CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra
1.2 Use the Language of Algebra
Learning Objectives
By the end of this section, you will be able to:
- Use variables and algebraic symbols
- Identify expressions and equations
- Simplify expressions with exponents
- Simplify expressions using the order of operations
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 the table below.
| Greg’s age | Alex’s age |
|---|---|
Letters are used to represent variables. Letters often used for variables are .
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 1.1 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 and | ||
| Subtraction | the difference of and | ||
| Multiplication | The product of and | ||
| Division | 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 )? 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 and means add plus , which we write as .
- The difference of and means subtract minus , which we write as .
- The product of and means multiply times , which we can write as .
- The quotient of and means divide by , which we can write as .
EXAMPLE 1
Translate from algebra to words:
| a. |
| 12 plus 14 |
| the sum of twelve and fourteen |
| b. |
| 30 times 5 |
| the product of thirty and five |
| c. |
| 64 divided by 8 |
| the quotient of sixty-four and eight |
| d. |
| minus |
| the difference of and |
TRY IT 1.1
Translate from algebra to words.
Show Answer
- 18 plus 11; the sum of eighteen and eleven
- 27 times 9; the product of twenty-seven and nine
- 84 divided by 7; the quotient of eighty-four and seven
- p minus q; the difference of p and q
TRY IT 1.2
Translate from algebra to words.
Show Answer
- 47 minus 19; the difference of forty-seven and nineteen
- 72 divided by 9; the quotient of seventy-two and nine
- m plus n; the sum of m and n
- 13 times 7; the product of thirteen and seven
When two quantities have the same value, we say they are equal and connect them with an equal sign.
Equality Symbol
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 is greater than
is to the right of on the number line
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 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:
- >
- <
| a. |
| 20 is less than or equal to 35 |
| b. |
| 11 is not equal to 15 minus 3 |
| c. |
| > |
| 9 is greater than 10 divided by 2 |
| d. |
| < |
| plus 2 is less than 10 |
TRY IT 2.1
Translate from algebra to words.
- >
- <
Show Answer
- fourteen is less than or equal to twenty-seven
- nineteen minus two is not equal to eight
- twelve is greater than four divided by two
- x minus seven is less than one
TRY IT 2.2
Translate from algebra to words.
- <
- >
Show Answer
- nineteen is greater than or equal to fifteen
- seven is equal to twelve minus five
- fifteen divided by three is less than eight
- y minus three is greater than six
EXAMPLE 3
The information in (Figure 1) 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
| a. | |
| MPG of Prius____MPG of Mini Cooper | |
| Find the values in the chart. | 48____27 |
| Compare. | 48 > 27 |
| MPG of Prius > MPG of Mini Cooper |
| b. | |
| MPG of Versa____MPG of Fit | |
| Find the values in the chart. | 26____27 |
| Compare. | 26 < 27 |
| MPG of Versa < MPG of Fit |
| c. | |
| MPG of Mini Cooper____MPG of Fit | |
| Find the values in the chart. | 27____27 |
| Compare. | 27 = 27 |
| MPG of Mini Cooper = MPG of Fit |
| d. | |
| MPG of Corolla____MPG of Versa | |
| Find the values in the chart. | 28____26 |
| Compare. | 28 > 26 |
| MPG of Corolla > MPG of Versa |
| e. | |
| MPG of Corolla____MPG of Prius | |
| Find the values in the chart. | 28____48 |
| Compare. | 28 < 48 |
| MPG of Corolla < MPG of Prius |
TRY IT 3.1
Use Figure 1 to fill in the appropriate < >.
- MPG of Prius_____MPG of Versa
- MPG of Mini Cooper_____ MPG of Corolla
Show Answer
- >
- <
TRY IT 3.2
Use Figure 1 to fill in the appropriate < >.
- MPG of Fit_____ MPG of Prius
- MPG of Corolla _____ MPG of Fit
Show Answer
- <
- <
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.
| parentheses | |
| brackets | |
| braces |
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:
| Expression | Words | Phrase |
|---|---|---|
| the sum of three and five | ||
| minus one | the difference of and one | |
| 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:
| a. | This is an equation—two expressions are connected with an equal sign. |
| b. | This is an expression—no equal sign. |
| c. | This is an expression—no equal sign. |
| d. | This is an equation—two expressions are connected with an equal sign. |
TRY IT 4.1
Determine if each is an expression or an equation:
Show Answer
- equation
- expression
TRY IT 4.2
Determine if each is an expression or an equation:
Show Answer
- expression
- equation
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.
Exponential Notation
For any expression is a factor multiplied by itself times if is a positive integer.
The expression is read to the power.
For powers of and , we have special names.
The table below lists some examples of expressions written in exponential notation.
| Exponential Notation | In Words |
|---|---|
| to the second power, or squared | |
| to the third power, or cubed | |
| to the fourth power | |
| to the fifth power |
EXAMPLE 5
Write each expression in exponential form:
| a. The base 16 is a factor 7 times. | |
| b. The base 9 is a factor 5 times. | |
| c. The base is a factor 4 times. | |
| d. The base is a factor 8 times. |
TRY IT 5.1
Write each expression in exponential form:
Show Answer
415
TRY IT 5.2
Write each expression in exponential form:
Show Answer
79
EXAMPLE 6
Write each exponential expression in expanded form:
a. The base is and the exponent is , so means
b. The base is and the exponent is , so means
TRY IT 6.1
Write each exponential expression in expanded form:
Show Answer
TRY IT 6.2
Write each exponential expression in expanded form:
Show Answer
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.
