1

Module 1:

Essential Skills (Self Study)

Equation and Inequalities

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Numbers

Natural Numbers ( )

These are the numbers we use for counting 1, 2, 3, 4, … This list of numbers has no end and is therefore called infinite. (means “going on forever”).

The natural numbers are the numbers 1, 2, 3, 4, …. The integers are the natural numbers together with 0 and the negative integers. That is, the integers are …-3, -2, -1, 0, 1, 2, 3,…. This lesson deals mainly with natural numbers and, sometimes, integers. The study of integers is called number theory. The sum and product of integers (or natural numbers, for that matter), is an integer (natural number).

A prime number is any number greater than 1 that is divisible by only 1 and itself, e.g. 2, 3, 5, 7, 11 … Every natural number that is not prime (and >1) is called composite , e.g. 4, 6, 8, 9, 10, 12, …

Graphical representation

Before we draw graphs on the number line, remember that the number line is used to show the relative positions and sizes of real numbers. As we read the number line from left to right, the numbers are getting bigger, whereas the numbers are getting smaller as we read from right to left.

These are all sets.This means for example, 5 < 30, -1 < 3 ,

41

31 −<− and -20 < -5 ( Even though 20 is

numerically larger than 5).

Size of numbers increases >

Size of numbers decreases<

3030-5-20

e.g. set of p rimenumbers ≤ 10

•1 2

•3•

4 5•

6 7 8 9

e.g. set of natural numbersless than 7

••••• •9876543210-1

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Note: (whole numbers), is the same as except that it also includes 0.

Consecutive numbers eg 5 6 7

Integers ( )If we add negative whole numbers, (not fractions or decimals), and zero to the list of natural numbers we get the list of integers: …-4, -3, -2, -1, 0, 1, 2, 3, 4, …Integers that are divisible by two are called even numbers and those that are not are called odd numbers.

Graphical representation

Rational Numbers ( )A rational number is any number that can be written as a fraction with an integer for its numerator and a non -zero integer for its denominator . In mathematical symbols:

Examples of rational numbers: •

−− 33.0,25.0,11399,8,5.0,

31,

43

Rational numbers have decimal forms that are either terminating or repeating decimals.Terminating decimals are numbers like 75.0

43 = or 375.0

83 =

Repeating decimals are numbers like:•

== 6.0...666.032 or

••== 128574.0...7144285714285.0

73

(the 6 digits 428571 repeat in this number).

-4 -3 -2 -1 0

• ••-6 -5 -4 -3 -2

e.g. odd integers≥ -2 and < 4

•1 2

•3 4 5

•6

e.g. integersless than 3

••••• •43210-1

A rational number is a number that can be written in the form:

0,,, ≠� qZqpqp

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How to Show That A Repeating Decimal is A Rational Number.

Example 1:Show that 0.151515… is a rational number. This means we have to prove that 0.151515…

can be written in the form: .

SolutionLet ...151515.0=x (1)Then ...151515.15100 =x (2)

(2) - (1): ...151515.0...151515.15100 −=− xx1599 =x

9915=� x

which is in the form .

� ...151515.0 is a rational number.

Irrational NumbersAll numbers that are not rational are called irrational. Therefore irrational numbers are allnon-repeating and non-terminating numbers. All square roots of non-perfect squares areirrational.For example: ...562213414.12 =

...5979898.424 =

Also π is irrational. The value which the calculator gives is only to the number of decimalplaces which the calculator can display.

722≠π This is only a useful approximation often used in school textbooks.

Exercise

1. Separate the following numbers into 2 lists, rational and irrational:3 27,

11132,5,

5372,14.3,

4...,1462.23,8.1,

73,2.6 −−

• π

0,,, ≠� qZqpqp

0,,, ≠� qZqpqp

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Real Numbers, ( )All the numbers that have been described so far are real numbers. Real numbers can be saidto be all rational numbers as well as all irrational numbers.

i.e. Real Numbers = Rational Numbers � Irrational Numbers

Graphical representation:

Numbers that are not real are called non-real . These non-real numbers cannot be drawn onthe real number line.

Changing the Subject of t he Formula

Core concepts:• Changing the subject of the formula• Manipulating fractional (rational) equations

The subject of the formula is normally written on the left hand side of the formula.

Eg if 2 = + , then = 2 − . Note that only a is on the LHS

Why?root....squarepositivethetakeonlyweandLHStheonissthat NoteLet side.sideissquareaofareatheAlso, AssA =�=× .2

. + = then + ≠ .

. + = then + ≠ .

Example 1

Refer to the figure below:

e.g. realnumbers >-4and ≤ 1

e.g. realnumbers ≤ 3

<<

°-5 -4 -3 -2 -1 0 1 2 3 4 5 6

6543210-1-2-3-4-5•

• ><

>

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The area of a circle: A = =

=

= � .

Example 2

Refer to the figure below:

Calculate the total surface area and make b the subject of the formula.Note the box can be cut up into 6 rectangles.

