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Algebra 1
Algebra 1 is the second math course in high school and will guide you through
among other things expressions, systems of equations, functions, real numbers,
inequalities, exponents, polynomials, radical and rational expressions
...
Example
4⋅x−34
A variable, as we learned in pre-algebra, is a letter that represents unspecified
numbers
...
Example
Evaluate the expression when x=5
4⋅x−3
First we substitute x with 5
4⋅5−3
And then we calculate the answer
20−3=17
An expression that represents repeated multiplication of the same factor is called a
power e
...
5⋅5⋅5=125
A power can also be written as
53=125
Where 5 is called the base and 3 is called the exponent
...
53=5⋅5⋅5
31
3 to the first power
42
4 to the second power or 4 squared
53
5 to the third power or 5 cubed
26
2 to the sixth power
3
4⋅4
5⋅5⋅5
2⋅2⋅2⋅2⋅2⋅2
Evaluate the following expression when
x=2 and y=−3
x2−y+2x
Solution
(2)2−(−3)+2(2)=4+3+4
=11
...
Operations in the right order
When faced with a mathematical expression comprising several operations or
parentheses, the result may be affected by the order in which the various operations
are tackled e
...
4⋅7−2
the result is influenced if we take the multiplication first:
28−2=26
Or if we begin with the subtraction:
4⋅5=20
To avoid misunderstandings mathematicians have established an order of
operations so that we always arrive at the same result
...
Simplify the expressions inside parentheses ( ), brackets [ ], braces { } and
fractions bars
...
Evaluate all powers
...
Do all multiplications and division from left to right
...
Do all addition and subtractions from left to right
...
We
first work out how many hours the person work each day:
4+3=7
and then multiply that with the number of working days:
7⋅2=14
if we instead were to write this as an expression, we would need to use parentheses
in order to calculate the addition first:
(4+3)⋅2=14
Composing expressions
n the previous section we used an example where we wanted to know how many
hours a person works over a period of two days if he each day were to work 4 hours
before lunch and 3 hours after lunch
...
For example words like "sum", "increased by" and "plus" indicates that we
are to use addition
...
When we are writing subtractions and divisions the order in which we write is
important
...
When we have a mathematical problem as in the example below we can begin by
making a verbal model where we describe the situation in words and relate the
words by usage of mathematical symbols
...
Example
Tony is at a car rental service to rent a car
...
Write an expression for
the total cost of renting a car at this particular car rental service
...
When we measure how fast a car or something is moving we are usually comparing
quantities measured in different units like comparing distances with time
...
The unit rate is when the denominator of the
fraction is 1 unit
...
Calculate the unit rate
...
In a unit analysis you exchange the numbers and
variables in the expression with the corresponding units
...
Administration fee
+
(dollars)
price each day
(dollars÷days)
x
number of days
(days)
And to test what the resulting unit is we only keep the units in the expression:
dollars
dollars +
x days = dollars+dollars=dollars
day
Composing equations and 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
shown by the symbols: >, <, ≤ or ≥
...
When you substitute the variable in an open sentence with a number the resulting
statement is either true or false
...
Example
Is 3 a solution to
5x+14=24
Substitute x for 3
5⋅3+145⋅3+14
15+14=29≠24
FALSE!
Since 29 is not equal to 24, 3 is not a solution to the equation
...
Representing functions as rules and
graphs
Let's begin by looking at an example:
At a store the carrots cost $2
...
The price the customer pays is dependent on
how many pounds of carrots that he buys
...
We can write this as an equation
...
50⋅x
A function is an equation which shows the relationship between the input x and the
output y and where there is exactly one output for each input
...
As we stated earlier the price the costumer has
to pay, y, is dependent on how many pounds of carrots, x, that the customer buys
...
Input variable = Independent variable = Domain
Output variable = Dependent variable = Range
Functions are usually represented by a function rule where you express the
dependent variable, y, in terms of the independent variable, x
...
50⋅x
You can represent your function by making it into a graph
...
Again we use the example with the carrots
...
50
5
...
50
A pair of an input value and its corresponding output value is called an ordered pair
and can be written as (a, b)
...
We can thus write our values as ordered pairs
(0, 0) - This ordered pair is also referred to as the origin
(1, 2
...
5)
These ordered pairs can then be plotted into a graph
...
Every function is a relation, but not all relations are functions
...
If you are insecure whether your relation is a function or not you can draw a vertical
line right through your graph
...
The relation portrayed in the graph to the left shows a function whereas the relation in
the graph to the right is not a function since the vertical line is crossing the graph in two
points