MODUL AM015 2021 INTAKE

AM015 MODUL

MATHEMATICS FOR ACCOUNTING
2021 INTAKE

BIL TOPICS
CHAPTER 1 NUMBER SYSTEM AND EQUATIONS
CHAPTER 2 INEQUALITIES AND ABSOLUTE VALUES
CHAPTER 3 SEQUENCES
CHAPTER 4 MATRICES AND SYSTEMS OF LINEAR EQUATIONS
CHAPTER 5 FUNCTIONS AND GRAPHS
CHAPTER 6 POLYNOMIALS
CHAPTER 7 LIMITS
CHAPTER 8 DIFFERENTIATION
CHAPTER 9 APPLICATIONS OF DIFFERENTIATION

ASSESSMENT TOPICS
INDIVIDUAL ASSIGNMENT CHAPTER 3 & CHAPTER 4
GROUP ASSIGNMENT PART A: CHAPTER 6 & CHAPTER 7
PART B(GROUP DISCUSSION): CHAPTER 7
UPS 1 CHAPTER 1 & CHAPTER 2
UPS 2 CHAPTER 3 &CHAPTER 4
UPS 3 CHAPTER 5 & CHAPTER 6

PSPM I (100 MARKS)

PART A PART B

(3 QUESTIONS, 25%) (7 QUESTIONS, 75%)

CHAPTER 7 CHAPTER 1

CHAPTER 8 CHAPTER 2

CHAPTER 9 CHAPTER 5

CHAPTER 8

CHAPTER 9

LET’S STRIVE FOR A!!!

AM015/ 1. Number System and Equations

CHAPTER 1: NUMBER SYSTEM AND EQUATIONS

LECTURE 1 OF 3
At the end of lesson, student should be able to:

a) Define natural numbers (N), whole numbers (W), integers (Z), prime numbers,

rational numbers (Q) and irrational numbers ( Q ).

b) Represent the relationship of number sets in a real number system

diagrammatically showing N ⊂ W ⊂ Z ⊂ Q and Q ∪ Q =  .

c) Represent open, closed and semi-open intervals and their representations on
the number line.

d) Find 'union' and 'intersection' of two or more intervals with the aid of number
line.

e) Express the rules of indices
f) Solve equations involving indices.

1.1 REAL NUMBERS

The Real Number (  )

 The real number consists of rational numbers and irrational numbers.

Natural Numbers (N)
 is the set of counting number.

 including prime and non prime number N  1, 2,3,...

The Whole Numbers(W)

 are the set of natural numbers together with the number 0. W  0,1, 2,3,...

Prime Number
 are the natural number that greater than 1 and can only be divided by himself

and 1. Prime number  2,3,5,7,11...

Integers (Z)
 The whole numbers together with the negative of counting numbers form the
set of integers and denoted by Z.

Z  ...  3, 2, 1, 0,1, 2,3,...

Even Number

 Are the integers in the set ..., 6, 4, 2, 0, 2, 4, 6,... which can be

represented in the general form {2k : k Z}

Odd Number

 Are the integers in the set ..., 5, 3, 1, 1, 3, 5,... which can be

represented in the general form {2k 1: k Z}

A Rational Number ( Q )

 is any number that can be represented as a ratio (quotient or fraction) of two

integers and can be written as Q a ; a, b  Z, b  0 .
 
 b

 Rational number can also be expressed as terminating or repeating decimals.

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AM015/ 1. Number System and Equations
 For example 5, 3 , 0.25 and 0.333… .

2

 Irrational Number Q

 is the set of numbers whose cannot be written as a fraction.
 their decimal representations are neither terminating nor repeating.

 for example 3, 5 , 1.41421356…and  .

Example 1
Rewrite the following numbers into decimal numbers.

a) 0.13

b) 1.236
Example 2
Rewrite the following numbers into bar form.

a) 0.454545...

b) 2.5373737...

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AM015/ 1. Number System and Equations

Relationship of Number Sets

The set of real numbers, denoted by the symbol  , consists of all rational numbers

and irrational numbers. Relationships among the subsets of real numbers are
illustrated in the diagram below.

Real number

Rational Number Irrational Numbers

Integers Non- Integers

Negatives of natural Whole Numbers
numbers

Zero Natural numbers

prime numbers Non-prime numbers

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AM015/ 1. Number System and Equations

The relationship of number sets in a real number system can also be represented by
the following Venn Diagram


QQ

W 

From the diagram, we can see that:

1. N W  Z  Q  

2. Q Q  

Example 3 7 , 1 , 2, 0, 3, 4, 5.1212…}, identify the set of
Given S = {-9, 3

a) whole numbers

b) integers

c) rational numbers

d) irrational numbers

e) prime number

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AM015/ 1. Number System and Equations

The Number Line
The set of numbers that corresponds to all point on number lines is called the set of
real number. The real numbers on the number line are ordered in increasing
magnitude from the left to the right
For example for –3.5, 2 and  can be shown on a real number line as

3

4 3 2 1 01 2 34
3.5
2 
3
Example
Symbol Description 3=3
a=b a equal to b 4 < 4
a<b a less than b 5>0
a>b a greater than b

Note: The symbols ‘<’ or ‘>’ are called inequality sign

All sets of real numbers between a and b, where a  b can be written in the form of
intervals as shown in the following table.

