We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Guias de estudo > Prealgebra

Using Variables and Algebraic Notation

Learning Outcomes

  • 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

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.

Exercises

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
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>a. For example, 7<11 is equivalent to 11>7.a>b is equivalent to b<a. For example, 17>4 is equivalent to 4<17.\begin{array}{l}a<b\text{ is equivalent to }b>a.\text{ For example, }7<11\text{ is equivalent to }11>7.\hfill \\ a>b\text{ is equivalent to }b<a.\text{ For example, }17>4\text{ is equivalent to }4<17.\hfill \end{array}

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.

Exercises

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

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

Exercises

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

  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(\right)
brackets []\left[\right]
braces {}\left\{\right\}
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}

Licenses & Attributions