Prealgebra 2e

# 2.1Use the Language of Algebra

Prealgebra 2e2.1 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

### Be Prepared 2.1

Before you get started, take this readiness quiz.

$Add:43+69.Add:43+69.$
If you missed this problem, review Example 1.19.

### Be Prepared 2.2

$Multiply:(896)201.Multiply:(896)201.$
If you missed this problem, review Example 1.48.

### Be Prepared 2.3

$Divide:7,263÷9.Divide:7,263÷9.$
If you missed this problem, review Example 1.64.

### Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $2020$ years old and Alex is $23,23,$ so Alex is $33$ years older than Greg. When Greg was $12,12,$ Alex was $15.15.$ When Greg is $35,35,$ Alex will be $38.38.$ No matter what Greg’s age is, Alex’s age will always be $33$ 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 $33$ 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 $g.g.$ Then we could use $g+3g+3$ to represent Alex’s age. See Table 2.1.

Greg’s age Alex’s age
$1212$ $1515$
$2020$ $2323$
$3535$ $3838$
$gg$ $g+3g+3$
Table 2.1

Letters are used to represent variables. Letters often used for variables are $x,y,a,b,andc.x,y,a,b,andc.$

### 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 $a+ba+b$ $aplusbaplusb$ the sum of $aa$ and $bb$
Subtraction $a−ba−b$ $aminusbaminusb$ the difference of $aa$ and $bb$
Multiplication $a·b,(a)(b),(a)b,a(b)a·b,(a)(b),(a)b,a(b)$ $atimesbatimesb$ The product of $aa$ and $bb$
Division $a÷b,a/b,ab,baa÷b,a/b,ab,ba$ $aa$ divided by $bb$ The quotient of $aa$ and $bb$

In algebra, the cross symbol, $×,×,$ is not used to show multiplication because that symbol may cause confusion. Does $3xy3xy$ mean $3×y3×y$ (three times $yy$) or $3·x·y3·x·y$ (three times $xtimesyxtimesy$)? 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 $55$ and $33$ means add $55$ plus $3,3,$ which we write as $5+3.5+3.$
• The difference of $99$ and $22$ means subtract $99$ minus $2,2,$ which we write as $9−2.9−2.$
• The product of $44$ and $88$ means multiply $44$ times $8,8,$ which we can write as $4·8.4·8.$
• The quotient of $2020$ and $55$ means divide $2020$ by $5,5,$ which we can write as $20÷5.20÷5.$

### Example 2.1

Translate from algebra to words:

1. $12+1412+14$
2. $(30)(5)(30)(5)$
3. $64÷864÷8$
4. $x−yx−y$

### Try It 2.1

Translate from algebra to words.

1. $18+1118+11$
2. $(27)(9)(27)(9)$
3. $84÷784÷7$
4. $p−qp−q$

### Try It 2.2

Translate from algebra to words.

1. $47−1947−19$
2. $72÷972÷9$
3. $m+nm+n$
4. $(13)(7)(13)(7)$

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

### Equality Symbol

$a=bis readais equal toba=bis readais equal tob$

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 $bb$ is greater than $a,a,$ it means that $bb$ is to the right of $aa$ on the number line. We use the symbols $“<”“<”$ and $“>”“>”$ for inequalities.

### Inequality

$a is read $aa$ is less than $bb$

$aa$ is to the left of $bb$ on the number line

$a>ba>b$ is read $aa$ is greater than $bb$

$aa$ is to the right of $bb$ on the number line

The expressions $abab$ 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 $a≤b,a≤b,$ it means $a or $a=b.a=b.$ We read this $aa$ is less than or equal to $b.b.$ Also, if we put a slash through an equal sign, $≠,≠,$ it means not equal.

We summarize the symbols of equality and inequality in Table 2.2.

Algebraic Notation Say
$a=ba=b$ $aa$ is equal to $bb$
$a≠ba≠b$ $aa$ is not equal to $bb$
$a $aa$ is less than $bb$
$a>ba>b$ $aa$ is greater than $bb$
$a≤ba≤b$ $aa$ is less than or equal to $bb$
$a≥ba≥b$ $aa$ is greater than or equal to $bb$
Table 2.2

### 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 2.2

Translate from algebra to words:

1. $20≤3520≤35$
2. $11≠15−311≠15−3$
3. $9>10÷29>10÷2$
4. $x+2<10x+2<10$

### Try It 2.3

Translate from algebra to words.

