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Study Guides > Mathematics for the Liberal Arts Corequisite

Introduction to The Language of Algebra

Learning Outcomes

  • Use Variables and Algebraic Symbols
    • Use variables to represent unknown quantities in algebraic expressions
    • Identify the variables and constants in an algebraic expression
    • Use words and symbols to represent algebraic operations on variables and constants
    • Use inequality symbols to compare two quantities
    • Translate between words and inequality notation
  • Identifying Expressions and Equations
    • Identify and write mathematical expressions using words and symbols
    • Identify and write mathematical equations using words and symbols
    • Use the order of operations to simplify mathematical expressions
  • Simplifying Expressions Using the Order of Operations
    • Use exponential notation
    • Write an exponential expression in expanded form

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 2323, so Alex is 33 years older than Greg. When Greg was 1212, Alex was 1515. When Greg is 3535, Alex will be 3838. 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 gg. Then we could use g+3g+3 to represent Alex’s age. See the table below.
Greg’s age Alex’s age
1212 1515
2020 2323
3535 3838
gg g+3g+3
Letters are used to represent variables. Letters often used for variables are x,y,a,b, and cx,y,a,b,\text{ and }c.

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 a plus ba\text{ plus }b the sum of aa and bb
Subtraction aba-b a minus ba\text{ minus }b the difference of aa and bb
Multiplication ab,(a)(b),(a)b,a(b)a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right) a times ba\text{ times }b The product of aa and bb
Division a÷b,a/b,ab,b)aa\div b,a/b,\frac{a}{b},b\overline{)a} aa divided by bb The quotient of aa and bb
In algebra, the cross symbol, ×\times , is not used to show multiplication because that symbol may cause confusion. Does 3xy3xy mean 3×y3\times y (three times yy ) or 3xy3\cdot x\cdot y (three times x times yx\text{ times }y )? 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 33, which we write as 5+35+3.
  • The difference of 99 and 22 means subtract 99 minus 22, which we write as 929 - 2.
  • The product of 44 and 88 means multiply 44 times 88, which we can write as 484\cdot 8.
  • The quotient of 2020 and 55 means divide 2020 by 55, which we can write as 20÷520\div 5.

Example

Translate from algebra to words:
  1. 12+1412+14
  2. (30)(5)\left(30\right)\left(5\right)
  3. 64÷864\div 8
  4. xyx-y
Solution:
1.
12+1412+14
1212 plus 1414
the sum of twelve and fourteen
2.
(30)(5)\left(30\right)\left(5\right)
3030 times 55
the product of thirty and five
3.
64÷864\div 8
6464 divided by 88
the quotient of sixty-four and eight
4.
xyx-y
xx minus yy
the difference of xx and yy

TRY IT

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

Equality Symbol

a=b[/latex]isread[latex]a[/latex]isequalto[latex]ba=b[/latex] is read [latex]a[/latex] is equal to [latex]b 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 aa, it means that bb is to the right of aa on the number line. We use the symbols <\text{<} and >\text{>} for inequalities.

a<ba<b is read aa is less than bb aa is to the left of bb on the number line

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right. a>b[/latex]isread[latex]a[/latex]isgreaterthan[latex]ba>b[/latex] is read [latex]a[/latex] is greater than [latex]b aa is to the right of bb on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right. The expressions a<b and a>ba<b\text{ and }a>b can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

a<b is equivalent to b>aa<b\text{ is equivalent to }b>a.

For example, 7<11 is equivalent to 11>77<11\text{ is equivalent to }11>7.

a>b is equivalent to b<aa>b\text{ is equivalent to }b<a.

.For example, 17>4 is equivalent to 4<1717>4\text{ is equivalent to }4<17

When we write an inequality symbol with a line under it, such as aba\le b, it means a<ba<b or a=ba=b. We read this aa is less than or equal to bb. Also, if we put a slash through an equal sign, \ne, it means not equal. We summarize the symbols of equality and inequality in the table below.
Algebraic Notation Say
a=ba=b aa is equal to bb
aba\ne b aa is not equal to bb
a<ba<b aa is less than bb
a>ba>b aa is greater than bb
aba\le b aa is less than or equal to bb
aba\ge b aa is greater than or equal to bb

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

Translate from algebra to words:
  1. 203520\le 35
  2. 1115311\ne 15 - 3
  3. 9>10÷29>10\div 2
  4. x+2<10x+2<10

Answer: Solution:

