# Chapter 1

## Algebra Basics

### What is Algebra?

Algebra is a type of
mathematics where symbols are used to represent amounts that are
unknown. These unknown quantities are generally combined with
mathematical operations (addition, subtraction, multiplication,
division, square root, cube root, exponents, etc) to form statements
that describe the relationship of things that change over a period of
time. These statements are expressed using equations, expressions
and terms. Problems can be solved by translating words into
algebraic equations. The description of a problem using an equation
and other mathematical concepts is known as a

**mathematical model.**A mathematical model can be used to solve numerous types of problems in every day life.
Suppose
you are in a store looking for some new shirts. On one rack a brand
is selling for $17.95 per shirt and on another rack you see some
shirts selling for $20 for one and $15 for each additional shirt.
You decide that you need 4 new shirts, all of the first brand or all
of the second brand. What is the total cost and which is the better
deal?

You
can use a mathematical model to figure out the total cost of the
shirts. The procedure to solve such a problem can be thought of in
terms of mathematical symbols and variables. A

**variable**is an unknown quantity and is typically represented by a letter. In word problems, many times the variable is a letter which relates to the unknown quantity. For example, if you want to solve for the number of objects, use*n*. If you want the length or width of a two dimension figure, use*l*and*w*, respectively.
The
model used for the shirts on the first rack is “

*c*equals 17.95 times*n*” (*c*= 17.95*n*), where*c*is the total cost of shirts and*n*is the number of shirts. For the first brand of shirts, you will pay $17.95 times 4 for a total of $71.80.
The
model used for the shirts on the second rack is “

*c*equals 20 plus 15 times*a*” (*c*= 20 + 15*a*), where*c*is the total cost of shirts and*a*is the number of additional shirts purchased. The first shirt on the second rack is $20 plus $15 times 3 additional shirts for a total of $65. Therefore the second brand is the better buy when you want to buy 4 shirts.
An

**algebraic expression**is a combination of variables and/or numbers and mathematical operations (addition, subtraction, multiplication, division, raising to a power and finding a root). In the example above example, an**equation**is used to represent each model**because they contain an***=*sign. It's easy to remember the difference between an expression and an equation. The word "equation" contains the word**equate**, which means equals. An equation can be solved for the given variable or variables.
Words
and phrases can represent different mathematical operations. It's
important to be able to translate words and phrases into equations
when solving word problems.

Here
are some common words and phrases and their associated mathematical
operation.

sum
of, added to, increased, plus:

**addition**
minus,
decreased, less than, reduced by:

**subtraction**
twice,
product, multiplied by, times, of:

**multiplication**
quotient,
divided by, ratio, into:

**division**Types of Numbers

In
algebra you will be working with a wide variety of types of numbers.
The types of numbers are as follows:

Integers, natural numbers, whole numbers, rational numbers, irrational numbers.

The
set of all of these numbers form what is known as the

**real numbers**.**Natural numbers**are the easiest to understand. When you count, you start at 1 and count up to 2, 3, 4, 5, etc. assuming you are counting by 1. The

*natural*way to count is counting by 1. If you think of natural numbers in that sense, it's easy to remember that the natural numbers are the set of numbers {1,2,3,4,5,6,7,....}

When
thinking about

**whole numbers**think of the word*whole*and what it means. When thinking of a whole, you think of the entire thing. The statement*I ate the whole pie*means that there is no portion of the pie left over, no fraction of the pie. The set of whole numbers are basically the set of natural numbers with 0 added to the set {0,1,2,3,4,5,6,7,.....}**Integers**include all the whole numbers and also the negative of the whole numbers. So the set of integers are {...., -4, -3, -2, -1, 0, 1, 2, 3, 4, ....}. Even integers are integers that are divisible by 2, whereas odd integers are not divisible by 2.

**Rational numbers**are all numbers that can be represented as a ratio or a fraction. The easiest way to remember this is notice the word "ratio" is the first 5 letters in rational.

