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Title: Pre-Algebra
Description: This is Pre-Algebra notes for all the students.
Description: This is Pre-Algebra notes for all the students.
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Pre-Algebra / Introducing Algebra
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
...
Arithmetic operations should always
be carried out in the following order:
1
...
2
...
3
...
4
...
Example
Suppose you want to figure out how many hours a person works in two days
assuming that they work 4 hours before lunch and 3 hours after lunch each day
...
6+x=126+�=12
To evaluate an algebraic expression, you have to substitute a number for each
variable and perform the arithmetic operations
...
If we know the value of our variables, we can replace the variables with their values
and then evaluate the expression
...
6z+4x=?6�+4�=?
6⋅2+4⋅3=?6⋅2+4⋅3=?
12+12=24
Identify properties
In this section, you´ll learn how to identify the properties of multiplication and addition
and how you can use identification to help solve mathematical problems
...
Example
58+69+91=?58+69+91=?
In this example, you can add in any order you prefer
...
91+58+69=218 58+69+91=218 91+58+69=218 58+69+91=218
The same is true for multiplication:
5⋅4⋅30=600 4⋅30⋅5=600
Equations with variables
In this section, you will learn how to solve equations that contain unknown variables
...
You can solve an easy equation in your head by using the multiplication table
...
8⋅8=64 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)
...
One of these numbers will satisfy the
equation
...
x=2⇒x=7⇒x=8⇒14−2=1214−7=714−8=6WrongCorrectWrong�=2⇒14−2=12
������=7⇒14−7=7��������=8⇒14−8=6�����
Answer: x=7
You have already solved equations where the solutions are quite easy to see, by
using mental math or patterns
...
One way to do this is to use
inverse operations
...
An
inverse operation is an operation that reverses the effect of another operation
...
Example
With numbers
18+4=22 18+4=22
18+4−4=22−4 18+4−4=22−4
18=1818=18
With variables and numbers
x+4=22 �+4=22
x+4−4=22−4 �+4−4=22−4
x=18 �=18
We subtract 4 from both sides
...
This is a typical coordinate system:
The horizontal axis is called the x-axis and the vertical axis is called the y-axis
The center of the coordinate system (where the lines intersect) is called the origin
...
The coordinates of the origin are (0,
0)
...
A
point is named by its ordered pair of the form of (x, y)
...
To graph a point, you draw a dot at the coordinates that corresponds to the ordered
pair
...
The x-coordinate tells you how
many steps you have to take to the right (positive) or left (negative) on the x-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
...
To find out the coordinates of a point in the coordinate system you do the opposite
...
There is
your x-coordinate
...
Inequalities
Equations and inequalities are both mathematical sentences formed by relating two
expressions to each other
...
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 ≥
...
When you substitute a number for the variable in an open sentence, the resulting
statement is either true or false
...
Example
Is 3 a solution to this equation?
5x+14=24 5�+14=24
Substitute 3 for x
5⋅3+14 5⋅3+14
5+14=29≠24 5+14=29≠24
FALSE!
Since 29 is not equal to 24, 3 is not a solution to the equation
...
For example, take the expression
4⋅7−24⋅7−2
If we do the multiplication first
4⋅7=284⋅7=28, we arrive at the following answer:
28−2=26 28−2=26
If instead we begin by subtracting 7−2=57−2=5, we get:
4⋅5=20 4⋅5=20
In order to avoid confusion and to ensure that everyone always arrives at the same
result, mathematicians established a standard order of operations for calculations
that involve more than one arithmetic operation
...
Simplify the expressions inside parentheses ( ), brackets [ ], braces { } and
fractions bars
...
Evaluate all powers
...
Do all multiplications and divisions from left to right
...
Do all additions and subtractions from left to right
...
First, work out how many hours the person works each day:
4+3=7
4+3=7
and then multiply that by the number of days the person worked:
7⋅2=14 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(4+3)⋅2=14
Title: Pre-Algebra
Description: This is Pre-Algebra notes for all the students.
Description: This is Pre-Algebra notes for all the students.