Type of Interval Interval notation Solution set Representation on the number line
Closed ab
a,b x : a  x  b

Open a,b x : a  x  b a b
Half-open a b
Half-open a,b x : a  x  b a b
a,b x : a  x  b
Infinite interval a b
 ,b x : x  b a b
 ,b
a,  x : x  b
x : x  a
a,  x : x  a
x : x
(, )

Note:
The symbol  is not a numerical. When we write [a, ), we are simply referring to the
interval starting at a and continuing indefinitely to the right

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AM015/ 1. Number System and Equations
Intersection and Union for Two Intervals
Since intervals are sets of real numbers, we can combine two or more intervals by
using the set of operation of union and intersection.

a) the union of A and B, denoted by A  B , is the set of all elements which

belong to A or to B.

b) the intersection of two sets A and B, denoted by A  B , is the set of

elements which belong to both A and B.
Example 4
Solve the following using the number line

a) 0, 5   4, 7
b) 5, 1 2, 4

c) , 7 7, 

d) , 4 4, 

Example 5

Given A  (9,) , set B  x : 7  x 13, x  Z. Solve the following using the number

line

a) A  B

b) A  B

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AM015/ 1. Number System and Equations

1.2 INDICES

Index indicates the number of times the base is used as a factor.

an

Where
The number a is called the base, n is called the index,
an is read as ‘a’ to the power ‘n’.

Rules of Indices

 1. am  an  a mn 2. am  an  am  amn
an

 3. am n  amn 4.  abm  ambm

5.  a m  am ,b  0 6. a0  1, a  0
 b  bm

1 m m
am
a n  n am  na
7. am   8.

9. am  an  m  n

Example 6 b) 0.04 3
Without using calculator, evaluate: 2

3

a) 9 2

Example 7 b) (a 3b) 2
Simplify: a 8b 2

35  36
a) 34

Example 8

By using the rule of indices, evaluate ( 3)3  1 1 .

27 4 3 4

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AM015/ 1. Number System and Equations

Example 9 b) 27x1  9x1

Find the value of x for the following equation:

a) 5x  125

Example 10 b) 4x  6(2x )  16
Solve the following equation:

 a) 52x1  6 5x 1

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AM015/ 1. Number System and Equations

LECTURE 2 OF 3

At the end of lesson, student should be able to:

a) State the laws of logarithms such as:

i. log MN  loga M  loga N
a

ii. loga M  log M  loga N and
N a

iii. N  N loga M

loga M

b) Change the base of logarithm using loga M  logb M .
log a
b

c) Solve equations involving logarithms.

1.2 LOGARITHMS

Logarithm is a number y,(y  0) for any given base a a  0 and is written as

loga y  x where y  a x and x .

Common Logarithm

Common logarithm is a number y (y > 0) for base 10 and is written as log10 y  x
where y  10x and x .

Note : log10 y  log y  lg y

Natural Logarithm of x
The natural logarithms is a number y for base e and is written as loge y  x where
y  ex and x .

Note : loge y  ln y

Example 1
Write in the logarithmic form

a) 23  8

Example 2
Write in the indices form
a) log5 25  2

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AM015/ 1. Number System and Equations

The Laws of Logarithms

RULE 1

loga MN  loga M  loga N Product Rule

RULE 2

loga M  loga M  loga N Quotient rule
N

RULE 3

loga M N  N loga M Power Rule

The following table lists the general properties for logarithmic form.

Logarithms with base a Common logarithms Natural logarithms
ln1  0
1) log a 1  0 log1  0
log10 1 ln e 1
2) log a a  1

3) log a ax  x log10x  x lnex  x

4) aloga (x)  x 10log(x)  x eln(x)  x

5) log a  1    log a (N ) log  1    log(N ) ln  1    ln(N )
 N   N   N 

6) log a m  log a n  m  n logm  logn m  n ln m  ln n  m  n

Example 3

Express in terms of logx,logy and logz

a) logxyz

Example 4
Express the following in a form log a and log b.

a) log ab2

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AM015/ 1. Number System and Equations

Example 5
Write the following as single logarithms:
a) log2 8  log2 6  log2 12

Change of Base

loga M  logb M log a M  log M M  1
logb a log M a log M a

Example 6

Find the following expression to four decimal places.

a) log3 5 b) log5 10

Example 7 b) log2 x  log2  x  7  3

Solve the following equation:

 a) 22x3  25 2x  3  0

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c) log3 x  4logx 3 3  0 AM015/ 1. Number System and Equations

d) 2ln 4x  2  ln 9

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AM015/ 1. Number System and Equations

LECTURE 3 OF 3
At the end of lesson, student should be able to :

a) Explain the meaning of a surd and its conjugate.
b) Perform algebraic operations on surds.
c) Solve equations involving surds.

