Using 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 age	Alex’s age
\(12\)	\(15\)
\(20\)	\(23\)
\(35\)	\(38\)
\(g\)	\(g+3\)

Letters are used to represent variables. Letters often used for variables are \(x,y,a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c.\)

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 the previous tutorial, 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	\(a+b\)	\(a\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}b\)	the sum of \(a\) and \(b\)
Subtraction	\(a-b\)	\(a\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}b\)	the difference of \(a\) and \(b\)
Multiplication	\(a·b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right)\)	\(a\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}b\)	The product of \(a\) and \(b\)
Division	\(a÷b,a/b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},b\overline{)a}\)	\(a\) divided by \(b\)	The quotient of \(a\) and \(b\)

In algebra, the cross symbol, \(×,\) is not used to show multiplication because that symbol may cause confusion. Does \(3xy\) mean \(3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}y\) (three times \(y\)) or \(3·x·y\) (three times \(x\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}y\))? 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 \(5\) 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 \(4·8.\)
The quotient of \(20\) and \(5\) means divide \(20\) by \(5,\) which we can write as \(20÷5.\)

Example

Translate from algebra to words:

\(\phantom{\rule{0.2em}{0ex}}12+14\phantom{\rule{0.2em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}\left(30\right)\left(5\right)\phantom{\rule{0.2em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}64÷8\phantom{\rule{0.2em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}x-y\)

Solution

\(12+14\)

12 plus 14

the sum of twelve and fourteen

\(\left(30\right)\left(5\right)\)

30 times 5

the product of thirty and five

\(64÷8\)

64 divided by 8

the quotient of sixty-four and eight

\(x-y\)

\(x\) 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

\(a=b\phantom{\rule{0.2em}{0ex}}\text{is read}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is equal to}\phantom{\rule{0.2em}{0ex}}b\)

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 \(\text{“<”}\) and \(\text{“>”}\) for inequalities.

Inequality

\(a

\(a\) 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 \(a\) is greater than \(b\)

\(a\) is to the right of \(b\) 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 \(ab\) can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

\(\begin{array}{l}aa.\phantom{\rule{2em}{0ex}}\text{For example,}\phantom{\rule{0.2em}{0ex}}7<11\phantom{\rule{0.2em}{0ex}}\text{is equivalent to}\phantom{\rule{0.2em}{0ex}}11>7.\hfill \\ a>b\phantom{\rule{0.2em}{0ex}}\text{is equivalent to}\phantom{\rule{0.2em}{0ex}}b4\phantom{\rule{0.2em}{0ex}}\text{is equivalent to}\phantom{\rule{0.2em}{0ex}}4<17.\hfill \end{array}\)

When we write an inequality symbol with a line under it, such as \(a\le b,\) it means \(a

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

Algebraic Notation	Say
\(a=b\)	\(a\) is equal to \(b\)
\(a\ne b\)	\(a\) is not equal to \(b\)
\(a	\(a\) is less than \(b\)
\(a>b\)	\(a\) is greater than \(b\)
\(a\le b\)	\(a\) is less than or equal to \(b\)
\(a\ge b\)	\(a\) 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

Translate from algebra to words:

\(\phantom{\rule{0.2em}{0ex}}20\le 35\)
\(\phantom{\rule{0.2em}{0ex}}11\ne 15-3\)
\(\phantom{\rule{0.2em}{0ex}}9>10÷2\)
\(\phantom{\rule{0.2em}{0ex}}x+2<10\)

Solution

\(20\le 35\)

20 is less than or equal to 35

\(11\ne 15-3\)

11 is not equal to 15 minus 3

\(9>10÷2\)

9 is greater than 10 divided by 2

\(x+2<10\)

\(x\) plus 2 is less than 10

Example

The information in the figure below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol \(\text{=},\text{<},\text{or}\phantom{\rule{0.2em}{0ex}}\text{>}\) 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)

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

Solution

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

MPG of Versa____MPG of Fit
Find the values in the chart.	26____27
Compare.	26 < 27
MPG of Versa < MPG of Fit

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

MPG of Corolla____MPG of Versa
Find the values in the chart.	28____26
Compare.	28 > 26
MPG of Corolla > MPG of Versa

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	\(\left(\phantom{\rule{0.5em}{0ex}}\right)\)
brackets	\(\left[\phantom{\rule{0.5em}{0ex}}\right]\)
braces	\(\left\{\phantom{\rule{0.5em}{0ex}}\right\}\)

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

\(8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24÷\left\{13-2\left[1\left(6-5\right)+4\right]\right\}\)

Using Variables and Algebraic Symbols