Section 2.1 Simplifying Algebraic Expressions
The terms of an expression are its addends (i.e. quantities separated by plus signs, seen
or unseen).
There are two types of terms that can occur in an algebraic expression. They are:
 Constant terms (or constants) and
 Variable terms.
Each variable term consists of two parts:
1. A numerical coefficient (often called the coefficient), which is the numeric
factor (sign and number) in a term, and
2. A variable part
Constants are also considered to be a numerical coefficient.
Two terms are called like terms if they are both constant terms or if they both contain the
same variables with the same exponents on each variable.
Combining Like Terms
We can simplify an algebraic expression by combining (or adding) the like terms in the
expression. To do this we add the numerical coefficients of each set of like terms by
using the distributive law. This process is outlined below.
To simplify an algebraic expression by combining like terms:
1. Rearrange the terms using the associative and commutative laws of addition so
that like terms are next to each other, if needed. Be careful with the signs of the
coefficients when you do this.
2. Add the coefficients of the like terms using the following alternate forms of the
distributive property: ac  bc  a  bc and ac  bc  a  bc
In order to simplify algebraic expressions involving several operations and/or grouping
symbols, use the properties of real numbers and order of operations.
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Section 2.2 The Addition and Multiplication Properties of Equality
Recall that the solution of an equation is a number that, when substituted for the variable,
results in a true equation. To solve an equation means to find all of the solutions to the
equation. We have found the solution to an equation when we reduce the equation to the
form variable term = constant, where the coefficient on the variable term is positive one.
A linear equation in one variable can be written in the form ax  b  c where a, b, and
c are real numbers and a  0 .
We can use arithmetic to help us determine the solution to a linear equation in one
variable using the following principles of equations.
The addition principle of equations states that if we add the same number to both sides
of the equation we do not change the solution of the equation. In other words, if a  b is
true, then a  c  b  c is also true.
When using the addition principle, we often say that we “add the same number to both
sides of an equation.” We can also “subtract the same number from both sides of an
equation,” since subtraction can be regarded as adding the opposite. This fact leads to the
following principle.
The subtraction principle of equations states that if we subtract the same
number from both sides of the equation we do not change the solution of the
equation. In other words, if a  b is true, then a  c  b  c is also true.
The multiplication principle of equations states that if we multiply both sides of the
equation with the same number (except for zero) we do not change the solution of the
equation. In other words, if a  b is true, then a  c  b  c is also true when c  0 .
When using the multiplication principle, we often say that we “multiply both sides of the
equation by the same number.” We can also “divide both sides of the equation by the
same number,” since division can be regarded as multiplying by the reciprocal. This fact
leads to the following principle.
The division principle of equations states that if we divide both sides of the
equation with the same number (except for zero) we do not change the solution of
the equation. In other words, if a  b is true, then a  c  b  c is also true when
c 0.
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Section 2.3 Solving Linear Equations
When the terms in an equation contain numerical coefficients that are fractions or
decimals, the difficulty in solving the equation is often increased. However, we can “clear
the equation of fractions or decimals” by applying the multiplication principle as follows.
To clear an equation of fractions, multiply every term in the equation by the least
common denominator (LCD) of the fractional coefficients.
To clear an equation of decimals, multiply every term in the equation by 10 n ,
where n is the most number of decimal places on any coefficient in the equation.
When solving equations, we often need to use more than one principle. However, the
order that we use these principles is important. The following procedure provides an
accurate method for solving linear equations.
An Equation-Solving Procedure
1. Use the multiplication principle to clear any fractions or decimals in the equation.
(This step is optional, but can ease computations. You can even do this step after
step 2.)
2. If necessary, use the distributive law to remove parentheses on each side of the
equal symbol. Then simplify each side of the equation by combining the like
terms on each side.
3. Use the addition or subtraction principle, as needed, to get all variable terms on
one side of the equation and all of the constant terms on the other side of the
equation.
4. Combine like terms on both sides of the equation again, if necessary.
5. Use the multiplication or division principle to reduce the coefficient on the
variable term to positive one.
6. Check your solution by substituting it back into the equation to see if it produces a
true equation.
When solving linear equations in one variable, the number and types of solutions that will
occur is one of the following possibilities.
 One Solution Exists – In this case you will find the variable is equal to one
number when solving the equation.
 No Solution Exists – In this case you will find an equation that is always false.
 Infinite Solutions Exist – In this case you will find an equation that is always true
for any real number. Thus the solutions are the set of real numbers.
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Section 2.4 An Introduction to Problem Solving
The goal of every mathematics course is to develop problem solving skills. There are
some steps one can take to make problem solving easier or more manageable.
Five Steps for Problem Solving in Algebra
1. Familiarize yourself with the problem.
2. Translate to mathematical language. This often means writing an equation.
3. Carry out some mathematical manipulation. This often means solving an
equation.
4. Check your possible answer in the original problem.
5. State the answer clearly.
Of these five steps the hardest to implement is the first one. The following list gives some
hints on how to get started with step one.
To Become Familiar with a Problem
1. Read the problem carefully. Try to visualize the problem.
2. Reread the problem, perhaps aloud. Make sure you understand all important
words.
3. List the information given and the question(s) to be answered. Choose a variable
(or variables) to represent the unknown and specify what the variable represents.
4. Look for similarities between the problem and other problems you have already
solved.
5. Find more information. Look up a formula in a book, at a library, or on-line.
Consult a reference librarian or an expert in the field.
6. Make a table that uses all the information you have available. Look for patterns
that may help in the translation.
7. Make a drawing and label it with known and unknown information, using specific
units if given.
8. Think of a possible answer and check the guess. Observe the manner in which the
guess is checked.
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Section 2.5 Formulas and Problem Solving
A formula is an equation that represents the relationship between two or more quantities.
They have letters that represent quantities that can vary, but they may also have letters
that represents constants.
If we are given the values(s) for all but one of the variable term(s) we can evaluate any
formula for the remaining variable term by substituting the known values into the
equation and solving for the remaining term.
Sometimes, we need to “rearrange” the terms in a formula. This “rearrangement” is
called solving for a letter in the formula. For example, let’s say that we want to solve for
the letter x in a formula. We want to rearrange the formula to be in form x = expression,
where the expression on the right-hand side of the equation is obtained from the
rearrangement of the original formula and does not contain the variable x.
We have a procedure we can follow to solve for a letter in a formula. It is outlined below.
To Solve a Formula for a Given Letter
1. If the formula contains fractions, use the multiplication principle to remove them.
2. Use the distributive property to remove parentheses if they occur.
3. Get all terms with the letter for which you are solving for on one side of the equation
and all other terms on the other side using the addition and/or subtraction principles.
4. Combine like terms, if possible. If the terms containing the variable you are solving
for are not like terms, you must factor the expression containing the variable of
interest.
5. If necessary, multiply or divide by the coefficient of the variable of interest to reduce
its coefficient to positive one.
Notice that the steps above are similar to those used in Section 2.3 to solve equations.
The main difference is the possible necessity of factoring in step 3.
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Section 2.6 Percent and Mixture Problem Solving
The percent symbol, %, means “parts per hundred.” Hence, “n parts per hundred” is
denoted as
p% 
p
.
100
For problem solving with percents, it is a necessary skill to be able to convert from
percent notation to decimal notation and vice versa.
 To convert from percent notation to decimal notation, move the decimal point two
places to the left and drop the percent symbol.
 To convert from decimal notation to percent notation, move the decimal point two
places to the right and write a percent symbol.
Solving Percent Problems
There are many real world situations that involve the use of percent. We can solve many
problems representing these situations through the use of the basic percent equation.
The Basic Percent Equation: Percent · base = amount, where the percent is in decimal
notation and the base is the original quantity we are finding the percentage of.
Alternatively, the basic percent equation can be written in the following form where the
percent does not have to be converted to a decimal.
p
amount

