example 2 Solution:

Express the following in the simplest form :

a) 18 x 2 y 5 b) 3n2  9n  27n
3x4 y

6

7/7/2022

Example 1 Solution:
Solve the following equations.

a 5x  125

b  16 x  8
 81  27

 c 2x4x1  82x1

7

7/7/2022

Example 2 Solution:
Solve the following equations.

a  125x  25x23x1  0

b 22t  4(2t )  32  0

CONCLUSION

Rules of Indices

1 . a m  a n  a mn

2. am  a mn
an

 3 . a m n  a m n

4 .  a b n  a n b n

5.  a n  an , b0
 b  bn

8

TOPIC 1 At the end of the lesson, students should be
able to:
NUMBER SYSTEMS AND (a) Explain the meaning of a surd and its
EQUATIONS
conjugate.
1.3 SURD (b) Express surd in simplified form
(c) Perform algebraic operations on surds
(d) Solve equations involving surds

What is a SURD? RULES OF SURD
A surd is an irrational number in the form
of n a where n  and a  which is 1. a  b  ab
expressed in terms of root signs.
2. a  a
na bb

Irrational number which are SURD: 2, 7, 3 5 3. a b  c b  a  c b

Is 9 known No, because 3 is not an NOTE : a and b are positive real numbers
as a surd? irrational number.

RATIONALISING THE DENOMINATOR NOTE: The choice of multiplier is by no
means accidental. When more
When square roots occur in quotients, it complicated expressions need
is customary to rewrite the quotient so
that the denominator contains no square rationalising the multiplier is simply the
conjugate of the original denominator.
roots.
a b a b These
In rationalising the denominator of a expressions
quotient, be sure to multiply both the a ba b are conjugates
numerator and the denominator by the of one another

same expression.

a  numerator
b denominator

   Surd conjugate of a  b is a  b EXAMPLE

 a  b a  b Simplify :

a a b a bb a) 2 3  5 2  7 3  2 2
 a b b) 5 7  3 5  8 7  5

 a  b a  b  ab c)  3  22 3  2
 2

d) 3 3  5 2

SOLUTION c)  3  22 3  2

a) 2 3  5 2  7 3  2 2

b) 5 7  3 5  8 7  5

 2 EXAMPLE

d) 3 3  5 2 Rationalise the following :

(a) 14 (b) 1
7 7 2

(c) 2  3
2 1

SOLUTION (b) 1 (c) 2  3
7 2 2 1
(a) 14
7

EXAMPLE SOLUTION

Solve the following equations : (a) 3x 1  x 1

(a) 3x 1  x  1
(b) x  2  x  3  5
(c) 3x  5  x  2  x  6

(b) x  2  x  3  5

(c) 3x  5  x  2  x  6

At the end of the lesson, students should be able to:
(a) Define Logarithms.

TOPIC 1 (b) State and apply the laws of logarithms:

NUMBER SYSTEMS AND i) loga MN = loga M + loga N;
EQUATIONS
ii) loga  M  = loga M - loga N;
1.4 LOGARITHMS  N 

iii) loga MN = N loga M; and

iv) loga M =  logb M  .
logb a

(c) Solve equations involving logarithms.

Definition If a, n R+ , a  1.
If a, n  R+ and n = a x, then log a n = x
(1) log10 n  log n
where a 1. (2) loga a  1

If a=10, log 10 n = x “common logarithm” (3) loga 1  0 (4) aloga x  x

If a=e , log e n = x “natural logarithm” (5) loge n  ln n
(6) ln e 1

Laws of Logarithms Laws of Logarithms

If a, M, N  R+ , a  1 and b  R, then If a, M, N  R+ , a  1and b  R, then
(a) loga MN  loga M  loga N
(b) loga M  loga M  loga N
Example N

Example

Laws of Logarithms Laws of Logarithms

If a, M, N  R+ , a  1and b  R, then If a, M, N  R+ , a  1and b  R, then

c loga M b b loga M d loga M   logb M 
 logb a 
Example  

Example

Example Solution

(a) Express logb 2xy  5logb y 1 as a single a) logb 2xy  5logb y 1
logarithm.