Some
examples of rational numbers are as follows:

4
1/2, 16/5,
7/2 , 4.35, -12.2, -35

**Terminating**and

**repeating decimals**are also rational because every terminating and repeating decimal can be written as a fraction.

For
example 2/3 = .666..... (repeating)

2/11 = .18181...
(repeating)

45% = 0.45
(terminating)

120.5 = 120 1/2 = 241/2
(terminating)

**Irrational numbers**are decimals that do not terminate and do not repeat. A very common irrational number is

**Pi (π)**= 3.1415926...... many square roots are also irrational numbers, although not all of them. Those that can be simplified such as √9 = 3, √16 = 4, etc are rational.

### Fractions and Decimals

**Fractions**and

**decimals**represent the same thing in different ways. If you have half of a pie, you can represent that as the fraction ½ or the decimal 0.50. A quarter is ¼ or 0.25. Converting decimals to fractions can be done simply by moving the decimal point over to the right until you are past the last digit. That number becomes the numerator of the fraction (the number above the bar in a fraction). Then count how many decimal places you moved and that number represents a power of 10 which goes in the denominator of the fraction (the number under the bar in a fraction). For example, take 0.25 and move the decimal point over 2 places to the right to get 25. The 25 becomes the numerator of the fraction. Since you moved the decimal 2 places, that is the 2

^{nd}power of 10 or 10

^{2 }= 10 times 10, which is 100. The fraction becomes 25/100, which is simplified to ¼ since 25 divides evenly into both 25 and 100.

**Other examples:**

0.001
changed to a fraction becomes 1/1000 since the decimal point is moved
3 places to the right, 10

^{3}= 1000.
0.37
changed to a fraction becomes 37/100 since the decimal point is
moved 2 places to the right, 10

^{2}= 100.
2.426
changed to a fraction becomes 2426/1000, which can be simplified to
1213/500 since 2 divides evenly into both 2426 and 1000.

To change a fraction
to a decimal we must divide the numerator by the denominator. For
practice, try long division with fractions and check your work on a
calculator.

Sometimes quantities
will be represented as percentages, which can be converted to
decimals by moving the decimal point over 2 places to the left. An
easy way to remember to move the decimal 2 places is to think of the
word

*cent*. That is part of percent. A cent is 1/100 of a dollar. One percent (represented as 1%) is 1/100 and 100 is 10^{2}. Recall that the power corresponds to the number of decimal places you move.**Example:**

Change
35% to a decimal.

Take
35 and move the decimal point over two places to the left to get
0.35.

•

**Note that the decimal point is to the right of the last number in a whole number.**
From
this point you can change 0.35 into a fraction.

35/100
simplifies to 7/20 since 5 divides evenly into both 35 and 100.

•

**Note that all numbers ending in 5 are divisible by 5.****Example:**

Change
524% to a decimal.

Take
524 and move the decimal point over two places to the left to get
5.24.

From
this point you can change 5.24 into a fraction.

524/100
= 131/25 since 4 divides evenly into both 524 and 100.

•

**Note if you don't know that 4 divides evenly, you can start by dividing the numerator and denominator by 2 since all even numbers are divisible by 2.**
Fractions with a
numerator larger than the denominator are known as

**improper fractions**. These fractions can be changed into a**mixed number**, which is a combination of a whole number and a**proper fraction**(numerator is smaller than the denominator).**Example:**

18/5,
32/7, and 9/4 are
improper fractions.

2
½, 5

^{7}/8 and 11 2/3 are mixed numbers.
To change 18/5 into a
mixed number you divide 5 into 18, which divides into 3 times since 5
times 3 is 15. The remainder is 3, which turns into 3/5 as a fraction
(the remainder over the denominator). The mixed number is then 3
3/5. To change a mixed number to an improper fraction, multiply the
denominator of the fraction by the whole number then add the
numerator. Take that result and put it over the denominator.

**Example:**

Change
4 3/7 to an improper fraction.

7(4)
= 28 (denominator times whole number)

28
+ 3 = 31 (add the numerator)

31/7
(result over the denominator)

You
can also think of a fraction as a

**ratio**between two numbers. The fraction 1/3 can also be written as 1 to 3 or 1:3. To find an equivalent fraction you can simply multiply the numerator and denominator by the same number.**Example:**(1/3)(2/2) = 2/6

(1/3)(3/3)
= 3/9

(1/3)(4/4)
= 4/12

(1/3)(5/5)
= 5/15

These are all

**equivalent fractions**. This technique is used when you find a common denominator when adding or subtracting fractions.

