1.2 Use the language of algebra

Use variables and algebraic symbols

Suppose this year Greg is 20 years old and Alex is 23. You know that 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     s and the 3 is a constant    . The ages change (“vary”) but the 3 years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by 3 years, 3 is the constant .

In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age g , then we could use $g+3$ to represent Alex’s age. See [link] .

The letters used to represent these changing ages are called variables . The letters most commonly used for variables are x , y , a , b , and c .

To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using.

There are four basic arithmetic operations : addition , subtraction , multiplication , and division . We’ll list the symbols used to indicate these operations below ( [link] ). You’ll probably recognize some of them.

We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.”

• The difference of 9 and 2 means subtract 9 and 2, in other words, 9 minus 2, which we write symbolically as $9-2.$
• The product of 4 and 8 means multiply 4 and 8, in other words 4 times 8, which we write symbolically as $4·8.$

In algebra, the cross symbol, $\times ,$ is not used to show multiplication because that symbol may cause confusion. Does 3 xy mean $3\times y$ (‘three times y ’) or $3·x·y$ (three times x times y )? To make it clear, use $·$ or parentheses for multiplication.

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

On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>.”

Inequality

$\begin{array}{}\\ \\ a<b\text{is read “}a\text{is less than}b\text{”}\hfill \\ a\text{is to the left of}b\text{on the number line}\hfill \end{array}$

$\begin{array}{c}a>b\text{is read “}a\text{is greater than}b\text{”}\hfill \\ a\text{is to the right of}b\text{on the number line}\hfill \end{array}$