0 Introduction

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Chapter 0 Introduction

0.1 Real Numbers and Algebraic Expressions

Numbers are the foundation of mathematics. The most common numbers in mathematics

are the real numbers. The real numbers can be written as a decimal, either repeating, such

as

1 1 3 70.3333 0.3, 0.166 0.16, 0.75, 1.75

3 6 4 4

or non repeating, such as

2 1.4142135 , 3.14159

Properties of real numbers

The real numbers are closed under operations of addition and multiplication. For any real

numbers a, b and c following properties hold:

1. Closure properties a b and ab are real numbers, where ab a b

2. Commutative properties a b b a , and ab ba

3. Associative properties ( ) ( )a b c a b c , and ( ) ( )ab c a bc

4. Identity properties 0 0a a a , and 1 1a a a ; where 0 is additive

identity, and 1 is multiplicative identity

5. Inverse properties ( ) ( ) 0a a a a , and 1 1

1a aa a

; where a is

additive inverse of a, 1

a is multiplicative inverse of 0a .

For every nonzero real number a, 11a

a

6. Distributive properties ( )a b c ab ac

7. If a b then a c b c , and ( ) , ( )a a a b ab

Subsets of real numbers

Natural numbers set {1,2,3,4,5,6, }N

Whole numbers set {0,1,2,3,4,5,6, }Z

Integers set { 3, 2, 1,1,2,3, }I

Rational numbers set 2 7

| 0, and areintegers 0,1,2,3, , ,3 11

pQ q p q

q

Irrational numbers set | isnot a rationalnumber { 2, , }Q x x

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Properties of inequality relations

For any real numbers a, b and c we have the following properties

Transitive property If a b and b c then a c

Additive property If a b then a c b c

Multiplicative property If a b and 0c then ac bc

If a b and 0c then ac bc

Trichotomy property Either a b or a b or a b

Definition of absolute value

For any real number x

if 0

if 0

x xx

x x

Properties of absolute value

For any real numbers x and y, following are always true

0x x x x y xy , 0xx

yy y

and the triangle inequality property is x y x y

Note that, the graph of absolute value function is y-axis symmetric, so it is an even

function.

Expressions

An expression is a combination of terms like 5 4, 2 3 , 5x y x y , where the real

numbers x and y are called variables. The meaning of 5x is 5 times x. The given

examples are also known as algebraic expressions. 5, 10, 13.2 could be treated as

constant expressions.

A combination of variables and numbers using the operations of addition, subtraction,

multiplication or division as well as powers or roots, is called an algebraic expression.

Special constant expressions

The greatest integer value is written as x and is defined for a real number x as the

largest integer that is less then or equal to x. Examples are 12.5 12, 0.9 1

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Floor and ceiling value: x is the floor means round down to the lower integer; x is

the ceiling means round up.

The floor value of x is also known as the greatest integer value of x, not greater than x.

For example,

floor(2.5) = 2, floor(-2.5) = -3, floor(2) = 2, floor(-2) = -2

ceil(2.5) = 3, ceil(-2.5) = -2, ceil(2) = 2, ceil(-2) = -2

The ceiling value x returns the smallest integer value of x not less than x.

Formal mathematical definition:

Greatest integer value or floor value max{ | }x x n Z n x , and the ceiling value

min{ | }x n n x . Note that x x iff x is an integer.

Two important relations: 1. Show that x x for any real number x

2. Show that 2 2

x xx , where x is an integer.

Exercise Set 0.1

In problems 1-10, write down each of the following without absolute value sign (do not

simplify), x is a real number:

1. 4 3 2. 13 3 3. 4 4. 3

5. 2 10 6. 3 8 1 7. 3 2 8. 24 3x

9. 3 15x , if 5x 10. 3 4x x , if 3 4x

In problems 11-13, use order operation to simplify the expressions (always perform

division before multiplication). One may remember PEDMAS (P for parentheses, E for

exponents, D for division, M for multiplication, A and S for addition and subtraction).

11. 15 5 4 6 8

6 ( 5) 8 2 12. 221 3 7 25 5 2 13.