1.2 SURD
A number expressed in terms of root sign is radical or a surd. Surds cannot be
evaluated exactly.
For example,
“ 4  2 and 3 27  3” = radical.
“ 7 or 3 71 ” = irrational numbers (surds)

Rules of Surds

BIL PROPERTY EXAMPLE
1 a a a
2. a  b  ab a, b  0 5 5 5

2  3  2(3)  6

3. a  a ,a,  0, b  0 3 3
22
bb
2 2  5 2  2  5 2  7 2
4. a b  c b  a  c b

5 a b c b  a c b 6 2  3 2  6  3 2  3 2

 6  2
2 2  3 22 6 3

a  b  a  2 ab  b

7  a  b a  b  5  2 5  2

 2  2  2  2

a b 5 2

 a b 52

3

Remark: a b  a  b b)  3  4 2  3  2 2 

Example 1
Simplify:
a) 5 3  27

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AM015/ 1. Number System and Equations

Rationalizing The Denominator is a rational

  The conjugate of a  b is a  b where a  b a  b  a  b

number.

To rationalize a denominator
Multiply the numerator and the denominator of the same expression that will result in
the denominator to become a rational number.

If Denominator Multiply by To Obtain Denominator Free
contains the factor the conjugate
3 from surds
3
1 3  2  3 3 3
1 3
3

 12  2

3  1 3  2

3 2 3 2  32  2
5 3 5 3
2 927

   5 2  3 2  5  3  2

Example 2
Rationalize and simplify:

1 b) 17  5
17  5
a)

7 2

c) 5  3  1 3 d) 1 2  1 2
5  3 1 3 1 2 1 2

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AM015/ 1. Number System and Equations

Surd Equation
Example 3
Solve each of the following equation:

a) 3x 1 1  x

b) x  x  2  2  0

c) t  7  t  2  2t  3

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AM015/ 1. Number System and Equations

EXERCISE:

1. Represent the followings on the real number line and state its type.

a) [-1, 4] b) {x : 2  x  5}

c) [2, ) d) {x : x  0, x  }

e) (10, ) f) (,4)

2. Given X   5,10 and Y  0,. Find

a) X  Y Ans: 0,10
b) X  Y Ans: 5,

3. Simplify: 6y4
Ans: x 2
18x 2 y 5
a) 3x 4 y a4x

b)  a2b3 3  x  2b1  Ans:
 x1 y2  a2 y3 
   b10 y9

1 1 Ans: x 1
x
x2  x 2
Ans: 1
c) 1 x10 y9

x2 Ans: pq
q p
d) (3x 2 y 3 )3
27x 4

p 1  q 1
e) p 2  q 2

4.Find the value of x for the following equation:

2 Ans: x  8
Ans: x  1 or x  2
a) x3  4
2
b) 49x2  1
73x2

5.Write the following as single logarithms:

a) 1 log 25  2log 3  2log 6 Ans: log20
2

b) 2ln  x  7  ln x Ans: ln  x  72 
 
 x 

d) log a 3xy  5log a y  2 Ans: log a  3xa2 
 y4 
 

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6. Solve the following equation: AM015/ 1. Number System and Equations

 a) 4x1 5 2x 1 0 Ans: x  2or x  0
Ans: x  1.7095
b) 2x1  3x Ans: x  ln4
c) e2x 3ex  4  0 Ans: x  3
Ans: x  1.4307
d) x5e3lnx  4x  21 Ans: x  9
e) 5x  10 Ans: x  1

f) lnx  2  ln3x 16 2

g) log8 x3  log x 8  2 , x  8 Ans: 3 5
Ans:8 7
7. Simplify: Ans: 5 42
a) 45 Ans: 4 21  3
b) 6 7  2 7
c) 3 6  5 7

d) 3 4 7  3

8. Rationalize and simplify: Ans: 6 2
a) 12

2

b) 3 Ans: x  3
23 2

c) 2  5  5 Ans: 1  5 5
3 5 44

9. Solve each of the following equation: Ans: x  13
a) 2x 1  5  0

Page 17 of 17

AM015/1 .Number System And Equations

TUTORIAL CHAPTER 1 : NUMBER SYSTEM AND EQUATIONS

TUTORIAL 1 OF 3

1. Determine whether each statement is true or false

True/false

a) Every counting number is an integer
b) Zero is a counting number
c) Negative six is greater than negative three
d) Some of the integers is natural numbers

e) 3,5  x : 3  x  5

2. List the number describe and graph them on the number line

(a) The counting number smaller than 6 (b) The integer between -3 and 3

Given S  3,0, 7, 1 , e, 4,8
3
3. identify the set of

(a) Natural numbers (b) whole numbers (c) integers

(d) Rational numbers (e) irrational numbers

4. Write each of the following inequalities in interval notation or set notation and show them
on the real number line.

(a) x : 2  x  6 (b) x : 3  x  7 (c) x : x  3

(d) x : x  1 (e) (4, 4) (f) (,5]

(g) (2, 0)  (3, 6) (h) [6, 2)  [3, 7) (i)  ,33,

Page 1 of 6

AM015/1 .Number System And Equations

TUTORIAL 2 OF 3

1. Evaluate

1 2  100   3 11 1. 1
3 2
(a) 27 (b)  27  3 (c) 9 2.8 2 9 3 .27 2
8 9
(d) 1 (e) 1  2

