# Pre Algebra

__PRE-ALGEBRA__

Pre-Algebra
is the first math course in high school and will guide you through among other
things integers, one-step equations, inequalities and equations, graphs and
functions, percent, probabilities.

This
Pre Algebra math course is divided into 3 chapters and each chapter is divided
into several lessons. Each lesson includes theory and examples. I hope that you
will enjoy studying Pre Algebra.

__1. INDRODUCING ALGEBRA__

Ø __Operations in the correct order :__

When you are faced with
a mathematical expression that has
several operations or parentheses, the solution may be affected by the order in
which you tackle the operations.

For example, take the
expression:

4 . 7
- 2

If we do the Multiplication
first, we arrive at the following answer :

28 -
2 = 26

If instead we begin by
subtracting , we get :

4 . 5
= 20

In order to avoid confusion
and to ensure that everyone always arrive at the same result, mathematicians
established a standard order of operations for calculations that involve more
than one arithmetic operations. Arithmetic operations should always be carried
out in the following order:

a) Simplify the expression inside parentheses
( ), brackets [ ], braces { } and

fraction bars.

b) Evaluate all powers.

c) Do all multiplications and divisions from
left to right.

d) Do all additions and subtractions from
left to right.

**Example :**

Suppose you want to figure
out how many hours a person works in 2 days assuming that they work 4 hours
before lunch and 3 hours after lunch each day.

First, work out how many
hours a person works each day:

4 +
3 = 7

and then multiply that by
the number of days the person worked:

7 .
2 = 14

If we were to write this
example as one expression, we would need to use parentheses to make sure that
people calculate the addition first:

(4 + 3) . 2 =
14

Simplify
the following expression :

2 .
[(8 - 3) + 6 . (2)] - 3

2 . [ 5 + 12 ] - 3

2 . 17 - 3

34 - 3

31

Ø __Evaluate
Expressions :__

A variable is a letter, for example x, y or z , that
represents an unspecified number.

6 + x
= 12

To evaluate an algebraic
expression, you have to substitute a number for each variable and perform the
arithmetic operations. In the example above, the variable x is equal to 6 since
6 + 6 = 12.

If we know the value of our variables, we can
replace the variables with their values and then evaluate the expression.

**Example :**

Calculate the following
expression for x = 3 and z = 2

6z + 4x = ?

Solution : Replace x with 3
and z with 2 to evaluate the expression.

6 . 2 + 4 . 3 = ?

12
+ 12
= 24

Evaluate
the expression for x = 2, y = 5 and z = 4

4x + ( 7 - z ) - 6y

4 . 2 + ( 7 - 4 ) - 6 . 5

8 + 3 - 30

11 - 30

-19

Ø __Identifying properties :__

In this section, you will learn how to
identify the properties of multiplication and addition and how you can use identification to help solve mathematical
problems.

To solve rather hard problems without using a
calculator you have to identify all expressions that have the same operation.

**Example
:**

** **58 + 69 + 91
= ?

In this example, you can add in any order you
prefer. The sum of the expression will not change if you prefer to add the
numbers in a different order

91 + 58 + 69 = 218

58 + 69 + 91 = 218

The same is true for multiplication :

5 . 4 . 30 = 600

4 . 30 . 4 = 600

Show that the following equation holds true

3 . a . 4 = a . 4 . 3

12 . a = a
. 12

12a = 12a

Ø __Equations
with variables:__

In this section, you will learn how to solve
the equations that contain unknown variables. You will learn how to solve
equations mentally by using the multiplication table and you will also learn
how to identify a solution to an equation with given numbers as well as by
using inverse operations.

You can solve an easy equation in your
head by using the multiplication table.

**Example:**

** **8 . x = 64

Which number should you multiply 8 by to get
a product of 64? By using the multiplication table, we know that the number is
8.

8 . 8 = 64

When we solve an equation, we figure out what
value of x ( or any other variable) makes the statement true ( satisfies the equation).

**Example:**

Which of the
following numbers is a solution to the equation? x = 2, 7, or 8?

14 − x = 7

Here you are given
the numbers 2, 7 and 8. One of these numbers will satisfy the equation. If you
don't know the solution right away, you can investigate which of the given
numbers gives results in the correct answer by plugging in the different values
of x.

x = 2 ⇒ 14 − 2 = 12 Wrong

x = 7 ⇒ 14 − 7 = 7 Correct

x = 8 ⇒ 14 − 8 = 6 Wrong

**Answer: x = 7**

You have already solved equations where
the solutions are quite easy to see, by using mental math or patterns. Most
equations are harder to solve and you have to simplify the equation before you
can see the solution. One way to do this is to use inverse operations.

An operation is, for example, addition,
multiplication, division and subtraction. An inverse operation is an operation
that reverses the effect of another operation. Addition and subtraction are
inverses of each other, just like division and multiplication are inverses.