38 ( 4)( 2 ) 16 2

4 3 6 2

In problems 14-17, evaluate the expressions

14. , 0x

xx

15. 3

, 33

xx

x 16.

2

3, if 4

9

xx

x 17.

2

3, 3

9

xx

x

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In problems 18-20, determine the value of the following expressions

18. 13.2 19. 13.2 20. 13.2 + 13.2

21. Classify the following numbers as whole number, rational number, and/or irrational

number:

31, 3.6666..., 13, 81, 64, 30, 5, 0

3

Simplify the expressions (22-24)

22. 2[5 7( 2) 2]x x 23. 3( 7) 4x x 24. 1

(12 8) ( 4 )4

x x

0.2 Exponents and Scientific Notation

Suppose a is a real number and n is a positive integer (meaning 0n ), then na is

defined as n times of a, like

total

n

n

a a a a a

n is called the exponent of the base a. Note that 00 is undefined.

Important properties of exponents

1. m n m na a a

2. , 0m

m n

n

aa a

a

3. ( )m n mna a

4. 0 1, 0a a

5. 1

, 0n

na a

a

6. ( )n n nab a b

7. , 0

n n

n

a ab

b b

8. 2, 2, :nn a a n a a

9. , 0, 0

n na b

a bb a

Examples Simplify with positive exponents

1. 2 25 5 25 25 625

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2. 100

100 99

99

55 5

5

3. 3

2 2 2 2 2 2 2 65 ( 5 )( 5 )( 5 ) 5 5 15625

4. 0 0105 1, ( 5) 1

5. 2 2 23 5 3 5 9 25 225

6. 1/3 3 1/33 64 ( 64) ( 4 ) 4

7. 2 4 3 3 6 12 6 12(3 ) 3 27x y x y x y

8.

4 4 48 4 5

4 5 8 5 3 20 12

20 100 5 625

100 20

xy x y

x y xy x y x y

Scientific Notation

For a real number 1 10a , the form 10na , where n is an integer is called scientific

notation. As an example 95.37 10 is the scientific form of the number 5,370,000,000

(five thousand three hundred seventy million). On the other hand 35.37 10 is the

scientific form of the number 0.00537.

Definition A number in scientific notation is expressed as a number greater than or equal

to 1 and less than 10 multiplied by some power of 10.

Examples

1. Write in scientific notations a) 27300000, b) 0.0000273

Solution: a) 727300000 2.73 10

b) 50.0000273 2.73 10

2. Write in a single scientific notation a) 2 10(12.37 10 ) (3 10 )

b) 8

3 2

12 10

(2 10 ) (3 10 )

Solution: a) 2 10 8 9(12.37 10 ) (3 10 ) 37.11 10 3.711 10

b) 8 8

3

3 2 5

12 10 12 102 10

(2 10 ) (3 10 ) 6 10

Exercise Set 0.2

Simplify with positive exponents

1. 4 45 2 2. 4 4( 5) 2 3.

3

3

( 15)

3 4.

3

3

( 2)

3

5.

03

3

15

3 6.

3 2( 3 ) 7. 2 42 2 8. 0

3

( 8)

3 2

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9. 143

3

x

x 10. 7 14( )( )x x 11. 2 4 2( 3 )x y 12.

3

6

y

13.

3

2

10

x 14.

25

3

3x

y 15.

3 6

5 10

35

5

x y

x y 16.

25 4

10 2

10

30

x y

x y

17.

33 5

2 3

20

100

x y

x y 18.

30

60 1/ 2( )

x

x 19.

5

1/5 4 5( )

x y

x y 20.

3 2 2

4 1 3

( )

( )

x y

x y

Express the given numbers in scientific notations

21. 2860000000 22. 1220000 23. 0.0000000142 24. 0.00808

Simplify the numbers

25. 51.23 10 26. 51.23 10 27. 5

3

1.23 10

2.54 10 28. 51.23 10 40

0.3 Radicals and Rational Exponents

Radicals provide a convenient notation for nth roots. Suppose the number a is positive

and real, for a positive integer 2n the expression n a is the principal nth root of a.