22 3 6.3 3

2. Simplify the following expressions:

(a) 3n2 9n  27n 4n  2 n 1 n

(b) 83 164

n 1 (d) 5n1 10n  202n 23n

(c) 9 2  3n3  32 5

3. Solve the following equations:

(a) 3x  243 (b) 2x1  1 (c) 4x2 1281x
64 (f) (4x ) x  4  8x

(d) 2  1 (e) 7x2  4962x  0

4x 3 9

4. Solve the equations: (b) 32x1  26(3x )  9  0 (c) 4x  6(2x ) 16  0

(a) 2(22x )  5(2x )  2  0

5. Without using calculator, evaluate

(a) log 4 64 (b) log 1 4 (c) log 125 25 (d) ln e2
log 9
2
(g)
(e) e 2ln3 log 125
log 3
(f)

log 5

6. Simplify

(a) log2  log6  log4 (b) 2log3  log2 (c) 1 log 25  2log 3  2log 6
2

(d) ln y3  1 ln( x3 y6 )  5ln y (e) 2 ln x  4ln 1   3ln( xy)
3 y

7. Solve the equations:

(a) 22x  5 (b) 33x1  7 (c) (5x )(5x1 )  10 (d) e2lnx  9

(e) eln(1x)  2x (f) 3e2x  75 (g) 5x  40 (h) e2x  ex  2  0

e2

Page 2 of 6

8. Solve these equations: AM015/1 .Number System And Equations

(a) 2log(x  2)  log(2x  5) (b) 2ln x  ln3 ln(6  x)
(c) log 2 (2x  4)  2  log 2 (x2  6) (d) log( x2  6)  log( x 1)  1
(e) ln x  2  ln(1 x) (f) log 3 x  2log x 3  1
(h) log 2 2x  log 4 (x  3)
(g) log 2 x  log x 2  2
(i) log 2 (2x1  32 2)  x

TUTORIAL 3 OF 3

1.Express in terms of the simplest possible surds:

(a) 8 (b) 75 (c) 180 (d) 125

2.Simplify: (b) ( 2 1)( 2 1) (c) ( 3  2)( 3 1) (d) (2 5  3)(3 5  2)

(a) 2(3 2)

3.Rationalise the denominators and simplify in the form a  b c :

3 1 11
(a) 2  3 (b) 3  2 5 (c) 2  1 2 1

4.Solve the equations:

(a) 3x 1  x 1 (b) 4x  9 1  2 x (c) 4x 13  x 1  12  x

ANSWERS :TUTORIAL CHAPTER 1

TUTORIAL 1 OF 3

1 (a) True (b) false (c) false (d) true (e) false

2 (a) N = {1, 2, 3, 4, 5} graph (b) Z = {-2, -1, 0, 1, 2} graph

3 (d) 3, 0, 1 , 4, 8 e) 7,e
3
(a) 4,8 (b)0, 4,8 (c)3, 0, 4,8

4 (a) 2,6 ( b) 3,7 (c) ,3

(d) 1, (e) x : 4  x  4 (f) x : x  5

(g) x : 2  x  03  x  6 (h) x : 3  x  2 (i) { }

graph on real number line - your own answers

Page 3 of 6

AM015/1 .Number System And Equations

TUTORIAL 2 OF 3

11 9 27
(c) 1000
(a) 3 (b) 4 (d) 6 (e) 1

2 (a) 9 (b) 2n (c) 27 (d) 5
2

3 (c) 1 (d) -216 , 216 (e) -6, 2 ( f)  1 ,2
(a) -5 ( b) -7 3 2

4 (a) x  1,1 (b) x  2 (c) x  3

5 (a) 3 (b) 2 (c) 2 (d) 2 (e) 3e2 (f) 3 (g) 2
3

6 (d) ln x (e) ln y 
x
(a) log3 (b) log18 (c) log20

7 (a) 1.1610 (b) 0.9237 (c) 1.2153

(d) 3 (e) 1 (f) 1.6094 (g) 1.4756 (h) 0
3

8 (a) 3 (b) 3 (c) 5 e2 (f) 9 or 1 (i) 11
2 (e) (1  e2 ) 3 2
(d) 8, 2 (g)2 (h) 1

TUTORIAL 3 OF 3

1 a) 2 2 b) 5 3 c) 6 5 d) 5 5

2 a) 3 2  2 b) 1 c) 5  3 3 d) 24  5 5

3 b) 2 5  3 c) 2 2
11 11
a) 6  3 3

4 b) 25 c) 3
a) 5 4

Page 4 of 6

AM015/1 .Number System And Equations
EXTRA EXERCISE CHAPTER 1 : NUMBER SYSTEM AND EQUATIONS

1) (a) Simplify

 32n3 18 3 2n1

 5 3n 2

(b) Without using the calculator, evaluate

1

log10 8  log10 272  log10 5

3 log10 6  log10 5
2

(c) Solve

x  5  4x 13

2) Solve the equation 2log x 3  log 3 x  3  0

3) Given p  q 3  5 where p and q are integers. Find the value of p and q without
2 3

using calculator

4) (a) Solve the equation 9x  4  5(3x )

(b) Solve the equation 4x 16  6(2x )

5) (a) Solve the equation log 2 2x  log x 2  log 4 26

(b) Given 3  2 2  a  b 2 ,find a and b where a,b 

1

643 p6q 2r10

6) Simplify
2
8 3 p 2q 10r 2

7) Find the values of x which satisfies the equation log 2 (5  x)  log 2 (x  2)  3  log 2 (1 x)
8) Solve the equation 2(32x1)  3  7(3x )

11

9) (a) Evaluate   11  2  11  2
2 2
3  3 without using calculator.