**Example:**2/3 + 5/6

The
common denominator is 6, so we have to get 2/3 into an equivalent
fraction with a denominator of 6. We have to multiply the
denominator by 2 to get 6, so we must multiply the numerator by 2 as
well. Therefore,

2/3
+ 5/6

(2/3)(2/2)
+ 5/6

4/6
+ 5/6

Now we can add the fractions. Finding common denominators for adding and subtracting fractions will be discussed in further detail in a later chapter.

A

**proportion**is two or more equivalent fractions equal to each other.
Examples
of proportions include

2/3
=

*x*/6, 4/9 = 12/27, -1/3 = 2/6,*x*/8 = 6/16 and -3/2 = -12/8.### Prime Numbers, Composite Numbers and Absolute Value

A

**prime number**is a natural number that is only divisible by 1 and itself. Note that 1 is not considered a prime number. The first several prime numbers are 2,3,5,7,11,13,17,19,23,29 and 31.
A

**composite number**is a natural number that is not a prime number, therefore it is divisible by more than just itself and 1.
The
first several composite numbers are
4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28 and 30.

**Absolute value**is essentially the distance between a number and 0 and since distance is always a positive number, the absolute value of any number is always positive.

For
example:

Take
two numbers -4 and 4. The distance from -4 to 0 is 4 and the distance
from 4 to 0 is also 4, so the absolute value of -4 and 4 are both 4.

Absolute
value is noted as follows:

Absolute
value of 4... |4| = 4

Absolute
value of -4... |-4| = 4

Numbers
with the same absolute value are known as

**opposites**. Notice in the following examples that opposites add to 0.
Examples
of opposites:

5
and – 5, 5 + (-5) = 0

14
and -14, 14 + (-14) = 0

-4.3
and 4.3, -4.3 + 4.3 = 0

**Review Problems: Set 1**

**1.**The cost of a general admission ticket to a baseball game is $8. Groups of 20 or more receive a $20 discount. Write a mathematical model to describe the total cost of the tickets for groups of 20 or more.

**2.**Determine whether each of the following is an algebraic expression or an equation.

a. 3

*x +*6 = 14
b. (14

*x*– 32)/16
c.
6

*a*+14*b -*9*c*+ 1
d.

*y*= 13*x –*14
e.

*F =*1.8*C +*32**3**. Use variables to create a formula for the following scenarios:

a. The perimeter of
a rectangle is two times the length plus two times the width.

b. The area of a
triangle is one half the length of the base times the height.

c. The sales price
is the difference between the regular price and the amount of the
discount.

Classify each as rational
or irrational

**4.**1/3

**5.**√14

**6.**3 2/9

**7.**-14.8

8

**.**2π

**9.**15%

Change each fraction to a
decimal and determine whether the decimal is terminating or
repeating.

**10.**8/9

**11.**10/4

**12.**-4/22

13.

**.**24/40

Change each percent to a
decimal.

**14.**14%

**15.**435%

**16.**2.7%

Change each decimal to a
fraction.

**17.**0.85

**18.**0.096

**19.**0.146

Change each improper
fraction to a mixed number.

**20.**23/4

**21.**42/5

**22.**19/6

Change each mixed number
to an improper fraction.

**23.**4 4/7

**24.**10 1/8

**25.**7 2/9

### Real Numbers: Operations and Properties

**Order of Operations**

A

**mathematical operation**is a calculation by mathematical methods which include addition, subtraction, multiplication, division, raising a number to an exponent and taking absolute value. Oftentimes there is more than one operation in an expression. When this occurs there is a specific order in which the expression must be evaluated. First you perform all calculations inside parentheses, then evaluate exponents and roots, multiplication and division from left to right, then addition and subtraction from left to right.
The order of operations
is often difficult to remember, but I like to use a system I learned
when I was in school. Remember the simple sentence “Please excuse
my dear aunt Sally”. The first letter in each word is the first
letter in each operation in the correct order (PEMDAS).

Here's a few
examples on how evaluating the expression ignoring the order of
operations will give a different and wrong answer than using the
order of operations.

**Example:**Evaluate 4+ 3[-1 + 6(5-3)

^{2}+2]

Ignoring the order
of operations you could take 4+3 to get 7, then multiply by -1 to get
-7. From there add 6 to get -1. Next you would multiply by 5 to get
-5, subtract 3 to get --8 then square it to get 64. Finally add 2 to
get 66.