Remember that 1/ nn a a . The symbol is called a radical and indicates the principal,

or nonnegative square root. Thus 16 4 , the result is positive 4.

Rational exponents: For positive integers m and n, and for any real number b, we have / 1/ 1/( ) ( )m n m n n mb b b . When n is even, b must be positive. The number x y is

called the conjugate of x y and x y x y x y .

Examples:

1. 3/ 2 1/ 2 3 316 (16 ) 4 64

2. 2/3 1/3 2 2( 8) (( 8) ) ( 2) 4

3. 2/3 1/3 2 28 (8 ) (2) 4

4. 1/ 4 4 1/ 44 81 (81) (3 ) 3

5. 3 3 1/33 64/ 27 (4 /3 ) 4 /3

6. 13 52 5 13 6 13 2 13 4 13

7. Rationalize the denominator:

a) 3 3( 2 3)

3( 2 3)2 3 ( 2 3)( 2 3)

b) 2 3 (2 3)(4 3) 11 6 3

134 3 (4 3)(4 3)

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Exercise Set 0.3

Simplify the expressions

1. 1/3( 125) 2. 1/416 3. 4 625 4. 1/ 25 5. 1/ 7( 128)

6. 1/7128 7. 2/ 7(128) 8. 3 27 /125 9. 3 27 /125 10. 3 24 / 81

Rationalize the denominator

11. 2 5

7 3 12.

12 3

6 13.

3 11

3 11 14.

2 3

2 3 3 2

15. 4 5

1 2 16.

3

5 17.

10

7 3 18.

13

10 3

19. 5

1 3 20.

1 2

(2 3)(1 2)

Simplify the expressions

21. 5

31

2700

100

x

x 22.

8

2

40

10

x

x 23.

2 36 3 2x x x 24. 5 12

5 2

64

2

x

x

25. 3 3 3( 9 )( 81)x 26. 32 32 8x x

0.4 Polynomials

The combinations of algebraic expressions of the form 2 2 35 6, 3 9, 5 6, 9x x x y y y are called polynomial. A polynomial is a single

term or sum of two or more terms with variables having whole number exponents.

The general form of an n degree polynomial is 1 2

1 2 1 0( ) n n n

n n np x a x a x a x a x a , where the real numbers

0 1 1, , , 0n na a a a are the coefficients, n is a nonnegative integer. The coefficient

0na is the leading coefficient and the coefficient 0a is the constant coefficient.

Definition: Monomial A polynomial which when simplified has exactly one term is

called monomial. 4( ) 9P x x is a monomial.

Binomial A binomial is a simplified polynomial with two terms each with different

degrees or exponents. Similarly a trinomial has three terms with different exponents.

The degree of the polynomial is the highest degree of the polynomial.

Practice the following:

1. Find the coefficients of the polynomial 4 29 3 5 11x x x

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2. Find the degree and all coefficients of the polynomial 5 4 27 3 17 9 13x x x x

3. Given two polynomials 4 2 5 4 2( ) 3 5 11; ( ) 9 3 2 9P x x x x Q x x x x

a) Add the polynomials

b) Subtract first polynomial from the second

c) Multiply the polynomials

d) Divide second polynomial by the first polynomial

4. Use 2 2 2( ) 2a b a ab b to simplify the multiplication of two given

polynomials (2 3)x and (2 3)x

Exercise Set 0.4

1. Determine the polynomial and its degree

a) 22 5 13x x b)

22 5 13x x

c) 3 2 122 9x x

x d) 2 4 99 13 10x x x

e) 5 1/4 99 9 13 10x x x f) 2 4 9

2

9 13 10x x x

x

g) 2 4 9 3

2

9 13 10x x x x

x h)

2 4 9 2

2

9 13 10x x x x

x

i) 2 3x j) 2 2 3(9 4 2)x x

k) 4 513 2 9x x x l) 4 2503 9x x

m) 6 3256 32 1x x n)

2 4 919 13 10x x x

2. Perform the indicated operations, write the result in standard form of a

polynomial and indicate its degree.

a) 2 4 9 2 12( 9 13 10) (3 12 9)x x x x x

b) 2 5 2(3 9 1) (15 12 9)x x x x

c) 9 2 4 2 93 2 5 3 (15 3 ) 2( 9)x x x x x x

d) 2 4 9 2 12( 9 13 10)(3 12 )x x x x x

e) 33( 9)( 9)x x

f) ( 1)( 2)( 3)x x x

0.5 Factoring Expressions

Factoring a polynomial expression: The process of writing a polynomial as a product of

rational factors is called factoring of polynomials. In this section we will discuss some

tricks and techniques of factoring in an easiest and convenient way. Let us start with the

following examples.