Ø __Coordinate system and
ordered pairs__

A coordinate system is a
two-dimensional number line, for example, two perpendicular number lines
or *axes*.

This is a typical coordinate system:

The horizontal axis is called the
x-axis and the vertical axis is called the y-axisThe center of the coordinate
system (where the lines intersect) is called the origin. The axes intersect
when both x and y are zero. The coordinates of the origin are (0, 0).

An ordered pair contains the
coordinates of one point in the coordinate system. A point is named by its
ordered pair of the form of (x, y). The first number corresponds to the
x-coordinate and the second to the y-coordinate.

To graph a point, you draw a dot at
the coordinates that corresponds to the ordered pair. It's always a good idea
to start at the origin. The x-coordinate tells you how many steps you have to
take to the right (positive) or left (negative) on the x-axis. And the
y-coordinate tells you have many steps to move up (positive) or down (negative)
on the y-axis.

**Example :**

The ordered pair (3, 4) is found in
the coordinate system when you move 3 steps to the right on the x-axis and 4
steps upwards on the y-axis.

The ordered pair (-7, 1) is found in
the coordinate system when you move 7 steps to the left on the x-axis and 1
step upwards on the y-axis.

To find out the coordinates of a
point in the coordinate system you do the opposite. Begin at the point and
follow a vertical line either up or down to the x-axis. There is your
x-coordinate. And then do the same but following a horizontal line to find the
y-coordinate.

Ø __Inequalities__

Equations and
inequalities are both mathematical sentences formed by relating two expressions
to each other. In an equation, the two expressions are deemed equal which is
shown by the symbol =

x = y x is equal to y

Where as in an
inequality, the two expressions are not necessarily equal which is indicated by
the symbols: >, <, ≤ or ≥.

x > y x is greater than y

x ≥ y x is greater than or equal to y

x < y x is less than y

x ≤ y x is less than or equal to y

An equation or an
inequality that contains at least one variable is called an open sentence.

When you
substitute a number for the variable in an open sentence, the resulting
statement is either true or false. If the statement is true, the number is a
solution to the equation or inequality.

**Example :**

Is 3 a solution to this equation?

5 x + 14 = 24

Substitute 3 for x

5⋅3
+ 14 = 29 ≠ 24 FALSE!

Since 29 is not equal to 24, 3 is not
a solution to the equation.

**Example:**

Is the following inequality true or
false?

·
x − 4 > 12 , x = 13

13 − 4 > 12 → false

·
y + 5 < 13 , y = 6

6
+ 5 < 13 → true

__2. EXPLORE AND
UNDERSTAND INTEGERS__

Ø __Absolute Value__

In
this section you'll learn how to the find the absolute value of integers.

4 - 0 = 4

4 - 1 = 3

4 - 2 = 2

4 - 3 = 1

4 - 4 = 0

4 - 5 = -1

In
this pattern you can see that 4 - 5 is equal to a negative number. A negative
number is a number that is less than zero (in this case -1). A negative number
is always less than zero, 0. We can study this in a diagram by using two
examples:

0 - 4 = -4 and -1
- 3 = -4

This
kind of diagram is called a number line. There are some things that you need to
observe when you draw and/or use a number line. Zero, is always in the middle
and separates negative and positive numbers. On the left side of zero, you'll
find numbers that are less than zero, the negative numbers. On the right side
of zero, you'll find numbers that are greater than zero, the positive numbers.
The absolute value is the same as the distance from zero of a specific number.

On
this number line you can see that 3 and -3 are on the opposite sides of zero.
Since they are the same distance from zero, though in opposite directions, in
mathematics they have the same absolute value, in this case 3. The notation for
absolute value is to surround the number by straight lines as in the examples
below.

**Example:**

Simplify the following expression

|3| + |−3| = ?

3 + 3 = 6

Ø __Adding and subtracting
integers__

You already know how to add 3 + 4 and
so on. But there are many ways to add integers. One way to add integers is by
using a number line.

**Example :**

−4 + (−3) = ?

** **

We always start at zero. Our first
number is negative four (-4) so we move 4 units to the left. We then have plus
negative three (-3) which is the same as subtracting 3 so we move 3 more units
to the left. This gives us the value of negative seven, (-7).

−4 + (−3) = ?

−4 − 3 = −7

We do the same thing if we have a
positive integer, but instead we move to the right.

−4 + 3 = ?

−4 + 3 = −1

** **

You can also add integers and
variables.

**Example:**

13 x + (−2) x = ?

13 x − 2 x = (13−2) x = 11x

When subtracting something from
something we wish to find out the difference between the two numbers. When you
subtract a negative number from any number the difference is even bigger. The
distance from the seabed at a depth of 150ft and an airplane flying at 3000ft
altitude at sea level is

3000 − (−150) = 3000 + 150 = 3150 ft

Thus when we subtract negative
numbers, we get:

4 − (−3) = 4 + 3 = 7

Subtracting −3 is the same as adding
3.