1. $14≤2714≤27$
2. $19−2≠819−2≠8$
3. $12>4÷212>4÷2$
4. $x−7<1x−7<1$

### Try It 2.4

Translate from algebra to words.

1. $19≥1519≥15$
2. $7=12−57=12−5$
3. $15÷3<815÷3<8$
4. $y-3>6y-3>6$

### Example 2.3

The information in Figure 2.2 compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol $=,<,or>=,<,or>$ in each expression to compare the fuel economy of the cars.

Figure 2.2 (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

### Try It 2.5

Use Figure 2.2 to fill in the appropriate $symbol,=,<,or>.symbol,=,<,or>.$

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

### Try It 2.6

Use Figure 2.2 to fill in the appropriate $symbol,=,<,or>.symbol,=,<,or>.$

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

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. Table 2.3 lists three of the most commonly used grouping symbols in algebra.

Common Grouping Symbols
parentheses $()()$
brackets $[][]$
braces ${}{}$
Table 2.3

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

$8(14−8)21−3[2+4(9−8)]24÷{13−2[1(6−5)+4]}8(14−8)21−3[2+4(9−8)]24÷{13−2[1(6−5)+4]}$

### 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
$3+53+5$ $3plus53plus5$ the sum of three and five
$n−1n−1$ $nn$ minus one the difference of $nn$ and one
$6·76·7$ $6times76times7$ the product of six and seven
$xyxy$ $xx$ divided by $yy$ the quotient of $xx$ and $yy$

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
$3+5=83+5=8$ The sum of three and five is equal to eight.
$n−1=14n−1=14$ $nn$ minus one equals fourteen.
$6·7=426·7=42$ The product of six and seven is equal to forty-two.
$x=53x=53$ $xx$ is equal to fifty-three.
$y+9=2y−3y+9=2y−3$ $yy$ plus nine is equal to two $yy$ 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 2.4

Determine if each is an expression or an equation:

1. $16−6=1016−6=10$
2. $4·2+14·2+1$
3. $x÷25x÷25$
4. $y+8=40y+8=40$

### Try It 2.7

Determine if each is an expression or an equation:

1. $23+6=2923+6=29$
2. $7·3−77·3−7$

### Try It 2.8

Determine if each is an expression or an equation:

1. $y÷14y÷14$
2. $x−6=21x−6=21$

### Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4·2+14·2+1$ we’d first multiply $4·24·2$ to get $88$ and then add the $11$ to get $9.9.$ 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:

$4·2+14·2+1$
$8+18+1$
$99$

Suppose we have the expression $2·2·2·2·2·2·2·2·2.2·2·2·2·2·2·2·2·2.$ 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 $2·2·22·2·2$ as $2323$ and $2·2·2·2·2·2·2·2·22·2·2·2·2·2·2·2·2$ as $29.29.$ In expressions such as $23,23,$ the $22$ is called the base and the $33$ is called the exponent. The exponent tells us how many factors of the base we have to multiply.

$means multiply three factors of 2means multiply three factors of 2$

We say $2323$ is in exponential notation and $2·2·22·2·2$ is in expanded notation.

### Exponential Notation

For any expression $an,aan,a$ is a factor multiplied by itself $nn$ times if $nn$ is a positive integer.

$anmeans multiplynfactors ofaanmeans multiplynfactors ofa$

The expression $anan$ is read $aa$ to the $nthnth$ power.

For powers of $n=2n=2$ and $n=3,n=3,$ we have special names.

$a2is read as"asquared"a3is read as"acubed"a2is read as"asquared"a3is read as"acubed"$

Table 2.4 lists some examples of expressions written in exponential notation.

Exponential Notation In Words
$7272$ $77$ to the second power, or $77$ squared
$5353$ $55$ to the third power, or $55$ cubed
$9494$ $99$ to the fourth power
$125125$ $1212$ to the fifth power
Table 2.4

### Example 2.5

Write each expression in exponential form:

1. $16·16·16·16·16·16·1616·16·16·16·16·16·16$
2. $9·9·9·9·99·9·9·9·9$
3. $x·x·x·xx·x·x·x$
4. $a·a·a·a·a·a·a·aa·a·a·a·a·a·a·a$

### Try It 2.9

Write each expression in exponential form:

$41·41·41·41·4141·41·41·41·41$

### Try It 2.10

Write each expression in exponential form:

$7·7·7·7·7·7·7·7·77·7·7·7·7·7·7·7·7$

### Example 2.6

Write each exponential expression in expanded form:

1. $8686$
2. $x5x5$

### Try It 2.11

Write each exponential expression in expanded form:

1. $4848$
2. $a7a7$

### Try It 2.12

Write each exponential expression in expanded form:

1. $8888$
2. $b6b6$

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

### Example 2.7

Simplify: $34.34.$

### Try It 2.13

Simplify:

1. $5353$
2. $1717$

### Try It 2.14

Simplify:

1. $7272$
2. $0505$

### 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:

$4+3·74+3·7$
$Some students say it simplifies to 49.Some students say it simplifies to 25.4+3·7Since4+3gives 7.7·7And7·7is 49.494+3·7Since3·7is 21.4+21And21+4makes 25.25Some students say it simplifies to 49.Some students say it simplifies to 25.4+3·7Since4+3gives 7.7·7And7·7is 49.494+3·7Since3·7is 21.4+21And21+4makes 25.25$

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.

Order of Operations
Excuse Exponents
My Dear Multiplication and Division

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.

### Manipulative Mathematics

Doing the Manipulative Mathematics activity Game of 24 will give you practice using the order of operations.

### Example 2.8

Simplify the expressions:

1. $4+3·74+3·7$
2. $(4+3)·7(4+3)·7$

### Try It 2.15

Simplify the expressions:

1. $12−5·212−5·2$
2. $(12−5)·2(12−5)·2$

### Try It 2.16

Simplify the expressions:

1. $8+3·98+3·9$
2. $(8+3)·9(8+3)·9$

### Example 2.9

Simplify:

1. $18÷9·218÷9·2$
2. $18·9÷218·9÷2$

### Try It 2.17

Simplify:

$42÷7·342÷7·3$

### Try It 2.18

Simplify:

$12·3÷412·3÷4$

### Example 2.10

Simplify: $18÷6+4(5−2).18÷6+4(5−2).$

### Try It 2.19

Simplify:

$30÷5+10(3−2)30÷5+10(3−2)$

### Try It 2.20

Simplify:

$70÷10+4(6−2)70÷10+4(6−2)$

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

### Example 2.11

$Simplify:5+23+3[ 6−3(4−2) ].Simplify:5+23+3[ 6−3(4−2) ].$

### Try It 2.21

Simplify:

$9+53−[ 4(9+3) ]9+53−[ 4(9+3) ]$

### Try It 2.22

Simplify:

$72−2[4(5+1) ]72−2[4(5+1) ]$

### Example 2.12

Simplify: $23+34÷3−52.23+34÷3−52.$

### Try It 2.23

Simplify:

$32+24÷2+4332+24÷2+43$

### Try It 2.24

Simplify:

$62−53÷5+8262−53÷5+82$

### Section 2.1 Exercises

#### Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

1.

$16 − 9 16 − 9$

2.

$25 − 7 25 − 7$

3.

$5 · 6 5 · 6$

4.

$3 · 9 3 · 9$

5.

$28 ÷ 4 28 ÷ 4$

6.

$45 ÷ 5 45 ÷ 5$

7.

$x + 8 x + 8$

8.

$x + 11 x + 11$

9.

$( 2 ) ( 7 ) ( 2 ) ( 7 )$

10.

$( 4 ) ( 8 ) ( 4 ) ( 8 )$

11.

$14 < 21 14 < 21$

12.

$17 < 35 17 < 35$

13.

$36 ≥ 19 36 ≥ 19$

14.

$42 ≥ 27 42 ≥ 27$

15.

$3 n = 24 3 n = 24$

16.

$6 n = 36 6 n = 36$

17.

$y − 1 > 6 y − 1 > 6$

18.

$y − 4 > 8 y − 4 > 8$

19.

$2 ≤ 18 ÷ 6 2 ≤ 18 ÷ 6$

20.

$3 ≤ 20 ÷ 4 3 ≤ 20 ÷ 4$

21.

$a ≠ 7 · 4 a ≠ 7 · 4$

22.

$a ≠ 1 · 12 a ≠ 1 · 12$

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

23.

$9 · 6 = 54 9 · 6 = 54$

24.

$7 · 9 = 63 7 · 9 = 63$

25.

$5 · 4 + 3 5 · 4 + 3$

26.