1. Factor 25 15x x

Solution: Look at common factors in each term (if any). Then write the following

form diving each term by the common factor and use parentheses.

25 15 5 ( 3)x x x x

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2. Factor 2 15 54x x

Solution: We have the leading coefficient 1 and there is no term in common. Find

factors of 54 such that their sum is –15. We have such factors are –9 and –6. Then

write the following form 2 15 54 ( 9)( 6)x x x x . This method is known as

backward FOIL (F for first, O for outer, I for inner and L for last) method.

3. You try to factor 2 3 54x x

4. Factor 22 15 27x x Solution: We observe that the leading coefficient is 2. Then follow the steps:

Find factors from 2 27 54 such that their sum is –15 and write the form

2 62 15 27 9x x x x

and replace the box by the leading coefficient

to get 2 62 15 27 2 9 ( 3)(2 9)

2x x x x x x

5. You try to factor 22 5 3x x

6. Factor 2 10 25x x

Solution: 2 210 25 ( 5)( 5) ( 5)x x x x x

7. Factor 2 81x using the formula 2 2 ( )( )x a x a x b

Solution: 2 2 281 9 ( 9)( 9)x x x x

8. You try to factor 22 72x

9. Factor 4 3 1x x x by grouping.

Solution:

4 3 3

3 3 3 2 2

2

2 2

1 ( 1) 1( 1)

( 1)( 1) ( )( )

( 1)( 1)( 1)

( 1) ( 1)

x x x x x x

x x use x a x a x xa a

x x x x

x x x

10. You try to factor 73 192x x

11. Factor 2 22 2 28 98x c x

Solution:

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2 2 2 2

2 2

2 2 28 98 2( 14 49 )

2(( 7) )

2( 7 )( 7 )

x c x x x c

x c

x c x c

12. Use formula 3 3 2 2( )( )x a x a x xa a to factor 3(2 1) 8x

Solution:

3 3 3

2 2

2

(2 1) 8 (2 1) 2

(2 1 2)((2 1) (2 1)2 2 )

(2 3)(4 3)

x x

x x x

x x

Exercise Set 0.5

Factor completely

1. 2 13 36x x

2. 3 25 36x x x

3. 2 15 26x x

4. 23 24 36x x

5. 2 312 6 8ax ax ax

6. 2 3 3 212 48x y x y

7. 3 29 9x xa

8. 2 481 36x a

9. 2 ( ) 3 3x a b a b

10. 2 236 18x x

11. 22 72 648x x

12. 2 2( 3) ( 3)x x

13. 3 38 27x a

14. 210 17 7a a

15. 4 3 29 18 27x x x

16. 2 (2 5) 4(2 5)x x x

17. 2 2( 1) ( 1)x x x

0.6 Rational Expressions

The expressions of the type ( )

, ( ) 0( )

p xQ x

Q x, is called a rational expression. The

numerator and denominator of rational expressions are polynomial expressions.

Domain and Range: The set of all real numbers for which an expression is defined is

called the domain. On the other hand the set of all real numbers that is the value of the

expression constitutes the range of the expression.

It is well known that the domain of a polynomial expression is the entire real line. The

domain of a rational expression is the set of all real numbers for which the denominator is

nonzero.

