Use the Language of Algebra

2.1 Use the Language of Algebra

The topics covered in this section are:

  1. Use variables and algebraic symbols
  2. Identify expressions and equations
  3. Simplify expressions with exponents
  4. 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 g. Then we could use g+3 to represent Alex’s age. See the table below.

Greg’s AgeAlex’s Age
1215
2023
3538
gg+3

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

OperationNotationSay:The result is…
Additiona+ba plus bthe sum of a and b
Subtractiona-ba minus bthe difference of a and b
Multiplicationa \cdot b, (a)(b), (a)b, a(b)a times bThe product of a and b
Divisiona \div b, a/b,  \frac{a}{b}, b)aa divided by bThe quotient of a and b

In algebra, the cross symbol, \times,  is not used to show multiplication because that symbol may cause confusion. Does 3xy mean(three timesy) or 3 \cdot x \cdot y (three times x times y)? To make it clear, use \cdot 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 5+3.
  • The difference of 9 and 2 means subtract 9 minus 2, which we write as 9-2.
  • The product of 4 and 8 means multiply 4 times 8, which we can write as 4 \cdot 8.
  • The quotient of 20 and 55 means divide 20 by 5, which we can write as 20 \div 5.

Example 1

Translate from algebra to words:

  1. 12+14
  2. (30)(5)
  3. 64 \div 8
  4. x-y
Solution

1.
12+14
12 plus 14
the sum of twelve and fourteen
2.
(30)(5)
30 times 5
the product of thirty and five
3.
64 \div 8
64 divided by 8
the quotient of sixty-four and eight
4.
x-y
x minus y
the difference of x and y

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

Equality Symbol

a = b is read a is equal to 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 b is greater than a, it means that b is to the right of a on the number line. We use the symbols “<” and “>” for inequalities.

Inequality

a < b is read a is less than b

a is to the left of b 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 is read [latex]a[/latex] is greater than [latex]b[/latex]

a is to the right of [latex]b[/latex] 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 > 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 > a. For example, 7 < 11 is equivalent to 11 > 7.

a > b is equivalent to b < a. For example, 17 > 14 is equivalent to 4 < 17.

When we write an inequality symbol with a line number it, such as a \leq b, it means a < b or a = b. We read this a is less than or equal to b. Also, if we put a slash through an equal sign, \neq, it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic NotationSay
a = ba is equal to b
a \neq ba is not equal to b
a < ba is less than b
a > ba is greater than b
a \leq ba is less than or equal to b
a \geq ba is greater than or equal to b

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. 20 \leq 35
  2. 11 \neq 15 - 3
  3. 9 > 10 \div 2
  4. x + 2 < 10
Solution

1.
20 \leq 35
20 is less than or equal to 35
2.
11 \neq 15 - 3
11 is not equal to 15 minus 3
3.
9 > 10 \div 2
9 is greater than 10 divided by 2
4.
x + 2 < 10
x plus 2 is less than 10

Example 3

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

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 “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.
(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

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

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\left \{ \right \}

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\div {13-2[1(6-5)+4]}

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:

ExpressionWordsPhrase
3+53 plus 5the sum of three and five
n-1n minus onethe difference of n and one
6 \cdot 76 times 7the product of six and seven
\frac{x}{y}x divided by ythe quotient of x and y

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:

EquationSentence
3+5=8The sum of three and five is equal to eight.
n-1=14n minus one equals fourteen.
6 \cdot 7 = 42The product of six and seven is equal to forty-two.
x=53x is equal to fifty-three.
y+9=2y-3y plus nine is equal to two y 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:

  1. 16-6=10
  2. 4 \cdot 2 + 1
  3. x \div 25
  4. y + 8 = 40
Solution

1. 16+6=10This is an equation—two expressions are connected with an equal sign.
2. 4 \cdot 2 + 1This is an expression—no equal sign.
3. x \div 25This is an expression—no equal sign.
4. y + 8 = 40This 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 4 \cdot 2 + 1 we’d first multiply4 \cdot 2to 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:

4 \cdot 2 + 1

8 + 1

9

Supposed we have the expression 2 \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 2 \cdot 2 \cdot 2 as 2^{3} and 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 as 2^{9}. In expressions such as 2^{3}, 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.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.

means multiply three factors of 2

We say 2^{3} is in exponential notation and 2 \cdot 2 \cdot 2 is in expanded notation.