1.
203520\le 35
2020 is less than or equal to 3535
2.
1115311\ne 15 - 3
1111 is not equal to 1515 minus 33
3.
9>10÷29>10\div 2
99 is greater than 1010 divided by 22
4.
x+2<10x+2<10
xx plus 22 is less than 1010

TRY IT

[ohm_question]144653[/ohm_question] [embed] [embed]
In the following video we show more examples of how to write inequalities as words. https://youtu.be/q2ciQBwkjbk

Example

The information in the table below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol =,<, or >\text{=},\text{<},\text{ or }\text{>} in each expression to compare the fuel economy of the cars. (credit: modification of work by Bernard Goldbach, Wikimedia Commons) This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled
  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

Answer: Solution

1.
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
2.
MPG of Versa____MPG of Fit
Find the values in the chart. 26____27
Compare. 26 < 27
MPG of Versa < MPG of Fit
3.
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
4.
MPG of Corolla____MPG of Versa
Find the values in the chart. 28____26
Compare. 28 > 26
MPG of Corolla > MPG of Versa
5.
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

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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.

8(148)213[2+4(98)]24÷{132[1(65)+4]}\begin{array}{cc}8\left(14 - 8\right)21 - 3\\\left[2+4\left(9 - 8\right)\right]\\24\div \left\{13 - 2\left[1\left(6 - 5\right)+4\right]\right\}\end{array}

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 3 plus 53\text{ plus }5 the sum of three and five
n1n - 1 nn minus one the difference of nn and one
676\cdot 7 6 times 76\text{ times }7 the product of six and seven
xy\frac{x}{y} 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.
n1=14n - 1=14 nn minus one equals fourteen.
67=426\cdot 7=42 The product of six and seven is equal to forty-two.
x=53x=53 xx is equal to fifty-three.
y+9=2y3y+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

Determine if each is an expression or an equation:
  1. 166=1016 - 6=10
  2. 42+14\cdot 2+1
  3. x÷25x\div 25
  4. y+8=40y+8=40
Solution
1. 166=1016 - 6=10 This is an equation—two expressions are connected with an equal sign.
2. 42+14\cdot 2+1 This is an expression—no equal sign.
3. x÷25x\div 25 This is an expression—no equal sign.
4. y+8=40y+8=40 This is an equation—two expressions are connected with an equal sign.
 

try it

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Learning Outcomes

  • Use the order of operations to simplify mathematical expressions
  • Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents

Simplify Expressions Containing Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify 42+14\cdot 2+1 we’d first multiply 424\cdot 2 to get 88 and then add the 11 to get 99. 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:

42+14\cdot 2+1 8+18+1 99

Suppose we have the expression 2222222222\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 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 2222\cdot 2\cdot 2 as 23{2}^{3} and 2222222222\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 as 29{2}^{9}. In expressions such as 23{2}^{3}, 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.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as means multiply three factors of 2\text{means multiply three factors of 2} We say 23{2}^{3} is in exponential notation and 2222\cdot 2\cdot 2 is in expanded notation.

Exponential Notation

For any expression an,a{a}^{n},a is a factor multiplied by itself nn times if nn is a positive integer. an means multiply n factors of a{a}^{n}\text{ means multiply }n\text{ factors of }a At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as The expression an{a}^{n} is read aa to the nth{n}^{th} power.
For powers of n=2n=2 and n=3n=3, we have special names.

a2a^2 is read as "aa squared"

a3a^3 is read as "aa cubed"

  The table below lists some examples of expressions written in exponential notation.
Exponential Notation In Words
72{7}^{2} 77 to the second power, or 77 squared
53{5}^{3} 55 to the third power, or 55 cubed
94{9}^{4} 99 to the fourth power
125{12}^{5} 1212 to the fifth power
 

example

Write each expression in exponential form:
  1. 1616161616161616\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16
  2. 99999\text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}
  3. xxxxx\cdot x\cdot x\cdot x
  4. aaaaaaaaa\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a

Answer: Solution

1. The base 1616 is a factor 77 times. 167{16}^{7}
2. The base 99 is a factor 55 times. 95{9}^{5}
3. The base xx is a factor 44 times. x4{x}^{4}
4. The base aa is a factor 88 times. a8{a}^{8}

 

try it

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In the video below we show more examples of how to write an expression of repeated multiplication in exponential form. https://youtu.be/HkPGTmAmg_s

example

Write each exponential expression in expanded form:
  1. 86{8}^{6}
  2. x5{x}^{5}

Answer: Solution 1. The base is 88 and the exponent is 66, so 86{8}^{6} means 8888888\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8 2. The base is xx and the exponent is 55, so x5{x}^{5} means xxxxxx\cdot x\cdot x\cdot x\cdot x