(b) Express log a b3c in terms of

loga,logband logc.

Solution Example

b) log a b3c Solve the following equations:
a) log3(x2 10) 1 log3(x  2)

b) 2ln x  ln(6  x)  ln 3

a) log3(x2 10) 1 log3(x  2)

b) 2ln x  ln(6  x)  ln 3 Example

Solve the following equations:

a) 2 log x 2  log 2 x 1

b) log4 x2  log x 16  5

Solution b) log4 x2  logx 16  5

a) 2 log x 2  log 2 x 1

Example

Solve the following equations.

a) 5x  20 b) 2x  5

TUTORIAL

TOPIC 1.0 NUMBER SYSTEM AND EQUATIONS

1.1 Real Number

1 Given S  3.5, 3  2,0, 4, 7,,1.43,8,  2 , 2 3,0.68 , identify the set of
 5 

(a) natural number

(b) whole number

(c) integers

(d) rational numbers

(e) irrational number

(f) real number

Ans: refer to lecturer

1.2 Indices

1 Find the value of

(i)  0.49 1 Ans: 10
2 7

1 1 2 Ans: 625
(ii) 1252  25 4 


1 Ans: 343

 (iii) 7x  49x x

(iv)  27 y 1  y 1 Ans: 1
3 3 3

2 Simplify x3
Ans: 64 y6w3
(i)  xy2 3
  Ans: 49m10
 4w 

   (ii) m3n2 4 7m1n4 2

1

3 5 TUTORIAL

p 2  p4 71

 (iii) p6 3 Ans: p 4

 7  x  1 5
6
Ans: 7  x2
(iv)  8
3 Ans:  7
 7  x 5

3 Solve the following equations Ans: 1
2
(i) 9  3x   1 4x5
 3  Ans: 3
Ans: x  1,1
(ii) 4x  22x   1 3x1 Ans: x  6,2
16 

(iii) 4x  2x 6 16

 (iv) 32x1  3 10 3x

(v) ex2  e262x  0

(vi) x22x  2x  0 Ans: x  1,1

1.3 Surd
4 Express the following in simplest possible surd.

(i) 12 Ans: 2 3

(ii) 54 Ans: 3 6

(iii) 180 Ans: 6 5

(iv) 1183 Ans: 13 7
5 Rationalise the denominators and simplify

2

2 TUTORIAL
(i) 5
10
5 63 Ans: 5
(ii) 6 12
5 21
7 7 Ans: 12
(iii) 7  7
4 7
6 Simplify Ans: 3

(i) 3 2 22  3 2  Ans: 5 2  6
Ans:8 5
(ii) 8 5  2 20  80
Ans: 13 2 42
 2
55 3 3
(iii) 6  7
Ans:
(iv) 5 66
5 33
Ans: 2 6
7 Express the following as a single fraction 5
(i) 1  1
6 1 6 1 Ans: 28
11
(ii) 5  3  5  3
5 3 5 3 Ans: 18

1 1  Ans: 2 8 3  26
2 11
2
   (iii)
2 5 2 5

(iv) 1  13  2
2 3 1 13  2

3

8 Solve the following equations. TUTORIAL
(i) 7  m  5  m
Ans: 2
(ii) x  2  2x  3  8 Ans: 11

(iii) q  5  q  2  3 q Ans: q  6

1.3 Surd Ans: 2.704
9 Evaluate Ans:1
Ans: 4
(i) 4log3log25log4
7
log 40  log 1  log 0.5 Ans: 4
(ii) 8
1
(iii) log6 36  log6 1  log6 6 Ans: 2
6 Ans: 8