Examples

1. Find the domain of the following expressions and write in interval notations

a) 2 1

1

x

x b)

23 2 1

2

x x

x c)

2

5

1

x

x d)

2

2

2 1

3

x

x

Solution:

a) For domain we must have 1 0 ( ,1) (1, )x x

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b) For domain we must have 2 0 ( , 2) ( 2, )x x

c) For domain we must have

2 1 0 1 ( , 1) ( 1,1) (1, )x x x

d) For all real values of x, the denominator 2 3 0x , thus the

entire real line is the domain and ( , )x

2. Simplify the following rational expressions and find their domain

a) 1 3

1x x b)

2 1 5

3 7

x x

x c)

32 1x

x d)

1 3

1xx

Solution:

a) 1 3 1 3 4 1

1 ( 1) ( 1) ( 1)

x x x

x x x x x x x x.

The domain is where 0,1 ( ,0) (0,1) (1, )x x

b) 22 1 5 7(2 1) 5 (3 ) 7 5

3 7 7(3 ) 7(3 )

x x x x x x x

x x x

The domain is where 3 ( , 3) ( 3, )x x

c) 23 2 3

2 1x x

xx x

The domain is where 0 ( ,0) (0, )x x

d) 1 3 1 3

1 ( 1)

x x

xx x x

The domain is where 1, 0 (0,1) (1, )x x x

Exercise Set 0.6

1. Find domain of the following functions

a) 21

13 x b)

2 3

3

x c)

2

2 16

64

x

x d)

2

3 3

2 3

x

x x

e) 2

5

2 11 5

x

x x f)

2

13

169

x

x g)

3

3

9 4

x

x x h)

2

100

5 14x x

2. Simplify and determine the domain

a) 1 2 2 13

1 2 1

x x x

x x x b)

2 6( 2)1

2 6

x x

x x

c) x b a a b a

a x b a a b d)

2 2

2 2

2(45 2 ) 3( 9)7

9 3

x x

x x

e)

22 21 5 2 1

92

x x x

x x f) ( )

x bc x ac x aba b c

b c c a a b

g) 2 4 3

23

x x

x h)

2

1

2 1

x

x x

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3. Simplify and find domain

a) 2

2

2 8 3

9 2

x x x

x x b)

2 3 2

12 6 2

8 32 3

x x

x x x

c) 2 3 9( 9)

2 5

xx

x d)

2 2

2

25 4 5

4 4 10 25

x x x

x x x

e) 2

2 3 4 2 10

5 25 8 12

x x x

x x x f)

2

2 3 4 2 10

5 25 8 12

x x x

x x x

0.7 Inequalities

The expressions related with the following symbols , , , are called inequalities.

2 3 2x , 2 4 3

23

x x

x are examples of inequalities. The first one is called linear

inequality, while the second is a rational inequality. Linear inequalities in one variable

When we write an equation, we usually use equality symbol „=‟ to indicate that two

algebraic expressions are equal. But in inequality we use two expressions, one is either

smaller or greater then the other. Some cases we use to indicate that the expression in

the left side of this symbol is either less than or equal to the expression in the right side.

Similarly has meaning that the expression in the left side is either greater than or equal

to the expression in the right side.

A solution of an inequality is any replacement of the variable that makes the inequality

true. The solution set of an inequality is the set of all solutions of the inequality. The

solution could be entire real line or part(s) of the real line.

Inequalities are said to be equivalent if they have the same solutions. The usual strategy

for solving inequalities algebraically is to use the following basic principles to transform

a given inequality into an equivalent one whose solutions are known.

Following operations on an inequality produces the equivalent inequality

1. Add or subtract the same quantity on both the sides

2. Multiply or divide both sides of the inequality by the same non zero positive

number

3. Multiply or divide both sides of the inequality by the same nonzero negative

number and reverse the direction of the inequality

Examples: Solve each of the given inequality problem and write solution in interval

notation

1. 10 5 2 3x x

Solution:

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10 5 2 3

5 3 2 10

8 8

1 (1, )

x x

x x

x

x x

2. 3

12 2 32

x x

Solution:

312 2 3

2

33 2 12

2

910

2

20 20,

9 9

x x

x x

x

x

3. 5 3(4 5 ) 2(2 3 )x x

Solution:

5 3(4 5 ) 2(2 3 )

5 12 15 4 6

21 3

1 1,

7 7

x x

x x

x

x

4. 2 2 4 6x

Solution:

2 2 4 6

6 2 10

3 5 (3 5]

x

x

x x

Polynomial inequalities

The linear inequalities are also the examples of polynomial inequalities of degree one.

