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…
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:

Solution
 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.

1. 18 plus 11; the sum of eighteen and eleven
2. 27 times 9; the product of twenty-seven and nine
3. 84 divided by 7; the quotient of eighty-four and seven
4. p minus q; the difference of p and q

TRY IT 1.2

Translate from algebra to words.

1. 47 minus 19; the difference of forty-seven and nineteen
2. 72 divided by 9; the quotient of seventy-two and nine
3. m plus n; the sum of m and n
4. 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:

1. >
2. <
Solution
 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.

1. >
2. <
1. fourteen is less than or equal to twenty-seven
2. nineteen minus two is not equal to eight
3. twelve is greater than four divided by two
4. x minus seven is less than one

TRY IT 2.2

Translate from algebra to words.

1. <
2. >
1. nineteen is greater than or equal to fifteen
2. seven is equal to twelve minus five
3. fifteen divided by three is less than eight
4. 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.

(credit: modification of work by Bernard Goldbach, Wikimedia Commons)
1. MPG of Prius_____ MPG of Mini Cooper
2. MPG of Versa_____ MPG of Fit
3. MPG of Mini Cooper_____ MPG of Fit
4. MPG of Corolla_____ MPG of Versa
5. MPG of Corolla_____ MPG of Prius
Solution
 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 < >.

1. MPG of Prius_____MPG of Versa
2. MPG of Mini Cooper_____ MPG of Corolla
1. >
2. <

TRY IT 3.2

Use Figure 1 to fill in the appropriate < >.

1. MPG of Fit_____ MPG of Prius
2. MPG of Corolla _____ MPG of Fit
1. <
2. <

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:

Solution
 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:

1. equation
2. expression

TRY IT 4.2

Determine if each is an expression or an equation:

1. expression
2. 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:

Solution
 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:

415

TRY IT 5.2

Write each expression in exponential form:

79

EXAMPLE 6

Write each exponential expression in expanded form:

Solution

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:

TRY IT 6.2

Write each exponential expression in expanded form:

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.

TRY IT 7.1

Simplify:

1. 125
2. 1

TRY IT 7.2

Simplify:

1. 49
2. 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.

• 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:

Solution
 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:

1. 2
2. 14

TRY IT 8.2

Simplify the expressions:

1. 35
2. 99

EXAMPLE 9

Simplify:

Solution
 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:

18

TRY IT 9.2

Simplify:

9

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.

TRY IT 10.1

Simplify:

16

TRY IT 10.2

Simplify:

23

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

EXAMPLE 11

.

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.

TRY IT 11.1

Simplify:

86

TRY IT 11.2

Simplify:

1

EXAMPLE 12

Simplify: .

Solution
 If an expression has several exponents, they may be simplified in the same step. Simplify exponents. Divide. Add. Subtract.

TRY IT 12.1

Simplify:

81

TRY IT 12.2

Simplify:

75

# Key Concepts

Operation Notation Say: The result is…
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 ( = ,<,  >).

Spurs Height Heat Height
Tim Duncan Rashard Lewis
Boris Diaw LeBron James
Kawhi Leonard Chris Bosh
Danny Green Ray Allen
1. Height of Tim Duncan____Height of Rashard Lewis
2. Height of Boris Diaw____Height of LeBron James
3. Height of Kawhi Leonard____Height of Chris Bosh
4. Height of Tony Parker____Height of Dwyane Wade
5. Height of Danny Green____Height of Ray Allen

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.

Mountain Elevation
Mt. Elbert
Mt. Massive
Mt. Harvard
Blanca Peak
La Plata Peak
Uncompahgre Peak
Crestone Peak
Mt. Lincoln
Grays Peak
Mt. Antero
1. Elevation of La Plata Peak____Elevation of Mt. Antero
2. Elevation of Blanca Peak____Elevation of Mt. Elbert
3. Elevation of Gray’s Peak____Elevation of Mt. Lincoln
4. Elevation of Mt. Massive____Elevation of Crestone Peak
5. Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

## 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?