If we have a plus sign before the
parentheses then we will not change the signs within the parentheses

If we have a minus sign before the
parentheses then we the signs within the parentheses will change.

Two negatives make one positive!

Ø __Multiplying
and Dividing integers__

You also have to pay attention to the
signs when you multiply and divide. There are two simple rules to remember:

When you multiply a negative number
by a positive number then the product is always negative.

When you multiply two negative
numbers or two positive numbers then the product is always positive.

This is similar to the rule for
adding and subtracting: two minus signs become a plus, while a plus and a minus
become a minus. In multiplication and division, however, you calculate the
result as if there were no minus signs and then look at the signs to determine
whether your result is positive or negative. Two quick multiplication examples:

3
⋅ (−4) = −12

3 times 4 equals 12. Since there is
one positive and one negative number, the product is negative 12.

(−3) ⋅
(−4) = 12

Now we have two negative numbers, so
the result is positive.

Turning to division, you may recall
that you can confirm the answer you get by multiplying the quotient by the
denominator. If you answer is correct then the product of these two numbers
should be the same as the numerator. For example,

12
/ 3 = 4

In order to check whether 4 is the
correct answer, we multiply 3 (the denominator) by 4 (the quotient):

3
⋅ 4 = 12

What happens when you divide two
negative numbers? For example,

(−12)
/ (−3) = ?

For the denominator (-3) to become
the numerator (-12), you would have to multiply it by 4, therefore the quotient
is 4.

So, the quotient of a negative and a
positive number is negative and, correspondingly, the quotient of a positive
and a negative number is also negative. We can conclude that:

When you divide a negative number by
a positive number then the quotient is negative.

When you divide a positive number by
a negative number then the quotient is also negative.

When you divide two negative numbers
then the quotient is positive.

The same rules hold true for
multiplication.

__3. INEQUALITIES AND ONE STEP EQUATION__

Ø __Different
ways to solve equation__

We have 4 ways of solving one-step
equations: Adding, Substracting, multiplication and division. If we add the
same number to both sides of an equation, both sides will remain equal.

**Example :**

x
= y

x
+ z = y + z

If we subtract the same number from
both sides of an equation, both sides will remain equal.

**Example :**

x
= y

x
− z = y − z

If we divide both sides of an
equation by the same number, both sides will remain equal.

**Example:**

x
= y

x
÷ z = y ÷ z

If we multiply both sides of an
equation by the same number, both sides will remain equal.

x
= y

x
⋅ z = y ⋅
z

Ø __Calculating
the area and perimeter__

The perimeter is the length of the
outline of a shape. To find the perimeter of a rectangle or square you have to
add the lengths of all the four sides. x is in this case the length of the
rectangle while y is the width of the rectangle.

The perimeter, P, is:

P
= x + x + y + y

P
= 2x + 2y

P
= 2 (x+y)

**Example:**

Find the perimeter of this rectangle:

P
= 7 + 7 + 4 + 4

P
= 2 ⋅ 7 + 2 ⋅
4

P
= 2 ⋅ (7+4)

P
= 2⋅ 11

P
= 22 in

The area is measurement of the
surface of a shape. To find the area of a rectangle or a square you need to
multiply the length and the width of a rectangle or a square

Area, A, is x times y.

A = x ⋅ y

**Example :**

Find the area of this square.

A = x ⋅ y

A = 5 ⋅ 5

A = 25 in2

There are different units for
perimeter and area The perimeter has the same units as the length of the sides
of rectangle or square whereas the area's unit is squared.

Ø __Solving
Inequalities__

When we add or subtract the same
number on both sides of the truth of the inequality doesn't change.

This holds true for all numbers:

x > y → x < y →

x + z > y + z → x + z < y + z →

x − z > y − z x − z < y − z

**Example:**

x + 3 > 9

x + 3 − 3 > 9 − 3

x
> 6

It is a little bit trickier when it
comes to division and multiplication

When we multiply or divide an
inequality by a positive integer, the truth of the inequality doesn't change.

x
> y →

x
⋅ z > y ⋅ z →

x/z
> y/z

If z>0

When we multiply or divide an
inequality by a negative integer, the sign of the inequality will be reversed
(changed).

x
> y →

x
⋅ z < y ⋅ z →

x
/z < y/z

If z<0

**Example :**

x
/ − 2 ≥ 3

x/
− 2 ⋅ −2 ≥ 3 ⋅
−2

x ≤ −6

Ø __Understanding
inequalities and equations__

To expand our knowledge in equations
solving, we need to solve some verbal problems. With verbal problems we mean
problems that we need to translate into an equation by ourselves.

**Example:**

Leonard earns $60 a day. After a few
days Leonard has earned $240. How many days has Leonard worked? We denote the
number of days he has worked as x.

$60 x = $240

$60 x 60 = $24060

x=4 To
earn $240 Leonard has to work 4 days.