Page 5 of 6

AM015/1 .Number System And Equations

(b) Show that

1  1 1
log p pq log q pq

10) Find the values of x that satisfies the equation (32x1)  32  28(3x )

11) Solve the equation log 2 x  log 4 8x  6

ANSWERS:EXTRA EXERCISE CHAPTER 1

1) (a)5 (b)1 (c)4 6) (pqr)8

2) x=9,x=3 7) x =3

3) p 175,q  100 8) x  1

4) (a) 0,1.262(3dp) 9) (a) 1 (b) shown
2
(b) 3
10) x=-1, x=2
5) (a) 2

(b) a  1,b  1

11) x=8

Page 6 of 6

AM015/ 2. Inequalities And Absolute Values 2021/2022

CHAPTER 2 :
INEQUALITIES AND ABSOLUTE VALUES

Lecture 1 of 2

2.1 INEQUALITIES

At the end of lectures, you should be able to
(a) relate the properties of inequalities
(b) find linear inequalities
(c) find quadratic inequalities

A. Relate the properties of inequalities

Inequalities involved the use of symbols: , , , 

Suppose a , b, and c are any numbers, we can state

the properties of inequalities as in the following table:

If a b, then Example

1 acbc,c 0 43

4232

2 ac bc,c 0 65
43
42 32

3 acbc,c0 86
43
a  b , c0 42 32
c c 8 6

*If both sides of an inequality are
multiplied or divided by negative

Page 1 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

number, the inequality sign must be 43
reversed.
4  3
2 2
2  1.5

4 1  b1, a,b  0 43
a
1  1
4 3

B. Find linear inequalities

A linear inequalities in one variable, x , is defined as

any relationship of the form:

ax b  0,ax b  0,ax b  0,ax b  0

where a and b are real numbers and a 0.

The solution to the inequality must be expressed in
solution set or interval form.

REMEMBER
These things will change (reverse) direction of the
inequality:

 Multiplying or dividing both sides by a negative
number

 Swapping left and right hand sides

Example
Solve the following inequalities:

(a) 3t+5t7

Page 2 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

(b) 1 x 5 1 x4
2 3

(c) 73x28

EXERCISE

Solve the following inequalities:

(a) 4x52x+9 (b) 11 y  y  4
4
x4
(c) 2  2 9 (d) 13x5x3 x9

Answers :

(a) x : x  7 (b)  y : y   5 (c) [14,8) (d) 2,3
 3


Page 3 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

C. Find quadratic inequalities

A quadratic inequality is an inequality of the form

ax2 bxc 0 where a , b and c are real number with
a 0. The inequality symbols , ,  and  may also

be used.

Quadratic inequalities can be solved by using
i. graphical approach;
ii. table of signs;
iii. positive number line.

I. Graphical approach

The graph of quadratic expression y  ax2 bxc is

sketched and points where the graph cuts the x -axis,
say pand q are noted.

Page 4 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

Example
Solve the following inequalities by using graphical

method.

(a) x2 6x50

(b) 67x3x2 0

Page 5 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

II. Table of signs

Example
Solve the following quadratic inequalities using table of
signs

(a) (x1)(x2)0

(b) x2 3x40

Page 6 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

III. Positive number line

Example
Solve the following quadratic inequalities by using
positive number line

(a) x2 2x150

(b)2x2 10 x

EXERCISE (b) 3x(x5)  2(2x3)

(a) (2x1)(x3) 4x

(a)  ,  3  1,  (b)  1 , 6 
 2   3 

MISCELLANEOUS EXERCISE

(a) x2 49 (b) 3x2 6x50

(c) 0 x2 98x (d) 64x2x2 3x32x

(a) {x : x  7}{x : x  7} (b) x : x   2 6 1 x : x  26 1 (c) 3, 9 (d)  3, 3
 3   3   2 

Page 7 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

Lecture 2 of 2

2.2 ABSOLUTE VALUES

At the end of lectures, you should be able to :

(a) define the absolute value of a and present it on

the number line and state the properties of

absolute values :

i. |a|≥0 ii. |-a|=|a| iii. |a+b|=|b+a|

iv. |a−b|=|b−a| v. |ab|=|a||b| vi. a  a

bb

where |b| ≠ 0

(b) solve absolute equations of the forms :

i.|ax+b|=c ii.|ax+b|=cx+d iii.|ax+b|=|cx+d|

iv.|ax²+bx+c|=d

(c) solve absolute inequalities of the forms:

i. |ax+b|<cx+d ii. |ax+b|>cx+d

A. Define the absolute value of a and present it on

the number line

The absolute value of a , a represents the distance of
a point a on the real number line from the origin.