Let = + +

+ = −

( + ) = −

=−

( + )

Example 3

Make x the subject of the formula:

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mamax

amamxamamxx

mxmaxaxamxa

mxaxa

+−=

−=+−=+

−=+−−=+

=−+

1

factorcommon)1(LHStoxintermstranspose

x)(a:LCDthebymultiply)(

Linear Inequalities and Set notation

Core concepts:• Solving linear inequalities• Set builder notation• Interval notation• Graphing inequalities

Consider the following:

It is clear that 132 =−x has only one solution while there many possible values of x that make 132 >−x true, e.g. ,20,

215,.3=x etc. In fact, any real number greater than 2

will satisfy the inequality.

Presentation of solutions of Inequalities

If a linear inequality has a solution, that solution will consist of so many real numbers that we cannot list them, so we have to describe the solution in other ways. These include set notation, interval notation and graphs (number lines).

Examples1. Illustrate: 0 < < 4 � on the number line.

Since x lies between 0 and 4 ( not included)So x = 1 ; 2 or 3. (Note that we use dots on the number line to represent the integers 1 , 2 and 3)

•••543210-1

><

132132

>−=−

xx

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2. Set-builder notation: }31:{ <<−� xRx Note x is an element of all real numbers that lies between - 1 and 3

or }31|{ <<−� xRx

Graphical:

Interval notation : (-1, 3) (an open interval)Note: The notation (-1, 3) mean that all real numbers between -1 and 3 (excluding -1 and 3) are in the interval.

3. Set-builder notation: }72:{ ≤≤� xRx

Graphical:Interval notation: [2, 7]

3. Set-builder notation: }03:{ <≤−� xRx

Graphical:Interval notation: [-3, 0) (half open interval)

4. Set-builder notation: }25:{ <<−� xZx

Graphical:Interval notation: Not possible. Interval notation cannot be used for integers; it isused for real numbers only.Listed (set notation): {-4, -3, -2, -1, 0, 1}Note: We cannot use interval for example 1 either.

Exercise:

Write each of the following in set-builder notation and graph on a real number line.1. [-2, 3)2. (-4, 2)3. (-10,-4) � [-2, ∞)4. [-3, 1) ∩ [-1, ∞)

oo

-2 43210-1><

••87654321

><

o

-5 -4 -3 -2 -1 0•

21><

•••••3-5 -4 -3 -2 -1 0

•21

><

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Word problemsCore concepts:

• Solving word problems• Using information to set up equations

Introduction:

• Maths is a concise language that uses symbols and notation.• Always read and do maths for understanding • Read carefully as certain information is implied

Steps in Method: Read the question. Get a picture of what it is about. What are you asked to find? Give these things letters. Say exactly what the variables represent. Translate information into equations and solve Do not forget to check your answer

Example 1

The sum of three consecutive natural numbers is 123. Find the largest of these numbers.

What are consecutive numbers? ( 5; 6; 7; 8 are consecutive numbers)Let the numbers be , ( + 1) ( + 2).Sum of the numbers: + ( + 1) + ( + 2) = 123 set up an equation and solve .

3 + 3 = 1233 = 120

= 40So, the numbers are 40; 41 and 42. The answer is 42 (Read the problem. Looking for the largest number!!)

Example 2.

In 2012 Daisy was 11 years and Mary was 30 years old. In which year will Daisy be half Mary’s age.

Is this an age old problem?

In 2013: Their ages would have been 12 and 31

In 2014: Their ages would have been 13 and 32

So, in x years their ages 12 + x and 30 +x.

But Mary’s will double : 2 (11 + x) = 30 + x … now solve for x.x = 8

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The answer is 2020 Why?

Example 3

The sum of two numbers is 54 and their difference is 6. Find the numbers.

Let the numbers be x and y.

Sum of numbers + = 54 …… (1)

− = 6 …….. (2)

(1) + (2) 2 = 60 ( by adding equations 1 and 2 we eliminate y)

= 30 = 24 .

Check!

Example 4

A water tank has 36 litres more water when it 30% empty than when it is 40% full. How many litres of water can the tank hold when it is full.

You are allowed to guess and check !!Let the tank hold 100 l of water when full.

So: When 30% empty (ie 70% full) …………….. 70l. When 40% full ………… 40 l.But the difference is 30l and not 36l?

Solution:Let the tank hold x litres of water when full. Now set up and equation and solve.

Let

full.whenwater oflitres120holdcantanktheie

0.4

1203

1036

363.07.036

=�

×=

==+

x

x

xxx

Example 4

The sum of two natural numbers is 20. When the numbers are increased by 3 and I, the product of the new numbers is quadrupled (multiplied by 4) the sum of the original numbers, Find the original numbers.

