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Teach
Yourself
Algebra
Vol 1
by
Fred Nolan
1

Introduction
In the vast landscape of mathematical understanding,
algebra stands as the gateway-a key to deciphering the
intricate patterns that govern our numerical world
...

Whether you are a student navigating the early
stages of algebraic exploration or a seasoned
mathematician seeking deeper insights, this book is
designed to be a companion on your Mathematical
voyage
...


This is the first volume and two other volumes will follow in
due course
...
In this
book he focuses on the basics of algebra
...


23

4

Algebra
Algebra can be defined as the part of mathematics in which
letters and other general symbols are used to represent
numbers and quantities in formulae and equations
...


There are various types of equations, but for now we will focus
on Linear equations
...

Below are some examples of linear equations:
 5x +6 = 17
 4(1 + 3x) = 26
 89x + 36 = x + 56
You will note that from the equations above the highest power
of the variable x is 1, this is always the case for linear equations
...


The unknown variables can be found by:
 Multiplying both sides by a number
 Dividing both sides by a number
 Adding a number to both sides
 Subtracting a number form both sides
We are going to solve some examples now to improve our
understanding of the concept
...

3x-19x = -9-7
-16x = -16
X=1
6

Example 2
2(2x+3) = 4(-7x+9)
4x+6 = -28x + 36
4x+28x = 36-6
32x = 30
Divide both sides by 32
32x = 30
32 32
X = 0
...

 Replace the unknown number in the word problem with
the letter you have chosen
...

Example 1
Four-Fifth of a number added to Two-Third of another
number equals 15
...
In three
years’, time she will be two times as old
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Solution
Let the first number be X, the next will be x + 1, third number
X + 2, fourth number x + 3, then our equation will be:
X + x+1 + x+2 + x +3 = 48

4x + 6 = 48
4x = 48 – 6
4x = 42
X = 42/4
X = 10
...
5,
X+1= 10
...
5,
X+2 = 10
...
5,
X=3 = 10
...
5
10
...
5, 12
...
5 respectively
...

Let the unknown number be x, we then add x to the numerator
and denominator of 4/5
4+x =7
5 +x

9

Multiply both sides by 9(5+x)

9(5+x) 4+x = 9(5 + x) 7
5+x

9

9(4+X) = (5+X)7
36 + 9X = 35 + 7X
Collect like terms
9x-7x = 35-36
2x = -1
X = -1/2
The unknown number is -1/2
...
Solve the following equations
12

a) 3x+3 = 6x-9

b) 2/5x -7/8x = 4/5
c)

3 = 5
5+x

7-x

d) 3x-7 - 8x-5 = 4
2

5

7

2
...
In 10 years time,
he will be twice as old
...
When 150 is subtracted from 4 times a certain number,
the result is equal to one-third of the original number
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4
...
Find the
number
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Jane is three years older than Victor, Stanley is half of
Jane’s age
...
How old is
Victor?

13

6
...
What is
the number?
7
...
What is the number?
8
...

9
...
Seven years ago, the
mother was four times as old as her son
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The sum of three numbers is 30
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Find the numbers
...
The statement looks like the usual equation except
that the equality sign (=) is replaced by an inequality sign such
as:
< Less than
> Greater than
< Less than or equal to
> Greater than or equal to
Solving equalities is similar to solving linear equations, the
difference is the equality sign is changed to an inequality sign in
the solution
...

 We subtract the same number from both sides
...

 We multiply or divide both sides by a negative number as
far as we change the inequality sign i
...

Now we are going to try some examples
15

Example 1
5x + 7 < 10
5x < 10 -7 (collect like terms)
5x < 3
X < 3/5
Example 2
4x + 3 > 8 - 9x
4x + 9x > 8 – 3 (collect like terms)
13x > 5
X > 5/13
Example 3
3/x > 7
X (3/x) > x(7)
3 > 7x
The inequality sign changes when the variable moves to
the other side
7x < 3
X < 3/7
Example 4
16

4x + 6 < 7x
5

9

Multiply both sides by the LCM 45
45 * 4x + 6 <

45* 7x

5
9(4x +6)

9
<

5(7x)

36x + 54 < 35x
36x -35x + 54 < 0
X + 54 < 0
X > -54 (the inequality sign changes)

Linear Inequalities – Word problems

17

Word problems involving linear inequalities are similar to word
problems with linear equations, the difference is the inequality
sign
...
Find the range of values of the number
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Example 2
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Kelvin bought x notebooks at $5 each and x+6 Textbooks at $15
each
...

Solution
Notebooks cost $5 each and Kelvin bought x of them, the total
value of the notebooks are $5(x) = $5x

The total value of the Textbooks are $15(x+6) = $(15x + 90)
Since he spent less than $150 on both items, we have:
5x +15x + 90 < 150
20x + 90 < 150
20x < 150 – 90
20x < 60
Divide both sides by 20
X<3

Exercise 1b
19

Solve the following inequalities
1) x/7 < 5/8

2) x/3 – x/8 > 3/5
3) x+4/x-9 < 9/5
4) 5x -9 < 7(x+6)
5) Michael bought x oranges at $7 and 4x oranges at $9
...

His average speed does not exceed 120km/hours, find x
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8) One-fifth of a certain number is greater than the sum of
the number and 7, find the number
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The method used
may be multiplication, division, subtraction, addition, square
root, squaring among others
...

Let’s try some examples
Example 1
Make s the subject of the formula
V2 = u2 + 2as
u2 + 2as = V2
2as = V2 - u2
S = V2 - u2
2a
Example 2
Make m the subject of the formula
E = mc2
mc2 = E
m = E/c2

Example 3
21

Make t the subject of the formula
V = u + at
u + at = V
at = V – u
t = V-u
a
Exercises 1c
1) Make u the subject of the formula
V2 = u2 + 2as
2) Make k the subject of the formula
P = RK/V
3) Make m the subject of the formula
D = 1/3mk2
4) Make a the subject of the formula
a/x + b/y = a/y
5) Make i the subject of the formula
t = 2π

√

i/g

6) Make f the subject of the formula
22

T = πfd3
16

Solution to Exercises
Exercise 1A
1a) x = 4
b) x = -32/19
c) x = -1/2
d) x = -355/7

2) x = 5
3) x = 450/11
4) x = 13
5) x = 41/5 or 8
...
5 years, mother = 21 years
10) First number = 7, Second number = 9, Third number = 14

Exercise 1B
1) X < 35/8
2) X > 72/25
3) X < 101/4
4) X > -51/2
5) X = 30/43
6) X < 210
7) X < -20
8) X < -35/4

Exercises 1c
1) u =

√V – 2as
2

2) K = VP/R
3) m = 3D/K2

24

4) a = bxy
xy – y2
7) i = gt2
4π2
8) f = 16T
πd3

25



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