(iv) 1 log4 64  log4 0.25  log4 1
4

 1 ln 4

(v) e 2

(vi) 83log8 2

10 Express the following as a single logarithm. Ans: log2 1
m3
 (i) log2 m  3  log2 m2 9

 (ii)7 7
9 logp q  3log p r
 9

Ans: log p qr3

4

TUTORIAL

(iii) ln8  ln 9  ln 3  ln 1 Ans: ln192
24
Ans: logb c5
(iv) 5logb c  logb d  logb g dg
11 Find the possible value(s) of if
Ans: x  2,8
(i) log2 x  3logx 2  4 Ans: 3

 (ii) log2 4x2  7x 15 1 2log2 x 2
Ans: x  3
(iii) x5e3lnx  4x  21 Ans: x  3 e
(iv) 3ln 2x  3 ln 27
2
(v) log3 x  4logx 3  3 Ans: x  3, 1

81

12 Solve the following equations Ans: 2.58
Ans: 0.973
(i) 6  2x Ans: 2.31
(ii) e2x  7 Ans: 4.70
(iii) ex3  2 Ans: 0.371
(iv) 2x1 13 Ans: 0.419
(v) 75x2  32x1
(vi) 3x  63x1  0 Ans: 0.2519,0.8614
13 Solve

 (i) 52x 3 14 5x 8  0

(ii) 22x1  2x3  6  0 Ans: x  0,1.585

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TOPIC 2 LEARNING OUTCOMES

INEQUALITIES AND ABSOLUTE VALUES At the end of the lesson, students should be able to:
2.1 INEQUALITIES (a) relate the properties of inequalities.
(b) represent intervals, solution sets and their

representations on the number line.
(c) determine union and intersection up to three

intervals with the aid of number line.

INEQUALITIES TYPE OF INTERVAL

An inequality is a relation between two expressions that are not equal Type of interval Notation Inequality Representations on a
number line
and containing the symbols > , ≥ , < or ≤ .
ab
Symbols Meaning Open interval  a, b x : a  x  b ab
Closed interval  a, b x : a  x  b ab
> greater than Half-open interval  a, b x : a  x  b ab
Half-open interval  a, b x : a  x  b
≥ greater than or equal to
< less than

≤ less than or equal to

TYPE OF INTERVAL NOTE

Type of interval Notation Inequality Representations on a Empty circles, : the end points of an open interval on the number line.
number line Dense circles, : the end points of a closed interval on the number line.
Infinite interval
Infinite interval  a,  x : x  a a ∞
Infinite interval
Infinite interval a, x : x  a a ∞
Infinite interval  , b x : x  b -∞ b

 , b x : x  b -∞ b

, x :   x   - ∞ ∞

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Example 1: Solution: Type of interval Representations on a
State the type of the following intervals. Hence, represent it on number line
a real number line. Notation

(a) 1,4 (a) 1,4
(b) 2,5 (b) 2,5
(c) 2,2 (c) 2,2
(d) 1,
 1, 

(d)

Example 2: Solution:
Graph each of the following on a number line.
Inequality Representations on a number line
(a) x: 2  x  4, xZ
(b) x: x  5, xW (a) x : 2  x  4, xZ
(c) x: 3 x  5, x R x: x  5, xW

(b)

x: 3  x  5, xR

(c)

INTERSECTION AND UNION OF INTERVALS Example 3:
By using number lines, find each of the following.
Intersection and Union operation can be done on intervals.
For example if A = [1,6) and B = (-2,4) ; then (a) 0,5 4,7
(b) ,3 0,
B B (c) ,5 1,9
A A (d) 5,0 0,3

-2 1 4 6 -2 1 4 6

AB  1,4 AB  2,6

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Solution: Solution:

Example 4: Solution:
Graph the following intervals and solve each of the following.