EXAMPLE 7
Simplify: .
| Expand the expression. | |
| Multiply left to right. | |
| Multiply. |
TRY IT 7.1
Simplify:
Show Answer
- 125
- 1
TRY IT 7.2
Simplify:
Show Answer
- 49
- 0
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.
Order of Operations
When simplifying mathematical expressions perform the operations in the following order:
1. Parentheses and other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents
- Simplify all expressions with exponents.
3. Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
4. 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.
| 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:
| a. | |
| Are there any parentheses? No. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply first. | |
| Add. | |
| b. | |
| Are there any parentheses? Yes. | |
| Simplify inside the parentheses. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply. |
TRY IT 8.1
Simplify the expressions:
Show Answer
- 2
- 14
TRY IT 8.2
Simplify the expressions:
Show Answer
- 35
- 99
EXAMPLE 9
Simplify:
| a. | |
| 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. |
| b. | |
| 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. |
TRY IT 9.1
Simplify:
Show Answer
18
TRY IT 9.2
Simplify:
Show Answer
9
EXAMPLE 10
Simplify: .
| 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. |
TRY IT 10.1
Simplify:
Show Answer
16
TRY IT 10.2
Simplify:
Show Answer
23
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
EXAMPLE 11
.
| 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. | |
TRY IT 11.1
Simplify:
Show Answer
86
TRY IT 11.2
Simplify:
Show Answer
1
EXAMPLE 12
Simplify: .
| If an expression has several exponents, they may be simplified in the same step. | |
| Simplify exponents. | |
| Divide. | |
| Add. | |
| Subtract. | |
TRY IT 12.1
Simplify:
Show Answer
81
TRY IT 12.2
Simplify:
Show Answer
75
ACCESS ADDITIONAL ONLINE RESOURCES
Key Concepts
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | the sum of and | ||
| Multiplication | The product of and | ||
| Subtraction | the difference of and | ||
| Division | divided by | The quotient of and |
- Equality Symbol
- is read as is equal to
- The symbol is called the equal sign.
- Inequality
- < is read is less than
- is to the left of on the number line
- > is read is greater than
- is to the right of on the number line
| 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 |
- Exponential Notation
- For any expression is a factor multiplied by itself times, if is a positive integer.
- means multiply factors of
- The expression of is read to the power.
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.
Glossary
- expressions
- An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
- equation
- An equation is made up of two expressions connected by an equal sign.
Practice Makes Perfect
Use Variables and Algebraic Symbols
In the following exercises, translate from algebraic notation to words.
| 1. | 2. |
| 3. | 4. |
| 5. | 6. |
| 7. | 8. |
| 9. | 10. |
| 11. < | 12. < |
| 13. | 14. |
| 15. | 16. |
| 17. > | 18. > |
| 19. | 20. |
| 21. | 22. |
Identify Expressions and Equations
In the following exercises, determine if each is an expression or an equation.
| 23. | 24. |
| 25. | 26. |
| 27. | 28. |
| 29. | 30. |
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
| 31. | 32. |
| 33. | 34. |
In the following exercises, write in expanded form.
| 35. | 36. |
| 37. | 38. |
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.
| 39.
a. b. |
40.
a. b. |
| 41. | 42. |
| 43. | 44. |
| 45. | 46. |
| 47. | 48. |
| 49. | 50. |
| 51. | 52. |
| 53. | 54. |
| 55. | 56. |
| 57. | 58. |
| 59. | 60. |
| 61. | 62. |
| 63. | 64. |
Everyday Math
|
65. Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol ( = ,<, >).
|
66. Elevation In Colorado there are more than mountains with an elevation of over The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.
|
Writing Exercises
| 67.Explain the difference between an expression and an equation. | 68. Why is it important to use the order of operations to simplify an expression? |
Answers
| 1. 16 minus 9, the difference of sixteen and nine | 3. 5 times 6, the product of five and six | 5. 28 divided by 4, the quotient of twenty-eight and four |
| 7. x plus 8, the sum of x and eight | 9. 2 times 7, the product of two and seven | 11. fourteen is less than twenty-one |
| 13. thirty-six is greater than or equal to nineteen | 15. 3 times n equals 24, the product of three and n equals twenty-four | 17. y minus 1 is greater than 6, the difference of y and one is greater than six |
| 19. 2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six | 21. a is not equal to 7 times 4, a is not equal to the product of seven and four | 23. equation |
| 25. expression | 27. expression | 29. equation |
| 31. 37 | 33. x5 | 35. 125 |
| 37. 256 | 39.
a. 43 b. 55 |
41. 5 |
| 43. 34 | 45. 58 | 47. 6 |
| 49. 13 | 51. 4 | 53. 35 |
| 55. 10 | 57. 41 | 59. 81 |
| 61. 149 | 63. 50 | 65. a. > b. c. < d. < e. > |
| 67. Answer may vary. |
Attributions
This chapter has been adapted from “Use the Language of Algebra” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.