Page 8 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

The absolute value of real number a , written as a can

be defined as

a   a, if a0 or a  a2
 if a0
a,

Therefore, absolute value of every real number is non-

negative, a 0 for every a .

Example

Define the following absolute values

(a) x5

(b) 5 x

Page 9 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

Properties of absolute values

Properties Examples

1 a 0 5 50
0 0

2 a  a 5 5
5 5

3 ab  ba 53  8 8
35  8 8

4 ab  ba 53  2 2
35  2 2

5 ab  a b 53 15 15
53 5315

a  a , b 0 5  5
b b 3 3
6
5  5
3 3

B. Solve absolute equations of the form

i. |ax+b|=c
ii. |ax+b|=cx+d
iii. |ax+b|=|cx+d|
iv. |ax2+bx+c|=d

x  a  x  a or x  a

Page 10 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

Example
Solve absolute equations below

(a) 53 x5 3

(b) 1 x  2x5

Page 11 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

(c) 2x6  x1

(d) x2 2x4  4

EXERCISE

(a) x4 8 x (b) x2 2x 1 (c) 3x  5 x  2  4

(a) x  6 (b) x   2 1, 2 1, x  1 (c) x   7 , x  2, x  73 1, x   73 1
3 66

Page 12 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

Property of Absolute Value Inequalities

Inequalities Algebraic Graphical
of Interpretation Interpretation

Absolute a xa -a a
Value
xa and xa -a a
x a,a 0 xa  xa -a a
-a a
x a,a 0 xa or x a
xa  x a
x a,a0
a xa
x a,a0 xa and xa

xa  xa

xa or xa
xa  xa

C. Solve absolute inequalities of the forms

i. |ax+b|<cx+d ii. |ax+b|>cx+d

Example
Solve the following inequalities:

(a) 2x5 9

Page 13 of 14

AM015/ 2. Inequalities And Absolute Values 2021/2022

(b) 2x1 3x2

(c) 5x2 2x1

EXERCISE BORANG
MAKLUMBALAS
(a) 2x  4  0 PENSYARAH &
PELAJAR TERHADAP
NOTA KULIAH BAB 2

AM015

(b) 16 2 6 x (c) 5x6 41

(a) x  2 (b) ,2 14, (c)  https://goo.gl/f
Page 14 of 14 orms/5Rr5vSQ9

9qtFoue92

CHAPTER 2 INEQUALITIES AND ABSOLUTE VALUES 2021/2022

TUTORIAL 1 OF 3
At the end of tutorial, student should be able to
(a) Relate the properties of inequalities.
(b) Find linear inequalities.

1. Solve the following inequalities and write down the answer in solution set.

(a) 40x 10 (b) x  3  3x 5 (c) 3x 1 2 x 5 (d)  7  3  4x  6

2. Solve the following inequalities and write down the answer in interval form.

(a) 5x  3x 1 14 (b) x  5  x  4 (c) 2x 5  53x  x 9
23

(d)  2x  3 x 1 5(1 x)
32 6

TUTORIAL 2 OF 3

At the end of tutorial, student should be able to

(c) Find quadratic inequalities by

i. graphical approach; ii. table of signs; iii. positive number line

1. Find the ranges of values of that satisfy the following inequalities using graphical

approach and write down the answers in solution set.

(a) 2x2 3x  2  0 (b) 5x2  3x  2 (c) x2  4x  3

(d) 4x2 1 (e) x2  25 (f) 1 x4  x  x 11

2. Solve the following and express the answer in interval form.

(a)  x 12  9 (b) x 9  x2  x  20 (c) x  4  x2  x 12

(d) 3x  4  x2 6  9 2x

TUTORIAL 3 OF 3 (iv) ax2  bx  c  d.
At the end of tutorial, student should be able to
(a) Solve absolute equations of the forms:

(i) ax  b  c; (ii) ax  b  cx  d; (iii) ax  b  cx  d ;

(b) Solve absolute inequalities of the forms:
(i) ax b  cx  d; (ii) ax b  cx d.

1. Solve the given equations (b) x  2  7 (c) 2x 3  4  3
(e) x  2  10  3x (f) x2  7  2
(a) x  3 (h) x 1  2  3x (i) 2  3x  x  3

(d) 4  3x  5x  4

(g) x 1x 5  7

2. Solve the following absolute value inequalities

(a)1 4 3x  8 (b) 2 x  3x 10 (c) 2x 1  3x  2
(f) 2x 1  4x  6
(d) 3x  4  2x 1 (e) 5x  7  3x 1 (i) x  2  1