(a) ,5 3, 3,3

(b) 5,3 0,3 3,0

Solution:

TOPIC 2

INEQUALITIES AND ABSOLUTE VALUES
2.1 INEQUALITIES

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LEARNING OUTCOMES PROPERTIES OF INEQUALITIES

At the end of the lesson, students should be able to: Property Example
(d) solve linear inequalities including compound
1. If a  b and b  c , then a  c . If 2  5 and 5  9 , then 2  9 .
inequalities.
(e) solve quadratic inequalities by graphical approach. 2. If a  b , then a  c  b c and If 2  5 , then 2 1 51 and

ac bc . 2 1 51 .

3. If ab c0 ac  bc 25 30 , then 23 53
a abnd , then If a53nd.
and 2
and c  . 3
c

PROPERTIES OF INEQUALITIES LINEAR INEQUALITIES

Property Example A linear inequality in one variable can be written in the

4. If a  b and c  0 , then ac  bc If 2  5 and 3  0, then following forms: a ax  b  0
and a  b . b ax  b  0
cc 23  53 and 2  5. c ax  b  0
3 3 d ax  b  0

TIPS: with a and b are real numbers and a  0.
The sign of inequality must be changed when it is multiplied
or divided by a negative number.

Example 1: Example 2:
Find the values of x which satisfy the inequality 3x 12  0. Find the values of x which satisfy the inequality
Solution : 4x  3  2x 11 .

Solution :

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Example 3: Example 4:
Find the values of x which satisfy the inequality
10  5 3x  2 . Find the values of x which satisfy the inequality
x  20  3x 8  4 .
Solution :
Solution :

QUADRATIC INEQUALITIES Example 5:

A linear inequality in one variable can be written in the Find the values of x which satisfy the inequality
x2 5x  4  0 .
following forms: a ax2  bx  c  0
b ax2  bx  c  0
c ax2  bx  c  0
d ax2  bx  c  0

with a, b and c are real numbers and a  0 .

Solution: Example 6:

Find the values of x which satisfy the inequality
9x  4  2x2  0 .

5

Solution: 7/7/2022

TOPIC 2

INEQUALITIES AND ABSOLUTE VALUES
2.2 ABSOLUTE VALUES

LEARNING OUTCOMES LEARNING OUTCOMES

At the end of the lesson, students should be able to: At the end of the lesson, students should be able to:
(a) state the properties of absolute values. (c) solve linear absolute inequalities in the form of:
(b) solve absolute equations in the form of:
i. ax  b  cx  d
i. ax  b  cx  d ii. ax  b  cx  d
ii. ax  b  cx  d ; and iii. ax  b  cx  d ; and
iii. ax2  bx  c  d iv. ax  b  cx  d

ABSOLUTE VALUES DEFINITION ABSOLUTE VALUES
The absolute value of a, denoted by a is the distance of a point
on the number line whose coordinate is a from the origin. i. a  a if a0
if a0
  a


-∞ -a a a ∞
0 a
Example:
ii. a  a
i. 3  3
ii. 2  2

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Example 1: Solution:
Write the following without the absolute value symbol:
(a) x  5
(b) 2  x

PROPERTIES OF ABSOLUTE VALUES EQUATIONS INVOLVING ABSOLUTE VALUE

Property Example Geometrically, x  a, a  0 represents the distance of x from
2  2  0; 2  2  0 the origin is a units.
1. a  0 3  3  3
x x a ∞
a  a 23  32  5 5 -∞ -a 0
2. 2 3  1 1; 3 2  1 1
Therefore x  a is equivalent to x  a or x   a .
ab  ba 23  6  6; 2 3  23  6
3.
10  5  5; 10 =10  5
ab  ba 2 22
4.

ab  a b

5. a  a
6. b b

Example 2: Solution:
Solve the following equations.
(a) 2x 1  4x  3
(b) 2x 1  4x 3
(c) x2  6x  4  4

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Solution: Solution:

ABSOLUTE VALUE INEQUALITIES ABSOLUTE VALUE INEQUALITIES

Geometrically, x  a represents the distance between x and the Geometrically, x  a represents the distance between x and the
origin is less than a units. origin is greater than a units.

