Elementary Algebra
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Section 1.1: Fractions
Converting between improper and mixed fractions:
Multiplying fractions: One way to remember how to multiply fractions is to think of finding the area of a
rectangle:
General rule for multiplying fractions:
Reducing fractions:
 You can reduce the numerator and denominator if they are both divisible by the same number.
 A reliable method for reducing fractions is to factor the numerator and denominator and cancel
the common factors.
 Here is a list of the first twenty prime numbers to aid your factoring:
2
3
5
7
11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
 Implication for multiplication: cancel before you multiply
Dividing fractions:
General rule for dividing fractions:
Adding (and subtracting) fractions: The key to adding fractions is to remember that you can only add
together pieces that are the same size.
 If the denominators are not the same, then you have to change your fractions so that they have
the same denominators so that you are adding pieces that are the same size.
 You can multiply the denominators together to get a common denominator, but you will do the
least amount of work in the long run if you use the Least Common Denominator (LCD).
 To find the LCD, use every factor in both denominators, and if a factor is repeated, then use the
largest number of repeats.
General rule for adding or subtracting fractions:
Section 1.2: Exponents, Order of Operations, and Inequality
The language of exponents:
1. Exponents are a mathematical shorthand for the operation of repeated multiplication.
2. Unlike the operations of addition ( + ), subtraction (– ) , multiplication (  ), and division (  ),
mathematicians do not use a special symbol to represent the operation of repeated multiplication.
Instead, they represent repeated multiplication with a superscript number (the “exponent”) after the
number that gets repeatedly multiplied (the “base”):
3  3  3  3  3 = 35.
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Order of Operations:
There is an agreed-upon order of carrying out the calculations in the above expression to guarantee that
everybody gets the same answer. That order is called the “Order of Operations”, and they are:
 Step 0: Do any calculations within grouping symbols (like parentheses ( ), square brackets [ ], curly
brackets { }, or a fraction bar) first, from the innermost to the outermost. Fraction bars are also a
grouping symbol, so you have to do calculations above and below the fraction bar before you divide.
 Step 1: Do any calculations with exponents.
 Step 2: Do any multiplications or divisions from left to right.
 Step 3: Do any additions or subtractions from left to right.
Relational Symbols:







1
 0.5 , x = 7
2
1
 (“is not equal to”) Example:  0.3 , x + 2  x + 3
3
1
 (“is approximately equal to”) Example:  0.333
3
> (“is greater than”) Example: 5 > 2
< (“is less than”) Example: -1 < 1
 (“is greater than or equal to”) Example: 7  5, 7  7
 (“is less than or equal to”) Example: 3  21, 3  3
= (“is equal to”) Example:
Section 1.3: Variables, Expressions, and Equations
A variable is a symbol, usually a letter like x, y, or z, used to represent any unknown number.
An algebraic expression is a sequence of numbers, variables, grouping symbols, and symbols of operations
(addition, subtraction, multiplication, division, or exponents) arranged according to the order of operations.
Algebraic expressions are meant to capture the steps of a multi-step calculation without being restricted to fixed
numbers all the time.
Examples:
x+5
2m – 9
8p2 + 6(p – 2)
Evaluating an algebraic expression means to plug a given number into the expression and calculate a
numerical answer.
Examples:
2m – 9 evaluated for m = 10 is: 2(10) – 9 = 20 – 9 = 11
Equations:
We write an equation when we know what we want the answer to a multi-step to calculation to turn out
to be.
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A solution to an equation is a number that we can “plug into” each side of the equation and have both
sides be equal. If we are given a possible solution to an equation, all we have to do is plug it into the equation
and see if both sides are equal to determine if it truly a solution or not.
Section 1.4: Real Numbers and the Number Line
Natural (Counting) Numbers:
{1, 2, 3, 4, … }
Whole Numbers:
{0, 1, 2, 3, 4, … }
Integers:
{…, -3, -2, -1, 0, 1, 2, 3, 4, … }
Rational Numbers:
{x | x is a quotient of two integers with denominator not zero}
Set builder notation: {x | x has a certain property}
- useful for when it is inconvenient or impossible to list all possible elements of a set
Irrational Numbers:
{x | x is a point on the number line that can not be represented as a rational number}
Real Numbers:
{x | x is a rational or an irrational number}
The Number Line and Ordering of Numbers:
Any two real numbers can only be greater than, less than, or equal to each other…
For any two real number a and b, a is less than b if a is to the left of b on the number line.
For any two real number a and b, a is greater than b if a is to the right of b on the number line.
The additive inverse of a number x is the number that is the same distance from 0 on the number line as x, but
is on the opposite side of 0. (The additive inverse is the negative of a number.)
Double Negatives:
For any real number x: -(-x) = x.
Absolute Value:
 x if x  0
For any real number x: x  
.
 x if x  0
Section 1.5: Adding and Subtracting Real Numbers
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Adding Numbers on a Number Line:
Adding a positive number to a given number means move to the right of the given number by the positive
number.
Adding a negative number to a given number means move to the left of the given number by the absolute value
of the negative number.
Rule for Adding Numbers with the Same Sign:
Rule for Adding Numbers with the Different Signs:
New way to think of Subtracting Numbers:
Definition of Subtraction:
For any real numbers x and y,
x – y = x + (-y)
That is, to subtract y from x, add the additive inverse of y to x.
Rule for Subtracting Numbers:
Translating English phrases for adding or subtracting into arithmetic expressions:
Word or Phrase
Sum of
Added to
More than
Increased by
Plus
Difference between
Subtracted from
Less
Less than
Decreased by
Minus
Example
The sum of -3 and 4
5 added to -8
12 more than -5
-6 increased by 13
3 plus 2
The difference between 10 and 7
12 subtracted from 8
6 less 5
6 less than 5
12 decreased by 6
8 minus 5
Section 1.6: Multiplying and Dividing Real Numbers
Multiplication by zero: x  0 = 0
Multiplying Numbers with Different Signs: x(-y) = (-x)y = -xy
Multiplying Numbers with the Same Signs: (-x)(-y) = xy
Translation
-3 + 4
-8 + 5
-5 + 12
-6 + 13
2+2
10 – 7
12 – 8
6–5
5–6
12 – 6
8–5
Elementary Algebra
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Reciprocal or Multiplicative Inverse:
Pairs of numbers whose product is 1 are reciprocals, or multiplicative inverses of each other.
To find a reciprocal, just invert the fraction.
Zero has no multiplicative inverse.
1
The notation for a reciprocal using exponents is to raise the number to the power of -1: 51 
1  2
3
,  
5  3
2
Division:
Division can be written as multiplication by a fraction, which is the multiplicative inverse of the denominator:
x
x
1
0
is undefined. But,  0 is okay to do).
 x  (Note: y  0, because
0
y
y
y
Converting English Statements into Mathematical Statements:
English
Product of
Times
Twice
Of (used with fractions and %)
As much as (used with fractions)
Math