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.acceptablearesolutionsboththat checkPleaseor andor So

((2)inyfor sub.and(1)Fromandfor solvenow

(1)....20equationstwoupset anddatatheanalysecarefully,problemtheRead

:Solution

193117

0)1)(17(01718

803632180)21)(3(

80)120)(320

)2....(80)1()3(

2

2

====

=−−=+−

=−+−=−+

=+−+−=

=+×+=+

yyxx

xxxx

xxxxxxx

xyyxyx

yx

Exercise1. A group of friends share 21 sweets and 35 marbles equally. How many are there in

the group?

2. A farmer has 20 m of fencing and wishes to construct a rectangular garden. Find the maximum area of the garden.

3. Joseph travels (from town A to town B) at 80 km/hr and returns at 120 km/hr. Calculate the average speed for the journey. (answer is not 100 km/hr)

4. Khumalo visited farmer Brown who had chicken and goats. He counted 215 heads and 460 legs. How many goats were there?

5. The length of a rectangle must be 6m more than its width. Find possible values of the width, if the area of the rectangle must be less than 40 m2.

6. Refer to the figure below represents a rectangular prism with AC = 12 cm, CD = 3 cm and AF = 2 cm. Calculate the shortest distance from point A to B.

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Exponential Equations

Core concepts:• Solving Exponential Equations• Prime factors

16 = × × × = .

→ ←

Definitions : If n is a natural number , and a > 0, . = × × …. Laws:2. × = +

3. ÷ = −

4. 10 =a5. ( ) = ×

6. ( × ) = ×

7. ( ) =

8. − =Note: In the expression, na , we call a the base , and n the exponent, (or index).

Study the table below and observe the patterns: powers of 2. Complete table for powers of 3.Powersof 2

116

18

14

12 1 2 4 8 16

Exponent form

2− 4 2− 3 2− 2 2− 1 20 21 22 23 24

Pattern -4 -3 -2 -1 0 1 2 3 4

Powersof 3

13

1 9 27

Exponent formPattern

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Practice Work 1Express in exponential form and simplify if possible:a) 3 × 3 × 3 b) nn bb × c) (a + b) × (a + b) ×(a + b)d) 3b × 3b × 3b e) abc × abc × abc × … (to abc factorse

f) 52

45

2015

baba

g) 70 + 2− 1

Solution:a) 3 × 3 × 3 = 33 (Definition 1)

b) + = 2

c) (a + b) × (a + b) ×(a + b) = (a + b)3 (Definition 1)

d) 3b × 3b × 3b = (3b) 3 (Definition 1)= 33b3 (Law 4)= 27b3

f) exponentspositive with answers write...43

43 3

45

25

ba

ba =−

−

g) 8and4....laws211

211 =+

Practice Work Worked examples

a)

333

2725435432

3

=�=

=÷==×

x

x

x

x

b)

32x–1 = 27

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32x–1 = 33 2x – 1 = 3 The base on the LHS is the same as the base on the RHS. Equate the exponents.

2x = 4 x = 2

Worked Examples

yx

yxy

+

+

=×=×==×=×=

==

22 253594522

,5x

yx

253152.ofpowersas45and15express2and32If1.

1. 2 = 20 .

We know that 24 = 16 and 25 = 32. So, x lies between 4 and 5, ie 4 < < 5.

Nb. Use your calculator and write your answer to two decimal places.

≈ 4.3

3.

1246

22)2()2(

48

246

12223

122

−=�−=

==

=

−

−

−

xxx

xx

xx

xx

5.

95

41933

31

33

811

279

4326

433

26

1

23

−=

−=+=

=

=

−−++

+−

+

+−

+

x

x

xx

x

x

x

x

Exercise1. Simplify

1.1 6

54

xxx

1.2 2

32

12)4(

−

−

tt

1.32

1

32−

− ���

����

�ca

abc

Solve the ffg:

Check Point!

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2. 324 =x

3.64

41 =���

���

x

4. 61 =−x

5. 168

422

12

=×−

+

x

xx

6. 122 311 =× ++ xx

7. .:2781 yxyx ofvaluethefindIf =

8. If ,41 =+a

a find the value of :

aa 11.8 2 + 3

3 12.8a

a +a

a 13.8 −

9.

312 32 =−− x

xx ifEvaluate

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Nb ln x refers to natural logs…

Use your calculator to check log 10 = 1 , log 1000 = 3 etc..

Examples:

1. Log 2 + log 5 = log(2x5) = log 10 = 1

2. 2log 5 + 2log2 = log 25 + log 4 ……. Apply (3)= log (25x4) = log 100= 2.

3. Log 200 - log 2 = log( 200 / 2) = log 100 = 2

LOGARITHMS

16 = 24.... can be written as log 2 16 =416 = 42.... can be written as log 4 16 =2Read: log 16 to the base 2 is equal to 4and log 16 to the base 4 is equal to 2Nb log 100 = 2 (base 10) 100 = 10 2 log 1 000 000 = 6 ... why?� the log of a no is the exponent.

Properties: 1. log x(A)(B) = log xA +log xB

2. log xAB = log xA - log xB

3. log xAn = n ( log x A)

4. log BA = log xAlog xB

5. log A A = 1 ... why ?