-∞ -a 0 a∞ -∞ -a 0 a∞

Therefore x  a is equivalent to  a  x  a or x  a and x   a . Therefore x  a is equivalent to x  a or x   a .

ABSOLUTE VALUE INEQUALITIES Example 3:
Solve the following equations.
Inequality Equivalent Form Representations on a number (a) 3 x  2x 9
line (b) 2x  x 3
x a a  x  a (c) x  2 103x
x a a  x  a -a a (d) 52x  13x
x a x  a or x  a -a a
x a x  a or x  a -a a
-a a

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Solution: Solution:

Solution: Solution:

9

TUTORIAL

TOPIC 2.0 INEQUALITIES AND ABSOLUTE VALUES

2.1 Inequalities

(a) State the solution for each graph:

(i)
-3 6

(ii)
05

(iii)
-2 4

(iv)
17

(v)
-5 -1

(vi)
28

(b) Find the intersection and union for each pair of graphs: 
       

(i)
49

(ii)
-3 2

(iii) -1
-6

(iv)
-1 3

(v)
2

1

TUTORIAL

(c) Find the solution set of the following inequalities.

(i) 7 5x  3 Ans: x : x  4
 
5 

(ii) 4x  3  5 Ans: x : x  343
6

(iii) 5  1 x  2  1 x Ans: x : x   425
53

(iv) x  2  2x  7 Ans: x : x   25
64  
4 

(v) 3 x  3  5x  2 Ans: x : x  50 
54 13

(vi) 11 x  x  4 Ans: x : x   5
4  3

(d) Solve the inequalities. Write your answer in interval notation.

(i) 2  4x  6 18 Ans: 2,3

(ii) 3  7x  5  0 Ans:  1 , 5 
2 7 7 

(iii) 1 1 x  4  4 Ans: 25, 40
5
Ans: 6, 4
(iv) x  20  3x 8  4

(v) 4x 8  x 1 6x  2 Ans:  1 , 3
5

2

TUTORIAL

(e) Solve the inequalities. Ans: ,4 6,
(i) x2  2x  24
(ii) 3x2 5x 12  0 Ans: 3, 4 
3 
(iii) 4x2 12x 5  0
Ans:  1 , 5
(iv)  x 92  0  2 2 

(v) 3x2 5x  4  2x2  6x 8 Ans: ,9 9,

(vi)  x  22x  3 10x Ans: 3,4

2.2 Absolute values Ans:   1 , 6
(a) Solve  2
(i) 5 x 1  3
2 Ans:  8 , 4
(ii) 2  5x 17 55
(iii) 2x  3  4x  4
(iv) 3 2x  3x 5 Ans: 3,19
(v) 3x  4  7x 1 5
(vi) 2  4x  3x  4
(vii) x2  2x  4  4 Ans: 7
(x) 2 3x  x2  2 2

Ans: 2
Ans:  3 , 5

10 4
Ans:  2 ,6

7
Ans: 2,0,2,4

Ans: 4,3,0,1

3

TUTORIAL

(b) Solve the following inequalities Ans: x : x  2 or x  5
(i) 3 2x  7
Ans: x : 10  x  2 
(ii) 3x  4  6  2x 5

(iii) 3 x  2  x  3 Ans: x : x   270
4
Ans: x : x  3
(iv) x  4  2  3x
(v) x 1 1 1 x Ans: x : 3  x  0

3 Ans: x : x 10
(vi) 2 x  3x 10
Ans: x : 6  x  43
(vii) 5  x  2x 1

4

TOPIC 3.0 MATRICES AND 7/7/2022
SYSTEMS OF LINEAR
EQUATIONS LEARNING OUTCOMES

3.1 MATRICES At the end of this topic, students should
be able to :

(a) identify the different types of matrices.
(a) perform operations on matrices.

Matrix Definition or in compact form A=[ aij]

A matrix is a rectangular array of numbers
enclosed between brackets.