(g) x  2 1 (h) x  2  1

Page 1 of 3

CHAPTER 2 INEQUALITIES AND ABSOLUTE VALUES 2021/2022

EXTRA EXERCISES

1. Determine the solution set which satisfies the following inequalities [5M]
[4M]
(a) x x  2  3x  2 (b) 3 2x  5 [6M]
[6M]
2. Find the solution set of the inequality 3x  2  x [5M]
[6M]
3. (a) Determine the solution set for x(x  6)  3x  4. [6M]

(b) Solve 5x 1  3x and express your answer in interval notation. [5M]
2 [6M]

4. Determine the possible values of if | − 2| < 2 + 3. Express your answer in [5M]
interval number form [7M]
[6M]
5. (a) Find the value of x such that x  2  3
x 1 [5M]

(b) Determine the solution set of 2x2 13x  6 [4M]
6. (a) Determine the solution set which satisfies the inequality (1 x)(4  x)  x 11
[4M]
(b) Determine the possible values of x if 2x 1  4. Express your answer in
3

interval form

7. Solve (x  2)2  3  4(2x 1). Write your answer in the form of solution set.

8. Find the solution set for x 1  2(x 1)
2

9. Section A, 2019/2020

Determine the possible value of x if x  4  2x  4 . Express your answer in the
23

solution set form.
10. Section B, 2020/2021

(a) Given c 10 12 18. Find the solution set of c .

(b) Solve x2  4x  32 .

Page 2 of 3

CHAPTER 2 INEQUALITIES AND ABSOLUTE VALUES 2021/2022

ANSWERS

TUTORIAL 1 OF 3

1 (a) x : x   1  (b)x : x  4 (c)x : x  9 (d) x :  3  x 5
 4   4
 2 


2 (a)  , 15  (b) 54,  (c) (1,2) (d)  2 ,  
 2   3 

TUTORIAL 2 OF 3

1 (a) x :  1  x  2 (b) x : x   2  x : x  1 (c) x : x 1x : x  3
 2  5 

(d) x :  1  x 1  (e) x : x  5x : x  5 (f) x : 1 x  7
 2 2 

2 (a) ,2 4, (b) 5,33,4 (c) 4,2 2,3 (d) 5, 2

TUTORIAL 3 OF 3 (b) x : 9,5 (c) x : 2,5 (d) x : 0 (e) x :3

1 (a) x : 3,3

 (f) x : 3,  5, 5,3 (g){ : −2,2 − √2, 2 + √2, 6} (h)  x : 1 , 3  (i)  x :  1 , 5 
 2 4   4 2 

2 (a)  1, 11  (b) ,10  (c)  3 ,   (d)  , 3  5,  (e) x : x  3  x : x  4
 3  5  5  4

(f) x : 5  x  7  (g) ,3 1, (h)  (i) x : x 
6 2 

EXTRA EXERCISES

1. (a) x : x  1x : x  2 (b) x : x  4x : x 1

2. x : 1  x  1
2

3. (a) x : 4  x 1 (b) ,1

4.   1 ,  
 3

5. (a) x :  5 ,  1 (b) x : x  12x : x  6
2 4 
(b)  ,15    9 ,
6. (a) x : x  1x : x  7  2  2 

7. x :1  x  3

8. x : x  53

9. x : x 16

10. (a) c : c  4c : c 16 (b) x : x  4x : x  8

Page 3 of 3

AM015/3. SEQUENCES

CHAPTER 3: SEQUENCES

LECTURE 1 OF 2

At the end of lesson, student should be able to :
a) Express the nth term of a sequence.
b) Find the nth term and sum of the first n terms.
c) Solve problems involving arithmetic sequences in business and economics

DEFINITION OF A SEQUENCE
A sequence is a set of numbers arranged in a particular order.
Example,
(i) − 5, − 3, −1, 1, 3, 5,...
(ii) -1, 1, -3, 3, -5, 5
(iii) 3, 3, 3, 3, 3,…

Each number in the sequence is called a term.

Example : 3, 5, 7, 9

First term, T1 = 3
Second term, T2 = 5
T3 = 7
Third term,
Fourth term, T4 = 9

n Term Test Rule
13 2n+1 = 2×1 + 1 = 3
25 2n+1 = 2×2 + 1 = 5
37 2n+1 = 2×3 + 1 = 7

 Tn = 2n +1
The notation Tn represents the n th term, or general term of a sequence.

1 of 10

AM015/3. SEQUENCES

Example 1

Express the nth term of the following sequences.

a) 4, 7, 10, 13, 16… b) 1 , 3 , 5 , 7 , 9
2 4 6 8 10

Example 2

Given Tn = 1  , find the first three terms and the 12th term.
 
 3n + 1n=1

ARITHMETIC SEQUENCE

Definition of an Arithmetic Sequence
An arithmetic sequence or an arithmetic progression is a sequence of the form

a, a + d, a + 2d, a + 3d,... , a + (n −1)d , where a is the first term and d is the common

difference of the sequence.

a, a + d, a + 2d, a + 3d,...

