# The Language of Algebra

## Use the Language of Algebra


### Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is  years old and Alex is  so Alex is  years older than Greg. When Greg was  Alex was  When Greg is  Alex will be  No matter what Greg’s age is, Alex’s age will always be  years more, right? 
In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The  years between them always stays the same, so the age difference is the constant.
In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age  Then we could use  to represent Alex’s age. See .
Letters are used to represent variables. Letters often used for variables are 
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
In algebra, the cross symbol,  is not used to show multiplication because that symbol may cause confusion. Does  mean  (three times ) or  (three times )? To make it clear, use • or parentheses for multiplication.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
1. The sum     means add  plus  which we write as 
2. The difference     means subtract  minus  which we write as 
3. The product     means multiply  times  which we can write as 
4. The quotient     means divide  by  which we can write as 
When two quantities have the same value, we say they are equal and connect them with an equal sign.
An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that  is greater than  it means that  is to the right of  on the number line. We use the symbols  and  for inequalities.
The expressions  can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
When we write an inequality symbol with a line under it, such as  it means  or  We read this  is less than or equal to  Also, if we put a slash through an equal sign,  it means not equal.
We summarize the symbols of equality and inequality in .
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions.  lists three of the most commonly used grouping symbols in algebra.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

### Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

### Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify  we’d first multiply  to get  and then add the  to get  A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
Suppose we have the expression  We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write  as  and  as  In expressions such as  the  is called the base and the  is called the exponent. The exponent tells us how many factors of the base we have to multiply.
We say  is in exponential notation and  is in expanded notation.
For powers of  and  we have special names.
 lists some examples of expressions written in exponential notation.
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

### Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression:
Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.
It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

### Key Concepts

1. Equality Symbol

2. Inequality

1. Exponential Notation

Order of Operations When simplifying mathematical expressions perform the operations in the following order:


### Practice Makes Perfect

Use Variables and Algebraic Symbols 
In the following exercises, translate from algebraic notation to words.
Identify Expressions and Equations 
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions with Exponents 
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
Simplify Expressions Using the Order of Operations 
In the following exercises, simplify.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.