The general form of a matrix with m rows and • The numbers that makes up a matrix

n columns : are called its entries or elements, aij ,
where i indicates the row and j indicates
 a11 a12 a13  a1n 
 
 a21 a22 a23  a2n 

A   a31 a32 a33  a3n  m rows the column
      
  Example

am1 am2 am3  amn  Given 1 3
1 8
n columns a12  3 and a22  8

• The order or dimension of a matrix with 1. Row Matrix is a (1 x n) matrix [one row]
m rows and n columns is m x n. A = [ a11 a12 a13 … a1n]
Example:
Example
0 1 3 A=[1 2]
4 1 8 B=[1 0 7 8]

order: 2x1 order: 2x2

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2. Column Matrix is a (m x 1) matrix 3. Square Matrix is a (nxn) matrix which
has the same number of rows as
[ one column ] columns.

 a11 
 
 a21  Example

A=  a31  Example
 
 . 1 3
A = 1 8 , 2 x 2 matrix
 .
 2
 .
0  3  1 3 2
am1  A = 4 A =  5 
  B = 3 1 2 , 3 x 3 matrix

 7  2 3 1
 

4. Zero Matrix is a (m x n) matrix which 5. Diagonal Matrix is a square matrix where all

every entry is zero, and denoted by O . the elements are zero except those in leading

diagonal

a11 0 0  0

Example Let A =  0 a22 0  0 
 
0 a33 
0   0
 
0 0 0 0 0 0 0    
0 0
O = 0 0 0 O =  0 0  O=  0 0  amm 
 

0 0 0  0 0  The diagonal entries of A are a11,a22 ,….,amm

Example 6. Identity Matrix is a diagonal matrix
where all its diagonal entries are 1 and
denoted by I.

A= 2 0 1 0 0 Example
0 3 ,B =0 2 0
0 3 1 0 1 0 0
0 0 1 = I2x2

 0 1 0  = I3x3
 

 0 0 1 

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7. Lower Triangular Matrix is a square matrix 8.Upper Triangular Matrix is a square matrix
and aij = 0 for i < j and aij = 0 for i > j

Example Example

1 0 0 a 0 0 1 2 3 a b c 
P = 0 2 4
A = 3 2 0 ,B = b f 0 R = 0 d e 
3 2 3 0 0 3 
c d e
0 0 f 

Operations on Matrices Example 1

Addition And Subtraction Of Matrices Consider the matrix A and matrix B as below:

The addition or subtraction of two 1 23 −2 0 2
matrices is only defined when they = −1 0 2 and = 1 −1 0
have the same order.
0 −2 3 4 20

Find

(a) A + B ,

(b) B – A

Solution (a) Solution (b)

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Scalar Multiplication Example 2

If c is a scalar and A [aij ] 1 2  4 3
A3 4 B5 6
then cA  [ caij ]
,
.

Find 3A – 2B

Solution Extra Example
Solution
Given that matrix A and matrix B:

1 23 −2 0 2
= −1 0 2 and = 1 −1 0

0 −2 3 4 20

Find 2A + 3B.

Multiplication of Matrices

The product of two matrices A and B is
defined only when the number of
columns in A is equal to the number of
rows in B.
• If order of A is m x n and the order of B is

n x p, then AB has order m x p.
• AB≠BA

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Amxn Bnxp ABmxp ATTENTION !

The order of the A row and a column must
have the same number of
product is m x p
entries in order to be
m×n n×p multiplied.

These numbers must
be equal

Multiplication Of Two Matrices Example 3

 b1  0

  Let A  2 5 1 and B   3  ,
   
b
2  2 

Ra1 a2 a3  an C   b3  find AB.
  
Solution
 

bn  A1x3 B3x1 AB1x1

RC  [a1b1  a2b2  a3b3  anbn ]

Solution

Example 4

1 2 1 2 1 2

Let A  3 4 5 and B3 2 1 
 

 2 3 1 1 4 1

Find AB.