 

try it

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

example

Simplify: 34{3}^{4}

Answer: Solution

34{3}^{4}
Expand the expression. 33333\cdot 3\cdot 3\cdot 3
Multiply left to right. 9339\cdot 3\cdot 3
27327\cdot 3
Multiply. 8181

 

try it

[embed]
 

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+374+3\cdot 7

Some students say it simplifies to 49.Some students say it simplifies to 25.4+37Since 4+3 gives 7.77And 77 is 49.494+37Since 37 is 21.4+21And 21+4 makes 25.25\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & & & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }4+3\text{ gives 7.}\hfill & & \hfill 7\cdot 7\hfill \\ \text{And }7\cdot 7\text{ is 49.}\hfill & & \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }3\cdot 7\text{ is 21.}\hfill & & \hfill 4+21\hfill \\ \text{And }21+4\text{ makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}

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.
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

Simplify the expressions:
  1. 4+374+3\cdot 7
  2. (4+3)7\left(4+3\right)\cdot 7
Solution:
1.
4+374+3\cdot 7
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first. 4+374+\color{red}{3\cdot 7}
Add. 4+214+21
2525
2.
(4+3)7(4+3)\cdot 7
Are there any parentheses? Yes. (4+3)7\color{red}{(4+3)}\cdot 7
Simplify inside the parentheses. (7)7(7)7
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply. 4949
 

try it

[ohm_question]144748[/ohm_question] [ohm_question]144751[/ohm_question]
 

example

Simplify:
  1. 18÷92\text{18}\div \text{9}\cdot \text{2}
  2. 189÷2\text{18}\cdot \text{9}\div \text{2}

Answer: Solution:

1.
18÷9218\div 9\cdot 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. Divide. 22\color{red}{2}\cdot 2
Multiply. 44
2.
189÷218\cdot 9\div 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. 162÷2\color{red}{162}\div 2
Divide. 8181

 

try it

[ohm_question]144756[/ohm_question]
 

example

Simplify: 18÷6+4(52)18\div 6+4\left(5 - 2\right).

Answer: Solution:

18÷6+4(52)18\div 6+4(5-2)
Parentheses? Yes, subtract first. 18÷6+4(3)18\div 6+4(\color{red}{3})
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right. 3+4(3)\color{red}{3}+4(3)
Any other multiplication or division? Yes.
Multiply. 3+123+\color{red}{12}
Any other multiplication or division? No.
Any addition or subtraction? Yes. 1515

 

try it

[ohm_question]144758[/ohm_question]
In the video below we show another example of how to use the order of operations to simplify a mathematical expression. https://youtu.be/qFUvF5-w9o0   When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

example

Simplify: 5+23+3[63(42)]\text{Simplify: }5+{2}^{3}+3\left[6 - 3\left(4 - 2\right)\right].

Answer: Solution:

5+23+3[63(42)]5+{2}^{3}+ 3[6-3(4-2)]
Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets. 5+23+3[63(42)]5+{2}^{3}+ 3[6-3\color{red}{(4-2)}]
Subtract. 5+23+3[63(2)]5+{2}^{3}+3[6-\color{red}{3(2)}]
Continue inside the brackets and multiply. 5+23+3[66]5+{2}^{3}+3[6-\color{red}{6}]
Continue inside the brackets and subtract. 5+23+3[0]5+{2}^{3}+3[\color{red}{0}]
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents. 5+23+3[0]5+\color{red}{{2}^{3}}+3[0]
Is there any multiplication or division? Yes.
Multiply. 5+8+3[0]5+8+\color{red}{3[0]}
Is there any addition or subtraction? Yes.
Add. 5+8+0\color{red}{5+8}+0
Add. 13+0\color{red}{13+0}
1313

 

try it

[ohm_question]144759[/ohm_question]
In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols. https://youtu.be/8b-rf2AW3Ac  

example

Simplify: 23+34÷352{2}^{3}+{3}^{4}\div 3-{5}^{2}.

Answer: Solution:

23+34÷352{2}^{3}+{3}^{4}\div 3-{5}^{2}
If an expression has several exponents, they may be simplified in the same step.
Simplify exponents. 23+34÷352\color{red}{{2}^{3}}+\color{red}{{3}^{4}}\div 3-\color{red}{{5}^{2}}
Divide. 8+81÷3258+\color{red}{81\div 3}-25
Add. 8+2725\color{red}{8+27}-25
Subtract. 3525\color{red}{35-25}
1010

 

try it

[ohm_question]144762[/ohm_question]
 

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