Using the correct order
of operations you get the following

4
+ 3[-1 + 6(2)

^{2 }+2] (evaluate the 5-3 inside the parentheses)
4 + 3[-1 + 6(4) +2]
(evaluate 2

^{2})
4+ 3[-1 + 24 + 2]
(multiply 6 and 4)

4+ 3(25)
(add the numbers inside the brackets)

4 + 75
(multiply 3 and 25)

79
(add 4 and 75)

**Example:**Evaluate 5(6 + 4)

^{2}- 3(2 - 3

^{2}) + 4

Ignoring
the order of operations, you would take 5 times 6 to get 30, then add
4

^{2}to get 30 + 16 = 46. From there subtract 3 to get 43 then multiply by 2 to get 86. Next, subtract 3^{2}which would be 86 – 9 = 77. Finally add 4 to get 81.
Using the correct order
of operations you get the following

5(10)

^{2}- 3(2- 3^{2}) + 4 (evaluate the 6 + 4 inside the parentheses)
5(10)

^{2}- 3(2 - 9) + 4 (evaluate 3^{2}inside the parentheses)
5(10)

^{2}-3(-7) + 4 (evaluate 2 – 9 inside the parentheses)
5(100) - 3(-7) + 4
(evaluate 10

^{2})
500 - 3(-7) + 4
(multiply 5 and 100)

500 + 21 + 4
(multiply -3 and -7)

525
(add 500, 21 and 4)

•

**Note if you have absolute value symbols in the problem, perform them with the parentheses and other grouping symbols first.****Properties**

There are many different
properties of real numbers.

The

**commutative property of addition**states that changing the order when adding doesn't affect the sum.
Think of the word

*commute*and how one might commute to work or to school. This means that a person travels to and from work or school. The distance is the same both directions, assuming the same route is taken to and from. So for any two numbers,*a*and*b*,*a*+*b*=*b*+*a*demonstrates the commutative property.
The

**commutative property of multiplication**states that changing the order when multiplying doesn't affect the outcome. For any two numbers*a*and*b*,*a · b = b · a*.
•

**Note that***a · b*is often written as*ab*.
The

**associative property of addition**states that changing the grouping does not affect the outcome when adding. Think of the meaning of the word*associate*. When one associates with someone, he or she is grouped with that person. Same can be applied here. For any numbers*a,b*and*c, a +*(*b + c*)*=*(*a + b*)*+c*.
The

**associative property of multiplication**follows the same principle and states that changing the grouping does not affect the outcome when multiplying. Therefore, (*ab*)*c*=*a*(*bc*).
The

**distributive property of multiplication**shows how multiplication distributes over addition. For numbers*a,b*and*c*,*a*· (*b + c*) =*ab*+*ac*.
The
identity properties show how when you add 0 to a number or multiply 1
to a number you still get that number.

**Identity property of addition**is (*a*+ 0 =*a*) and the**Identity property of multiplication**is*a*(1)=*a*.
The

**inverse property of addition**shows how when you add a number to its inverse (or opposite), the result is 0. For example -4 + 4= 0.
The

**inverse property of multiplication**shows how when you multiply a number by its inverse, the result is 1. For example 2(½) = 1.
Division properties:

0/

*a*= 0 for all numbers of*a*, except when*a*is 0
0/

*a*is undefined if*a*= 0
0/0 is indeterminate

•

**Note that for all practical purposes, it's important to understand how to use the property more than knowing the name of the property.****Examples:**

**Commutative property of addition**

1 +
2 = 2 + 1

3
= 3

**Commutative property of multiplication**

2· 3 = 3 · 2

6
= 6

It
doesn't matter what order we add or multiply numbers together, the
result will be the same.

**Associative property of addition**

1 +
(2 + 3) = (1 + 2) + 3

1
+ 5 = 3 + 3 (from the order of operations, we add what's inside the
parentheses first)

6
= 6

**Associative property of multiplication**

2 ·
(3 · 4) = (2 · 3) · 4

2
· 12 = 6 · 4 (from the order of operations, we multiply
what's inside the parentheses first)

24
= 24

Grouping
the numbers differently does not affect the answer when adding or
multiplying numbers.