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Example 5 Solution

Given that

1 2 1 9 9 x

P  3 4 5 and P2  19 y 28
 9 19 18
 2 3 1

Find the values of x and y.

CONCLUSION

Types of matrices Operation

-Row -Addition
-Column -Subtraction
-Square -Multiplication
-Diagonal
-zero MATRICES
-Identity
-Lower triangle
-Upper triangle

TOPIC 3.0 MATRICES AND LEARNING OUTCOMES
SYSTEMS OF LINEAR
EQUATIONS At the end of this topic, students should
be able to :
3.1 MATRICES
(c) determine the transpose of a matrix.
(d) apply the properties of transpose.

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Transpose Matrix Example 6

Definition 2 BT
The transpose of a matrix A , written Let B  1 then

as AT, is the matrix obtained by 331
interchanging the rows and columns of A.
That is, the 1 3 3
i th column of AT is the i th row of A for If D  2 5 4 then DT
all i’s.
1 3 5

Example 7 Solution

Let 3 4 , and C  1 4
B  2 1 3 2

Find CTBT.

Properties of transpose Properties of Transpose
(A ± B)T = AT ± BT
(AT)T = A i == =.
(AB)T = BTAT  AT BT
(kA)T = kAT , k is a scalar Example (i)

12 3
If = 0 1 −1 , verify that

2 0 −3

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Solution Properties of Transpose
12 3
)  ± = ±
If = 0 1 −1 , then Example (ii)
2 0 −3
If 1 2 3 4 56
Solution
= 0 1 −1 and B= −1 0 1 ,

2 0 −3 5 30

verify that + = + .

Properties of Transpose Properties of Transpose

)   = , k is a scalar (iv) =
Example (iv)
Example (iii)

If B= 6 3 , verify that 2 =2 . Given the matrices
−1 2
131 −1 −2 0
Solution = 2 −1 0    = 4 0 1

042 1 32

show that = .

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Solution

CONCLUSION TOPIC 3.0 MATRICES AND
SYSTEMS OF LINEAR
Types of matrices Operation EQUATIONS

-Row -Addition 3.2 Determinant of Matrices
-Column -Subtraction
-Square -Multiplication
-Diagonal
-zero MATRICES
-Identity
-Lower triangle
-Upper triangle

Transpose

LEARNING OUTCOMES YOUR PREVIOUS KNOWLEDGE

At the end of this topic, students should Determinant of a 2x2 matrix
be able to :
(a) determine the minors and cofactors of a a b  ad  bc
A
matrix. cd
(b) determine the determinant of a matrix by

using expansion of the cofactors.

Example

35
A 1  3  5  8
1

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MINOR COFACTOR
The cofactor, c ij , of the element aij is
Let A be n x n matrix , A = aij nn
cij = (-1)i+j mij
The minor, mij , is the determinant of
(n -1) x (n -1) matrix obtained by

deleting the i th row and j th column of A

a11 a12 a13  a11 a12 a13 a22 a23
A a21 a22  m11  a21 a22 a23 a32 a33
a32 a23  a32 a33 
a31 a31
a33 

 a22a33  a23a32

EXAMPLE 1 SOLUTION

1 2 1 Minor of a11 , m11 , is the  1 2 -1 
determinant of 2x2 matrix 3 
Let A  3 4   1
 obtained by deleting the
2 , A   4 2
first row and first column  

 4 3

 


1 4 3  of A.

Find (i) the minors of ,     . m11 = 42 =

43

(ii) the cofactors of ,     .

2 1 1 2 -1  The cofactor, cij , of the element aij
m21 = 4 3 
 cij = (-1)i+j mij
cofactor of a11
3A  4 2
  c11 = (-1)1+1 m11

1 4 3 cofactor of a21

  c21 = (-1)2+1 m21


= (2 x 3) – (-1 x 4) =

32 1 2 -1 
m12 = 1 3 


3A  4 2
 

1 4 3

 


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