**2.1 Use the Language of Algebra**

The topics covered in this section are:

- Use variables and algebraic symbols
- Identify expressions and equations
- Simplify expressions with exponents
- Simplify expressions using the order of operations

**2.1.1 Use Variables and Algebraic Symbols**

Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 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 3 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 the table below.

Greg’s Age | Alex’s Age |
---|---|

12 | 15 |

20 | 23 |

35 | 38 |

Letters are used to represent variables. Letters often used for variables are and .

**VARIABLES AND CONSTANTS**

- A variable is a letter that represents a number or quantity whose value may change.
- A constant is a number whose value always stays the same.

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.

Operation | Notation | Say: | The result is… |
---|---|---|---|

Addition | plus | the sum of and | |

Subtraction | minus | the difference of and | |

Multiplication | times | The product of and | |

Division | , , , b)a | divided by | The quotient of and |

In algebra, the cross symbol, , is not used to show multiplication because that symbol may cause confusion. Does mean(three times) or (three times 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.

- The
*sum*5*of*3 means add 5 plus 3, which we write as .*and* - The
*difference**of*92 means subtract 9 minus 2, which we write as .*and* - The
*product**of*48 means multiply 4 times 8, which we can write as .*and* - The
*quotient**of*2055 means divide 20 by 5, which we can write as .*and*

**Example 1**

Translate from algebra to words:

**Solution**

1. |

12 plus 14 |

the sum of twelve and fourteen |

2. |

30 times 5 |

the product of thirty and five |

3. |

64 divided by 8 |

the quotient of sixty-four and eight |

4. |

minus |

the difference of and |

When two quantities have the same value, we say they are equal and connect them with an *equal sign*.

### Equality Symbol

is read is equal to

The symbol is called the 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.

### Inequality

is read is less than

is to the left of on the number line.

is read [latex]a[/latex] is greater than [latex]b[/latex]

is to the right of [latex]b[/latex] on the number line

The expressions and can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general

is equivalent to . For example, is equivalent to .

is equivalent to . For example, is equivalent to .

When we write an inequality symbol with a line number 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 the table below.

Algebraic Notation | Say |
---|---|

is equal to | |

is not equal to | |

is less than | |

is greater than | |

is less than or equal to | |

is greater than or equal to |

### Symbols and

The symbols and each have a smaller side and a larger side.

- smaller side larger side
- larger side smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

**Example 2**

Translate from algebra to words:

**Solution**

1. |

20 is less than or equal to 35 |

2. |

11 is not equal to 15 minus 3 |

3. |

9 is greater than 10 divided by 2 |

4. |

plus 2 is less than 10 |

**Example 3**

The information in the figure below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol, in each expression to compare the fuel economy of the cars.

- MPG of Prius_____ MPG of Mini Cooper
- MPG of Versa_____ MPG of Fit
- MPG of Mini Cooper_____ MPG of Fit
- MPG of Corolla_____ MPG of Versa
- MPG of Corolla_____ MPG of Prius

**Solution**

1. | MPG of Prius____MPG of Mini Cooper |

Find the values in the chart. | 48____27 |

Compare. | |

MPG of Prius MPG of Mini Cooper | |

2. | MPG of Versa____MPG of Fit |

Find the values in the chart. | 26____27 |

Compare. | |

MPG of Versa MPG of Fit | |

3. | MPG of Mini Cooper____MPG of Fit |

Find the values in the chart. | 27____27 |

Compare. | |

MPG of Mini Cooper MPG of FIt | |

4. | MPG of Corolla____MPG of Versa |

Find the values in the chart. | 28____26 |

Compare. | |

MPG of Corolla MPG of Versa | |

5. | MPG of Corolla____MPG of Prius |

Find the values in the chart. | 28____48 |

Compare. | |

MPG of Corolla MPG of Prius |

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. The table below lists three of the most commonly used grouping symbols in algebra.

**Common Grouping Symbols**

parentheses | |

brackets | |

braces |

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

**2.1.2 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:

Expression | Words | Phrase |
---|---|---|

3 plus 5 | the sum of three and five | |

minus one | the difference of and one | |

6 times 7 | the product of six and seven | |

divided by | the quotient of and |

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:

Equation | Sentence |
---|---|

The sum of three and five is equal to eight. | |

minus one equals fourteen. | |

The product of six and seven is equal to forty-two. | |

is equal to fifty-three. | |

plus nine is equal to two minus three. |

**Expressions and Equations**

- An
**expression**is a number, a variable, or a combination of numbers and variables and operation symbols. - An
**equation**is made up of two expressions connected by an equal sign.