**Distributive property of multiplication**

2(4
+ 5) = 2(4) + 2(5) (multiplied the 2 by 4 and then 2 by 5)

2(9)
= 2(4) + 2(5) (added numbers inside the parentheses on left side of
the = sign)

18
= 8 + 10 (multiplied on both sides of the equation)

18
= 18

**Identity property of addition and identity property of multiplication**

2 + 0 = 2,
2(1) = 2

When you add 0 to a
number, the result is that number. When you multiply 1 to a number,
the result is that number.

**Inverse property of addition and inverse property of multiplication**

2 + (-2) = 0,
2(1/2) = 1

When you add a
number to its opposite (the negative of that number), the result is
0. When you multiply a number by its inverse (switch numerator and
denominator, ex: 1/3 is inverse of 3/1), the result is 1.

### Using Operations With Real Numbers

When adding two
positive numbers, the result is a positive number. When adding two
negative numbers, the result is a negative number.

**Examples:**

4 + 9 = 13, -4 + -9 =
-13.

When subtracting two
positive numbers the result will be positive if the smaller number is
being subtracted from the larger number and negative if the larger
number is being subtracted from the smaller number. The result will
be zero is both numbers have the same value.

**Examples:**

10 - 15 = -5, 15 - 10 =
5, 10 - 10 = 0.

•

**If you have trouble with this concept consider a real life situation involving money. If you have $15 and give someone $10, you still have $5 left. If you have $10 and owe someone $15, you can pay $10 and still owe $5, this having a $5 deficit or -$5.**
When subtracting a
negative number, you add the second number to the first. The double
negative turns into a positive, or an addition. For example, -4 -
(-6) turns into -4 + 6, which is 2. Think of a number line and this
might make more sense. Typically when subtracting you move to the
left on the number line, the the double negative makes you move right
on the number line. It's as if you have two magnets with the
negatives against each other, it repels, pushing the magnets in
opposite directions.

**Examples:**

-3 - (-13) = -3 + 13 =
10, -15 - (-7) = -15 + 7 = -8

Multiplying and
dividing numbers with like signs results in a positive number,
whereas multiplying and dividing numbers of opposite signs yields a
negative number.

**Examples:**

24/6 = 4, -24/6 = -4,
(-2)(-7) = 14, (-2)(7) = -14

If you have a number
raised to a positive integer exponent, you multiply that number by
itself the number of times indicated in the exponent. If you have a
fraction raised to a positive integer exponent, you multiply the
fraction by itself the number of times indicated in the exponent.

**Examples:**

5

^{3 }= 5 (5)(5)
2

^{4}= 2(2)(2)(2)
(1/3)

^{3}= (1/3)(1/3)(1/3).
(3/5)

^{3}= (3/5)(3/5)(3/5)
When multiplying
fractions, multiply the numerators and the denominators separately.

**Examples:**

(4/7)(2/5) =
(4)(2)/(7)(5) = 28/10

(2/9)(5/6) = (2)(5)/(9
)(6) = 10/54

When
dividing fractions, multiply the first fraction by the

**reciprocal**of the second.**Example:**

(4/7) / (2/5) =
(4/7)(5/2) = 20/14

(2/9) / (5/6) =
(2/9)(6/5) = 12/45

Evaluating
Expressions

Sometimes you will
encounter problems where you have to evaluate an expression given the
values of the variables in the expression.

**Example:**Evaluate the expression for x = 4, y = 2 and z = -1

(2xy
– 3)

^{2 }+ 5z - 14
In a problem like
this, you substitute the values for the variables into the
expression. Sometimes people get confused and put the numbers in the
expression but fail to remove the variable. If you think of a

*substitute*, it's a replacement. A substitute teacher is in for the regular teacher. Both teachers are not present at the same time. In a sporting event, when a player is put in as a substitute, the other player leaves the game. So in the previous problem, evaluating would be as follows:
[2(

**4**)(**2**) - 3]^{2}+5(**-1**) -14
Following the order of
operations, you get

(16-3)

^{2 }+5(-1) -14 (parentheses)
13

^{2 }+ 5(-1) - 14 (parentheses)
169 + 5(-1) -14
(exponent)

169 - 5 -14
(multiplication)

150
(subtraction)

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