100
base
Applications
Merchandising
 Discount – When a retailer has a sale, the object’s price is reduced by a dollar
amount called a discount. Discount = Percent · Original Price. The new price is
found by subtracting the discount from the original price.
New Price = Original Price – Discount.
 Markup – Retailers often sell a product at a higher price than what they paid for it.
The amount they add to their cost is called the markup. Frequently, markup is a
percentage of their cost (often referred to as the wholesale price).
Markup = Percent · Wholesale Price. The price they sell the object for is the sum
of the wholesale price and the markup. New Price = Wholesale Price + Markup.
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Percent increase is used to represent how much a quantity has increased over its original
value. There are many real world applications of percent increase.
A basic equation can be used to solve these types of problems.
Basic Percent Increase Equation: 1  r  base  amount
where r represents the percent increase in decimal notation,
the base represents the original value, and
the amount is the quantity that the original value increased to.
Alternatively, the basic percent equation can be written in the following form where the
percent does not have to be converted to a decimal.
p
amount of increase

100
original value
Percent decrease is used to represent how much a quantity has decreased from its
original value. There are many real world applications of percent decrease.
A basic equation can be used to solve these types of problems.
Basic Percent Decrease Equation: 1  r  base  amount
where r represents the percent decrease in decimal notation,
the base represents the original value, and
the amount is the quantity that the original value decreased to.
Alternatively, the basic percent equation can be written in the following form where the
percent does not have to be converted to a decimal.
p
amount of decrease