First term: T1 = a

Second term: T2 = a + d = a + (2 −1)d

Third term: T3 = a + 2d = a + (3 − 1)d

The nth term of an arithmetic sequence is given by Tn = a + (n −1)d

The common difference, d = T2 −T1 OR
= T3 − T2 OR
= Tn −Tn−1

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Example 3
Given the arithmetic sequence 7 , 13 , 19 , … , 307

Find (i) the number of terms

(ii) the 20th term and

(iii) the nth term of the arithmetic sequence

Example 4
The 11th term of an arithmetic sequence is 52 and the 19th term is 92. Find
(a) the first term and common difference
(b) 100th term

Example 5
The nth term of an arithmetic sequence is 40 + 7n.
(a) Find the common difference.
(b) Which term of the sequence is 215 ?

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AM015/3. SEQUENCES

The Sum of the First n Terms of Arithmetic Sequence (Arithmetic
Series)

The terms of an arithmetic sequence can be added to form an arithmetic series. Let

Sn = a + (a+d) + (a+2d) +...+ a+ (n-1)d ……..(1)

Writing the sum in reverse order

Sn = a+(n-1)d +...+ (a+2d)+ (a+d) + a ……(2)

thus, (1) + (2)
2Sn =[ 2a + ( n-1 )d + … + 2a + ( n-1 )d ]
There are n identical terms on the right side of this equation

2Sn = n 2a+(n-1)d

 Sn = n 2a + (n −1) d 
2

Since the last term, Tn = l = a + (n −1)d

Sn = n a + a + ( n-1) d 
2

 Sn = n a +l

2

The relationship between Tn and Sn is, Tn = Sn − Sn−1

Example: T5 = S5 − S4

Example 6
Find the sum of all integers between 100 and 200 that are multiples of 6.

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Example 7
The tenth term of an arithmetic sequence is -25 and the sum of the first ten terms is
65. Find the first term , the common difference and the sum of the first twenty terms.

Example 8
The sum of the first 15 terms of Arithmetic Sequence is 255 and the sum of the next
15 terms is 705. Find the first term, the common difference and the 50th term.

APPLICATION OF ARITHMETIC SEQUENCES
Example 9
Iman settles her debt of RM4980 by paying RM50 at the end of the first month. And
for the following months, she pays RM8 more than the previous month. Find
(a) how long will Iman take to settle her debt,
(b) the amount of the last payment she makes.

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EXERCISE

1. An arithmetic series has 48 terms where the first term and the last term are
-16 and 98 respectively. Find the sum of this series.
( Answer: 1968 )

2. The sum of the first n terms of an arithmetic sequence is Sn = 7n2 − 8n .
Find
a) the first term
b) the common difference
c) the 100 th term
( Answer: -1, 14, 1385 )

3. The sum of the first 8 terms of an arithmetic sequence is 60 and the sum of
the next 6 terms is 108. Find the 25th term of this arithmetic sequence.

(Answer: 153 )
4

4. Hisyam has to repay his debt of RM10 360 by monthly instalments. If the first
instalment is RM100 and he agree to increase the amount by RM20 each
month after the first instalment, find the number of months he takes to settle
his debt and the final instalment.
( Answer: 28, 640 )

LECTURE 2 OF 2
At the end of lesson, student should be able to :

a) Find the nth term and sum of the first n terms
b) Solve problems involving geometric sequences in business and economics

GEOMETRIC SEQUENCE

Definition of a Geometric Sequence

A sequence in which any term can be obtained from the previous term by multiplying
by a constant is called a geometric sequence or geometric progression.

The geometric sequence are given by a, ar, ar 2 , ar3 ,..., ar n−1,...
where a is the first term and r is the common ratio.

The first term, T1 = a
Second term, T2 = ar
Third term, T3 = ar2
Fourth term, T4 = ar3

So the n th term, Tn = arn−1

The n th term is called general term of a geometric sequence.
The common ratio, r is given by

r = T2 = T3 = ... = Tn . 6 of 10
T1 T2 Tn−1

AM015/3. SEQUENCES
Example 1
Find the sixth and nth term for the geometric sequence 4, -8, 16, -32, …

Example 2
The third term of a geometric sequence is 15 and the sixth term is 120. Find the first
term, the common ratio and tenth term.

Example 3
In a geometric progression, the second term exceeds the first term by 20 and the
fourth term exceeds the second term by 15. Find the possible values of the first term.

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The Sum of the First n Terms of a Geometric Sequence
Consider the geometric sequence,
a, ar , ar 2, ar 3,...
Let the sum to n terms of a GS as Sn,
Sn = a + ar + ar 2 + ar 3 + ... + ar n−2 + ar n−1 − − − (1)

Multiply the equation by the common ratio r ,
r Sn = ar + ar 2 + ar 3 + ... + ar n−1+ ar n − − − − − − − (2)

(1) – (2), Sn − r Sn = a − ar n
or Sn (1 − r ) = a(1 − r n )

If r  1, Sn = a(1 − rn) and If r  1, Sn = a(r n − 1)
(1 − r) r −1

The relationship between Tn and Sn is, Tn = Sn − Sn−1

Example 4

Find the sum of a geometric series 1+ 1 + 1 + ... + 1
24 256

Example 5
The sum of the first 3 terms of a geometric progression is 7 and the sum of the next

4
three terms is 7 . Find the common ratio of the progression.

32

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