$6 · 3 + 5 6 · 3 + 5$

27.

$x + 7 x + 7$

28.

$x + 9 x + 9$

29.

$y − 5 = 25 y − 5 = 25$

30.

$y − 8 = 32 y − 8 = 32$

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

31.

$3 · 3 · 3 · 3 · 3 · 3 · 3 3 · 3 · 3 · 3 · 3 · 3 · 3$

32.

$4 · 4 · 4 · 4 · 4 · 4 4 · 4 · 4 · 4 · 4 · 4$

33.

$x · x · x · x · x x · x · x · x · x$

34.

$y · y · y · y · y · y y · y · y · y · y · y$

In the following exercises, write in expanded form.

35.

$5 3 5 3$

36.

$8 3 8 3$

37.

$2 8 2 8$

38.

$10 5 10 5$

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

39.
1. $3+8·53+8·5$
2. $(3+8)·5(3+8)·5$
40.
1. $2+6·32+6·3$
2. $(2+6)·3(2+6)·3$
41.

$2 3 − 12 ÷ ( 9 − 5 ) 2 3 − 12 ÷ ( 9 − 5 )$

42.

$3 2 − 18 ÷ ( 11 − 5 ) 3 2 − 18 ÷ ( 11 − 5 )$

43.

$3 · 8 + 5 · 2 3 · 8 + 5 · 2$

44.

$4 · 7 + 3 · 5 4 · 7 + 3 · 5$

45.

$2 + 8 ( 6 + 1 ) 2 + 8 ( 6 + 1 )$

46.

$4 + 6 ( 3 + 6 ) 4 + 6 ( 3 + 6 )$

47.

$4 · 12 / 8 4 · 12 / 8$

48.

$2 · 36 / 6 2 · 36 / 6$

49.

$6 + 10 / 2 + 2 6 + 10 / 2 + 2$

50.

$9 + 12 / 3 + 4 9 + 12 / 3 + 4$

51.

$( 6 + 10 ) ÷ ( 2 + 2 ) ( 6 + 10 ) ÷ ( 2 + 2 )$

52.

$( 9 + 12 ) ÷ ( 3 + 4 ) ( 9 + 12 ) ÷ ( 3 + 4 )$

53.

$20 ÷ 4 + 6 · 5 20 ÷ 4 + 6 · 5$

54.

$33 ÷ 3 + 8 · 2 33 ÷ 3 + 8 · 2$

55.

$20 ÷ ( 4 + 6 ) · 5 20 ÷ ( 4 + 6 ) · 5$

56.

$33 ÷ ( 3 + 8 ) · 2 33 ÷ ( 3 + 8 ) · 2$

57.

$4 2 + 5 2 4 2 + 5 2$

58.

$3 2 + 7 2 3 2 + 7 2$

59.

$( 4 + 5 ) 2 ( 4 + 5 ) 2$

60.

$( 3 + 7 ) 2 ( 3 + 7 ) 2$

61.

$3 ( 1 + 9 · 6 ) − 4 2 3 ( 1 + 9 · 6 ) − 4 2$

62.

$5 ( 2 + 8 · 4 ) − 7 2 5 ( 2 + 8 · 4 ) − 7 2$

63.

$2 [ 1 + 3 ( 10 − 2 ) ] 2 [ 1 + 3 ( 10 − 2 ) ]$

64.

$5 [ 2 + 4 ( 3 − 2 ) ] 5 [ 2 + 4 ( 3 − 2 ) ]$

#### 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 $83″83″$ Rashard Lewis $82″82″$
Boris Diaw $80″80″$ LeBron James $80″80″$
Kawhi Leonard $79″79″$ Chris Bosh $83″83″$
Tony Parker $74″74″$ Dwyane Wade $76″76″$
Danny Green $78″78″$ Ray Allen $77″77″$
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 $5050$ mountains with an elevation of over $14,000feet.14,000feet.$ The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

Mountain Elevation
Mt. Elbert $14,433′14,433′$
Mt. Massive $14,421′14,421′$
Mt. Harvard $14,420′14,420′$
Blanca Peak $14,345′14,345′$
La Plata Peak $14,336′14,336′$
Uncompahgre Peak $14,309′14,309′$
Crestone Peak $14,294′14,294′$
Mt. Lincoln $14,286′14,286′$
Grays Peak $14,270′14,270′$
Mt. Antero $14,269′14,269′$
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?

#### 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.

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