100
original value
Mixture Problems – involve problems where solutions or other items are mixed to
produce a desired concentration or other mixture. To solve these types of problems:
1. Organize the information into a table.
2. Use the table to construct a linear equation to solve for the desired quantities.
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Section 2.7 Further Problem Solving
One problem solving technique we have used in previous problems was to create a table
to organize the information in the problem and then use that information to create an
appropriate equation to solve for the requested quantities.
This problem solving technique can be used in multiple applications including the
following introduced in this section.
Distance, Rate, and Time Problems
The distance one travels is equal to the rate or speed of travel times the length of time one
has traveled. Mathematically we write the formula d  rt .
Money Problems
The total amount of money one has can be found by multiplying the quantity of bills or
coins of each denomination or value by its denomination or value.
Simple Interest Problems
Interest, denoted as I, is the amount of money paid for the use of someone else’s money.
The total amount borrowed (whether by an individual from a bank in the form of a loan
or by a bank from an individual in the form of a savings account) is called the principal
and is denoted by P. The rate of interest, denoted as r and expressed as a percent, is the
amount charged for the use of the principal for a given period of time, usually on a yearly
basis. The total amount the lender receives after loaning the money for a period of t years
is called the future value of the loan and is denoted as A.
There are two general types of interest: simple and compounded.
Simple interest is interest charged only on the principal. If a principal of P dollars is
borrowed for a period of t years at an annual interest rate r, expressed as a decimal, the
interest I charged is I  Pr t . The future value of the loan is A  P  Pr t  P1  rt  .
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Section 2.8 Solving Linear Inequalities
Definition
A linear inequality in one variable is an inequality that can be written in one of the
following forms, where a, b, and c are real numbers and a  0 .
ax  b  c
ax  b  c
ax  b  c
ax  b  c
A solution of an inequality is a value that makes the inequality a true statement.
The solution set of an inequality is the set of values that contains all of the solutions to
the inequality.
Inequalities containing one inequality symbol are called simple inequalities.
Inequalities containing two inequality symbols are called compound inequalities.
There are many ways to describe the solution set of an inequality. Three common
methods are:
1. Inequality notation
2. Graphical notation
3. Interval notation
Inequality Notation
xc
xc
xc
xc
a xb
a xb
a xb
a xb
Graphical Notation
Interval Notation
 , c
 , c
c, 
c, 
a, b
a, b
a, b
a, b
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Addition Property of Inequalities
Given any linear inequality, we can add or subtract any value from both sides of the
inequality without changing its solutions. In other words,
Given expressions a, b, and c,
If a  b , then a  c  b  c .
If a  b , then a  c  b  c .
If a  b , then a  c  b  c .
If a  b , then a  c  b  c .
Multiplication Property of Inequalities
Given expressions a, b, and c with c  0 (c is positive),
If a  b , then ac  bc .
If a  b , then ac  bc .
If a  b , then ac  bc .
If a  b , then ac  bc .
Given expressions a, b, and c with c  0 (c is negative),
If a  b , then ac  bc .
If a  b , then ac  bc .
If a  b , then ac  bc .
If a  b , then ac  bc .
This property holds true for division as well, because division is defined as multiplying
by the reciprocal of a number.
To Solve a Simple Linear Inequality in One Variable
1. Simplify each side of the inequality by removing parentheses, combining like
terms, eliminating fractions, etc.
2. Use the addition property to get the variable terms on one side of the inequality
and the constant terms on the opposite side.
3. Use the multiplication property to reduce the coefficient on the variable term to
positive one. Don’t forget to change the order symbol if, and only if, you
multiplied or divided by a negative number!
To Solve a Compound Linear Inequality in One Variable
1. Simplify all three parts (left, middle, and right) of the inequality by removing
parentheses, combining like terms, eliminating fractions, etc.
2. Use the addition property to get the variable terms in the middle of the inequality
and the constant terms on the left and right sides. Note: You must add or
subtract the same object from all three parts!
3. Use the multiplication property to reduce the coefficient on the variable term to
positive one. Note: You must multiply or divide by the same value to all three
parts! Don’t forget to change the order symbols if, and only if, you multiplied
or divided by a negative number!
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To solve word problems with inequalities, we need to be able to translate the words into
the correct inequality symbol. The following table represents the most common phrases
and how they should be translated.
Important Word Phrase
is at least
is at most
cannot exceed
must exceed
is less than
is more than
no more than
no less than
is between
Inequality Symbol



>
<
>


? depends…
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