The inequality relations where the expressions are polynomials are the polynomial

inequalities. For an instance the inequality 2 3 2x x is a polynomial inequality. In the

example the polynomial in the left side of is a quadratic polynomial which is less than

or equal to a linear polynomial in the right side. The solution of this polynomial will be

some portion of the real line. Later in this section we will discuss the solution of such

inequalities.

5. 3 22 15x x x

Solution: To solve 3 22 15 0x x x ,we solve the following equation and use real

line test

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3 2

2

2 15

(2 15) 0

( 3)(2 5) 0 0, 3, 2.5

x x x

x x x

x x x x x x

Test points -3 -1 2 4

-2.5 0 3

Sign of 3 22 15x x x - + - +

The solution is ( , 2.5] [0,3]x

6. 4( 15)( 2) ( 10) 0x x x

Solution: Like in example 6 solve for x and find that 15, 2, 10x x x . Set all

these points on x-axis and test the sign of 4( 15)( 2) ( 10)x x x considering

appropriate test points. The solution is ( , 15] [10, )x

Rational Inequalities

Inequalities involving rational expressions may be solved in much the same way as

polynomial inequalities.

Examples

7. Solve 9

53

x

x

Solution: We simplify and solve as follows

9 6 245 0 0

3 3

x x

x x and solve for x from 3 0,6 24 0 3,4x x x . We

now use test points on real line as in example 6 and test the sign of 6 24

3

x

x. We are

looking for positive answers. The solution is ( ,3) (4, )x

8. Solve 9

53

x

x

Solution: We simplify and solve as follows

9 6 245 0 0

3 3

x x

x x and solve for x from 3 0,6 24 0 3,4x x x . We

now use test points on real line as in example 6 and test the sign of 6 24

3

x

x. We are

looking for negative answers. The solution is (3,4]x . As we have equal sign as

well, we take closed interval sign but not for the x value 3, which is coming from

denominator.

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Absolute Value Inequalities

Let k be a nonzero positive number and x a real number then x k is equivalent to

k x k , on the other hand x k is equivalent to x k or x k . We present few

examples now.

9. Solve 1 3x

Solution: 3 1 3 2 4 [ 2,4]x x x

10. Solve 2 5 6x x

Solution: 2 26 5 6 0 5 6,x x x x or 2 25 6 5 6 0x x x x . Using

same technique as in example 6 we have the following steps:

For 20 5 6x x : solve for x from 2 5 6 0 ( 2)( 3) 0 2,3x x x x x .

From real line test we have the solution ( ,2] [3, )x

On the other hand 2 5 6 0x x : solve for x from 2 5 6 0 ( 6)( 1) 0 1,6x x x x x .

From real line test we have the solution [ 1,6]x . Thus the solution of

2 5 6x x is the common region [ 1,2] [3,6]x .

] [

2 3

[ ]

-1 6

11. Solve 2 5 6x x

Solution: We use the property “ x k is equivalent to orx k x k ”. We now

have 2 5 6x x or 2 5 6x x . Using line test method we find the solution as the

common region ( , 1] [2,3] [6, )x

12. Solve 4

33

x

x

Solution: Using the property “ x k is equivalent to k x k ” we write

4 43 3 3 0

3 3

x x

x x or

4 5 23 0 0

3 3

x x

x x or

4 130

3

x

x. We now

proceed to solve as in example 6.

For 5 2

03

x

x, we solve for x: 2.5, 3x . Using real line test we get

Test points 1 2.6 4

2.5 3

Sign of

5 2

3

x

x - + -

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16

The solution is then ( ,2.5] (3, )x

For 4 13

03

x

x, we solve for x: 3, 13/ 4x . Using real line test we get

Test points 1 3.1 4

3 13/4

Sign of

4 13

3

x

x + - +

The solution is then ( ,3) [13/ 4, )x .