2y


Quotient of
Divided by
Ratio of



Section 1.7: Properties of Real Numbers
Commutative Properties: You can add or multiply two numbers in any order. Algebraic way of saying this:
ab  ba
ab  ba
Associative Properties: If you have to add three numbers, you can add the first two first and then add the third
one, or you can add the last two first and then add the first one. The same applies for multiplication. Algebraic
way of saying this:
 a  b   c  a  b  c 
 ab  c  a  bc 
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Identity Properties: There are special numbers that when added to or multiplied by any real number leaves the
original number unchanged. These special numbers are called the identity elements and are 0 (for addition) and
1 (for multiplication).
a0  a
a 1  a
The identity property is the key behind how we simplify and add fractions.
Inverse Properties: Every number has a special number called its inverse such that when the inverse is added
to or multiplied by that number, the answer is the identity element. The additive inverse is the negative of a
number, while the multiplicative inverse is the reciprocal of a number.
a   a   0
a
1
1
a
Distributive Property: If you have to multiply a sum of numbers by a factor, you can instead sum up the
products of each addend with the factor. Algebraic way of saying this:
a  b  c   ab  ac
 b  c  a  ba  ca
Simplifying Algebraic Expressions by Removing Parentheses Using the Distributive Property:
Oftentimes, removing parentheses results in a simplified algebraic expression that involves one less
calculation...
Section 1.8: Simplifying Expressions
The distributive property for simplifying expressions:
a  b  c   ab  ac
 b  c  a  ba  ca
Combining like terms for simplifying expressions:
 A term is a number, a variable, or a product or quotient of numbers and variables raised to powers.
 The numerical coefficient is the number part of any term.
Example: Circle the numerical coefficient in each of the above terms.
 Each piece of a term that makes up a product is called a factor (just like in arithmetic).
Example: identify the factor in the above examples.
Elementary Algebra
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

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Terms are combined by addition or subtraction.
Like terms are terms with exactly the same variables and exponents (but possibly different coefficients),
and can be combined using addition or subtraction into a single term.
Recognizing and combining like terms is a key skill in simplifying expressions.
The distributive property “run in reverse” is the reason you can combine like terms:
Number trick example:
o Pick a number.
o Double it.
o Add 5.
o Now double that answer.
o Subtract four times your original number.
o The answer you get is 10, no matter what number you picked.