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 | the sum of | ||
Subtraction | the difference of | ||
Multiplication | The product of | ||
Division | The quotient of |
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 of 5 and 3 means add 5 plus 3, which we write as
.
- The difference of 9 and 2 means subtract 9 minus 2, which we write as
.
- The product of 4 and 8 means multiply 4 times 8, which we can write as
.
- The quotient of 20 and 55 means divide 20 by 5, which we can write as
.
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. |
the difference of |
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 |
---|---|
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. |
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 | |
2. | MPG of Versa____MPG of Fit |
Find the values in the chart. | 26____27 |
Compare. | |
MPG of Versa | |
3. | MPG of Mini Cooper____MPG of Fit |
Find the values in the chart. | 27____27 |
Compare. | |
MPG of Mini Cooper | |
4. | MPG of Corolla____MPG of Versa |
Find the values in the chart. | 28____26 |
Compare. | |
MPG of Corolla | |
5. | MPG of Corolla____MPG of Prius |
Find the values in the chart. | 28____48 |
Compare. | |
MPG of Corolla |
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 | |
the difference of | ||
6 times 7 | the product of six and seven | |
the quotient of |
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. | |
The product of six and seven is equal to forty-two. | |
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 multiply
to 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 | |
4. The base |
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 | |
And |
Some students say is simplifies to 25.
Since | |
And |
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:
- Parentheses and other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents
- Simplify all expressions with exponents.
- Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction
- 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. Please Excuse My Dear Aunt Sally.
ORDER OF OPERATIONS
Please | Parentheses |
Excuse | Exponents |
My Dear | Multiplication and Division |
Aunt Sally | Addition and Subtraction |
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.
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. | |
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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