Thus the solution of 4

33

x

x is the common region, which is

( ,2.5] [13/ 4, )x

0.8 Complex Numbers

Complex numbers are numbers of the form a ib , where a and b are real numbers and 2 1, 1i i .

Properties of complex numbers

Equality: If a ib c id then ,a c b d

Sum: ( ) ( ) ( ) ( )a ib c id a c b d i

Difference: ( ) ( ) ( ) ( )a ib c id a c i b d

Product or multiplication:

2( )( )

( ) ( )

a ib c id ac iad ibc i bd

ac bd i ad bc

Conjugate The complex number a ib is called the conjugate of a ib and

we have 2 2( )( )a ib a ib a b

Division or quotient:

2 2

( ) ( )a ib a ib c id ac bd i bc ad

c id c id c id c d

Powers of i: Remember that 2 1i . Now observe the following:

17 16 2 8 8( ) ( 1)i i i i i i i 35 34 2 17 17( ) ( 1)i i i i i i i 64 2 32 32( ) ( 1) 1i i 66 2 33 33( ) ( 1) 1i i

13. Express the following problems in the standard form a ib

0 Introduction

17

a) 2

3 4i b)

2 5

3

i c)

2

2 3

i

i d) 2(2 3 )i

Solution:

a) 2 2

2 2(3 4 ) 6 8 6 8

3 4 (3 4 )(3 4 ) 3 4 25 25

i ii

i i i

b) 2 5 2 5

3 3 3

ii

c) 2 2

2 (2 )(2 3 ) 7 4 7 4

2 3 (2 3 )(2 3 ) 2 3 13 13

i i i ii

i i i

d) 2 2(2 3 ) 4 12 9 4 12 9 5 12i i i i i

14. Simplify 45 5

Solution: 245 5 45 5 15 15i i i Error: 45 5 45( 5) 225 15

15. Find all solutions of 2 16 0x

Solution: 2

2 2

16 0

16 (4 )

4

x

x i

x i

0 Introduction

18

Exercise Set 0.7 - 0.8

Solve the following Inequalities and write

answers in interval(s), use real line test:

1. 13419 x

2. 6 (2 3) 4 5x x x

3. 6 (2 3) 4 5x x x

4. 3 1 5

32 2 6

x x

5. 4 7

2 53

xx

6. 4 7

2 5 03

xx

7. 4

3 45

x

8. 4

3 45

x

9. 5

14x

10. 5

14x

11. 2 1

53 4

x

x

12. 2 1

53 4

x

x

13. 3 4

2 1x x

14. 3 4

2 1x x

15. 4 3

2 1x x

16. 4 3

2 1x x

17. 4 3

2 1x x

18. 7 1

2 2x x

19. 7 1

2 2x x

20. 3

05

x

x

21. 3

05

x

x

22. 1

1 02

x

x

23. 062 xx

24. 22 5 12 0x x

25. 22 5 12 0x x

26. 2 12 0x x

27. 22 9 4 0x x

28. 22 9 4 0x x

29. 2 9 0x x

30. 2 4x x

31. 2 4 3 0x x

32. 2 16x

33. 2 16x

34. 2 16x

35. 2 5 6 0x x

36. 2 5 6 0x x

37. 2 6 0x x

38. 2 6 0x x

39. 2 5 6 0x x

40. 2 5 6

01

x x

x

41. 2 5 6

01

x x

x

42. 2 5 6

01

x x

x

0 Introduction

19

43. 2 5 6

02

x x

x

44. 2

2

3 2 3

4 2

x x x

x x

45. 2

2

3 2 3

4 2

x x x

x x

Hint: Do not cross multiply to solve. Use

3 12 0 0

1 1

x x

x x etc

46. 1 3x

47. 2 5 7x Hint: This inequality

has no solution, as the left side is

always positive.

48. 2 5 7x

49. 2 5 6x x

50. 2 1 8x

51. 23 6 3

5 5

x x

52. 2 5 6x x

53. Evaluate the following and write as

a ib :

a) (3 5 ) (7 ) 3i i i

b) (3 ) ( 17 ) 3i i i

c) (3 5 ) (7 2 )i i

d) (3 5 )(7 )i i

.