EXPONENTIAL NOTATION

For any expression a^{n}, a is a factor multiplied by it self n times if n is a positive integer.

a^{n} means multiply n 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 “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.

The expression a^{n} is read as a to the n^{th} power.

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

a^{2} is read as “a squared”

a^{3} is read as “a cubed”

The table below lists some example of expressions written in exponential notation.

Exponential NotationIn Words
7^{2}7 to the second power, or 7 squared
5^{3}5 to the third power, or 5 cubed
9^{4}9 to the fourth power
12^{5}12 to the fifth power

Example 5

Write each expression in exponential form:

  1. 16 \cdot 16 \cdot 16 \cdot 16 \cdot 16 \cdot 16 \cdot 16
  2. 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9
  3. x \cdot x \cdot x \cdot x
  4. a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a
Solution

1. The base 16 is a factor 7 times.16^{7}
2. The base 9 is a factor 5 times.9^{5}
3. The base x is a factor 4 times.x^{4}
4. The base a is a factor 8 times.a^{8}

Example 6

Write each expression in expanded form:

  1. 8^{6}
  2. x^{5}
Solution

  1. The base is 8 and the exponent is 6, so 8^{6} means 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8
  2. The base is x and the exponent is 5, so x^{5} means x \cdot x \cdot x \cdot x \cdot x

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

Example 7

Simplify: 3^{4}.

Solution

3^{4}
Expand the expression.3 \cdot 3 \cdot 3 \cdot 3
Multiply left to right.9 \cdot 3 \cdot 3
27 \cdot 3
Multiply.81

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:

4+3 \cdot 7

Some students say is simplifies to 49.

4+3 \cdot 7
Since 4+3 gives 7.7 \cdot 7
And 7 \cdot 7 is 49.49

Some students say is simplifies to 25.

4+3 \cdot 7
Since 3 \cdot 7 is 21.4+21
And 21 + 4 makes 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.
  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 MDear Aunt Sally.

ORDER OF OPERATIONS

PleaseParentheses
ExcuseExponents
MDearMultiplication and Division
Aunt SallyAddition and Subtraction

It’s good that ‘MDear’ 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

  1. 4+3 \cdot 7
  2. (4+3) \cdot 7
Solution

1.4+3 \cdot 7
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.4+3 \cdot 7
Add.4+21
25
2.(4+3) \cdot 7
Are there any parentheses? Yes.(4+3) \cdot 7
Simplify inside the parentheses.(7)7
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.49

Example 9

Simplify the expressions

  1. 18 \div 9 \cdot 2
  2. 18 \cdot 9 \div 2
Solution

1.18 \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.2 \cdot 2
Multiply.4
2.18 \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 \div 2
Divide.81

Example 10

Simplify: 18 \div 6 + 4(5-2).

Solution

18 \div 6 + 4(5-2)
Parentheses? Yes, subtract first.18 \div 6 + 4(3)
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.3 + 4(3)
Any other multiplication or division? Yes.
Multiply.3 + 12
Any other multiplication or division? No.
Any addition or subtraction? Yes.15

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

Example 11

Simplify: 5+2^{3}+3[6-3(4-2)].

Solution

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+2^{3}+3[6-3(4-2)]
Subtract.5+2^{3}+3[6-3(2)]
Continue inside the brackets and multiply.5+2^{3}+3[6-6]
Continue inside the brackets and subtract.5+2^{3}+3[0]
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.5+2^{3}+3[0]
Is there any multiplication or division? Yes.
Multiply.5+8+3[0]
Is there any addition or subtraction? Yes.
Add.5+8+0
Add.13+0
13

Example 12

Simplify: 2^{3}+3^{4} \div 3-5^{2}.

Solution

2^{3}+3^{4} \div 3-5^{2}
If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.2^{3}+3^{4} \div 3-5^{2}
Divide.8+81 \div 3-25
Add.8+27-25
Subtract.35-25
10
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