**Example 4**

Determine if each is an expression or an equation:

**Solution**

1. | This is an equation—two expressions are connected with an equal sign. |

2. | This is an expression—no equal sign. |

3. | This is an expression—no equal sign. |

4. | This is an equation—two expressions are connected with an equal sign. |

**5.1.3 Simplify Expressions with Exponents**

To simplify a numerical expression means to do all the math possible. For example, to simplify we’d first multiplyto get 8 and then add the 1 to ge 9. 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:

Supposed 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 2 is called the **base** and the 3 is called the exponent. The exponent tells us how many factors of the base we have to multiply.

means multiply three factors of 2

We say is in exponential notation and is in expanded notation.

**EXPONENTIAL NOTATION**

For any expression , is a factor multiplied by it self times if is a positive integer.

means multiply factors of

The expression is read as to the power.

For powers of and , we have special names.

is read as “ squared”

is read as “ cubed”

The table below lists some example of expressions written in exponential notation.

Exponential Notation | In Words |
---|---|

7 to the second power, or 7 squared | |

5 to the third power, or 5 cubed | |

9 to the fourth power | |

12 to the fifth power |

**Example 5**

Write each expression in exponential form:

**Solution**

1. The base 16 is a factor 7 times. | |

2. The base 9 is a factor 5 times. | |

3. The base is a factor 4 times. | |

4. The base is a factor 8 times. |

**Example 6**

Write each expression in expanded form:

**Solution**

- The base is 8 and the exponent is 6, so means
- The base is and the exponent is 5, so means

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

**Example 7**

Simplify: .

**Solution**

Expand the expression. | |

Multiply left to right. | |

Multiply. |

**2.1.4 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:

**Some students say is simplifies to 49.**

Since gives . | |

And is . |

**Some students say is simplifies to 25.**

Since is . | |

And makes . |

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.

**ORDER OF OPERATIONS**

When simplifying mathematical expressions perform the operations in the following order:

**P**arentheses and other Grouping Symbols- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

**E**xponents- Simplify all expressions with exponents.

**M**ultiplication and**D**ivision- Perform all multiplication and division in order from left to right. These operations have equal priority.

**A**ddition and**S**ubtraction- Perform all addition and subtraction in order from left to right. These operations have equal priority.

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. **P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally.

**ORDER OF OPERATIONS**

Please | Parentheses |

Excuse | Exponents |

My Dear | Multiplication and Division |

Aunt Sally | Addition and Subtraction |

It’s good that ‘**M**y **D**ear’ goes together, as this reminds us that **m**ultiplication and **d**ivision 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, ‘**A**unt **S**ally’ goes together and so reminds us that **a**ddition and **s**ubtraction also have equal priority and we do them in order from left to right.

**Example 8**

Simplify the expressions

**Solution**

1. | |

Are there any parentheses? No. | |

Are there any exponents? No. | |

Is there any multiplication or division? Yes. | |

Multiply first. | |

Add. | |

2. | |

Are there any parentheses? Yes. | |

Simplify inside the parentheses. | |

Are there any exponents? No. | |

Is there any multiplication or division? Yes. | |

Multiply. |

**Example 9**

Simplify the expressions

**Solution**

1. | |

Are there any parentheses? No. | |

Are there any exponents? No. | |

Is there any multiplication or division? Yes. | |

Multiply and divide from left to right. Divide. | |

Multiply. | |

2. | |

Are there any parentheses? No. | |

Are there any exponents? No. | |

Is there any multiplication or division? Yes. | |

Multiply and divide from left to right. | |

Multiply. | |

Divide. |

**Example 10**

Simplify: .

**Solution**

Parentheses? Yes, subtract first. | |

Exponents? No. | |

Multiplication or division? Yes. | |

Divide first because we multiply and divide left to right. | |

Any other multiplication or division? Yes. | |

Multiply. | |

Any other multiplication or division? No. | |

Any addition or subtraction? Yes. |

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

**Example 11**

Simplify: .

**Solution**

Are there any parentheses (or other grouping symbol)? Yes. | |

Focus on the parentheses that are inside the brackets. | |

Subtract. | |

Continue inside the brackets and multiply. | |

Continue inside the brackets and subtract. | |

The expression inside the brackets requires no further simplification. | |

Are there any exponents? Yes. | |

Simplify exponents. | |

Is there any multiplication or division? Yes. | |

Multiply. | |

Is there any addition or subtraction? Yes. | |

Add. | |

Add. | |

**Example 12**

Simplify: .

**Solution**

If an expression has several exponents, they may be simplified in the same step. | |

Simplify exponents. | |

Divide. | |

Add. | |

Subtract. | |

**Licenses and Attributions**

**Licenses and Attributions**

*CC Licensed Content, Original*

*Revision and Adaptation. Provided by: Minute Math. License: CC BY 4.0*

*CC Licensed Content, Shared Previously*

*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/2-1-use-the-language-of-algebra*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*