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Mathematics LibreTexts

1.2: Use the Language of Algebra

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Learning Objectives

By the end of this section, you will be able to:

  • Find factors, prime factorizations, and least common multiples
  • Use variables and algebraic symbols
  • Simplify and evaluate expressions using the order of operations
  • Translate an English phrase to an algebraic expression

Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.

Similarly, a multiple of 3 would be the product of a counting number and 3.

Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.

We could find the multiples of any number by continuing this process.

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

MULTIPLE OF A NUMBER

A number is a multiple of n if it is the product of a counting number and n.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, 15÷3 is 5, so 15 is 53

DIVISIBLE BY A NUMBER

If a number m is a multiple of n, then m is divisible by n.

If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:

DIVISIBILITY TESTS

A number is divisible by:

  • 2 if the last digit is 0, 2, 4, 6, or 8.
  • 3 if the sum of the digits is divisible by 3.
  • 5 if the last digit is 5 or 0.
  • 6 if it is divisible by both 2 and 3.
  • 10 if it ends with 0.

Example 1.2.1

Is 5,625 divisible by

  1. 2?
  2. 3?
  3. 5 or 10?
  4. 6?
Solution

a. Is 5,625 divisible by 2?

Does it end in 0, 2, 4, 6 or 8?No.5,625 is not divisible by 2.

b. Is 5,625 divisible by 3?

What is the sum of the digits?Is the sum divisible by 3?5+6+2+5=18Yes.5,625 is divisible by 3.

c. Is 5,625 divisible by 5 or 10?

What is the last digit? It is 5.5,625 is divisible by 5 but not by 10.

d. Is 5,625 divisible by 6?

Is it divisible by both 2 and 3?No, 5,625 is not divisible by 2, so 5,625 isnot divisible by 6.

try-it.png 1.2.1

Is 4,962 divisible by a) 2?   b) 3?   c) 5?   d) 6?   e) 10?

Answer

a) yes   b) yes   c) no   d) yes   e) no

try-it.png 1.2.2

Is 3,765 divisible by a) 2?   b) 3?   c) 5?   d) 6?   e) 10?

Answer

a) no   b) yes   c) yes   d) no   e) no

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since 8·9=72, we say that 8 and 9 are factors of 72. When we write 72=8·9, we say we have factored 72.

8 times 9 is 72. 8 and 9 are factors. 72 is the product.

Other ways to factor 72 are 1·72,2·36,3·24,4·18, and 612. The number 72 has many factors: 1,2,3,4,6,8,9,12,18,24,36, and 72.

FACTORS

If a and b are counting numbers, and a·b=m, then a and b are factors of m.

Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.

PRIME NUMBER AND COMPOSITE NUMBER

A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.

A composite number is a counting number that is not prime. A composite number has factors other than 1 and the number itself.

The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!

This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.

PRIME FACTORIZATION

The prime factorization of a number is the product of prime numbers that equals the number.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

example 1.2.2: How to Find the Prime Factorization of a Composite Number

Factor 48.

Answer

Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.

Step 2 ...

Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.

We say 22223 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.22223 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.

If we first factored 48 in a different way, for example as 6·8, the result would still be the same. Finish the prime factorization and verify this for yourself.

try-it.png 1.2.3

Find the prime factorization of 80.

Answer

22225

try-it.png 1.2.4

Find the prime factorization of 60.

Answer

2235

FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER

  1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree.
  3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. Write the composite number as the product of all the circled primes.

One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

LEAST COMMON MULTIPLE

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.

example 1.2.7: How to Find the Least Common Multiple Using the Prime Factors Method

Find the least common multiple (LCM) of 12 and 18 using the prime factors method.

Solution:
Step 1 is to write each number as a product of primes. The number 12 is written as a product of 2, 2 and 3. The number 18 is written as a product of 2, 3 and 3.
Figure_01_01_006b_img_(1).jpg" />
Step 3 is to bring down the number from each column. When a column has the same number at the top and the bottom, that number is brought down. When a column has only one number that number is brought down. The numbers brought down are 2, 2, 3 and 3.
Figure_01_01_006d_img.jpg" />

Notice that the prime factors of 12 (2·2·3) and the prime factors of 18 (233) are included in the LCM (2·2·3·3). So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

try-it.png 1.2.5

Find the LCM of 9 and 12 using the Prime Factors Method.

Answer

36

try-it.png 1.2.6

Find the LCM of 18 and 24 using the Prime Factors Method.

Answer

72

FIND THE LEAST COMMON MULTIPLE USING THE PRIME FACTORS METHOD

  1. Write each number as a product of primes.
  2. List the primes of each number. Match primes vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.
 

Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are x,y,a,b,c.

VARIABLE

A variable is a letter that represents a number whose value may change.

A number whose value always remains the same is called a constant.

CONSTANT

A constant is a number whose value always stays the same.

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.

OPERATION SYMBOLS
Operation Notation Say: The result is…
Addition a+b a plus b the sum of a and b
Subtraction ab a minus b the difference of a and b
Multiplication ab,ab,(a)(b),(a)b,a(b) a times b the product of a and b
Division a÷b, a/b, ab, b¯)a a divided by b the quotient of a and b;
a is called the dividend, and b is called the divisor

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.

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

For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.

The statements a<b or 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.

INEQUALITY SYMBOLS
Inequality Symbols Words
ab a is not equal to b.
a<b a is less than b.
ab a is less than or equal to b.
a>b a is greater than b.
ab a is greater than or equal to b.

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.  Note: all of these symbols are used differently in regard to sets (see Section 0.2) than as simple grouping symbols.  We have to be careful to understand the way they are being used, in any given context.

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]}

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. A sentence has a subject and a verb. In algebra, we have expressions and equations.

EXPRESSION

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

ExpressionWordsEnglish Phrase3+53 plus 5the sum of three and fiven1n minus onethe difference of n and one6·76 times 7the product of six and sevenxyx divided by ythe quotient of x and y

Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.

An equation is two expressions linked by 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.

EQUATION

An equation is two expressions connected by an equal sign.

EquationEnglish Sentencen1=14n minus one equals fourteen.6·7=42The product of six and seven is equal to forty-two.y+9=2y3y plus nine is equal to two y minus three.

Suppose we need to multiply 2 nine times. We could write this as 2·2·2·2·2·2·2·2·2. This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write 2·2·2 as 23 and 2·2·2·2·2·2·2·2·2 as 29. In expressions such as 23, the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.

The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.

EXPONENTIAL NOTATION

We say 23 is in exponential notation and 2·2·2 is in expanded notation.

an means multiply n factors of the number a.

The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.

The expression an is read a to the nth power.

While we read an as a to the nth power”, we usually read:

a2a squareda3a cubed

We’ll see later why a2 and a3 have special names.

Table shows how we read some expressions with exponents.

Expression In Words  
72 7 to the second power or 7 squared
53 5 to the third power or 5 cubed
94 9 to the fourth power  
125 12 to the fifth power  

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify 4·2+1 we would first multiply 42 to get 8 and then add the 1 to get 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:

42+18+19

By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.

SIMPLIFY AN EXPRESSION

To simplify an expression, do all operations in the expression.

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+37. Some students simplify this getting 49, by adding 4+3 and then multiplying that result by 7. Others get 25, by multiplying 3·7 first and then adding 4.  In order to avoid this issue, the convention is to use specific rules for simplifying an expression.

THE ORDER OF OPERATIONS

  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”.

ParenthesesPleaseExponentsExcuseMultiplication DivisionMy DearAddition SubtractionAunt Sally

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 1.2.10

Simplify: 18÷6+4(52).

Answer
  CNX_IntAlg_Figure_01_01_009a_img.jpg
Parentheses? Yes, subtract first. CNX_IntAlg_Figure_01_01_009b_img.jpg
Exponents? No.  
Multiplication or division? Yes.  
Divide first because we multiply and divide left to right. CNX_IntAlg_Figure_01_01_009c_img.jpg
Any other multiplication or division? Yes.  
Multiply. CNX_IntAlg_Figure_01_01_009d_img.jpg
Any other multiplication of division? No.  
Any addition or subtraction? Yes.  
Add. CNX_IntAlg_Figure_01_01_009e_img.jpg

example 1.2.11

Simplify: 30÷5+10(32).

Answer

16

example 1.2.12

Simplify: 70÷10+4(62).

Answer

23

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

example 1.2.13

Simplify: 5+23+3[63(42)].

Answer
  CNX_IntAlg_Figure_01_01_010a_img.jpg
Are there any parentheses (or other grouping symbols)? Yes. CNX_IntAlg_Figure_01_01_010b_img.jpg
Focus on the parentheses that are inside the brackets. Subtract. CNX_IntAlg_Figure_01_01_010c_img.jpg
Continue inside the brackets and multiply. CNX_IntAlg_Figure_01_01_010d_img.jpg
Continue inside the brackets and subtract. CNX_IntAlg_Figure_01_01_010f_img.jpg
The expression inside the brackets requires no further simplification.  
Are there any exponents? Yes. Simplify exponents. CNX_IntAlg_Figure_01_01_010g_img.jpg
Is there any multiplication or division? Yes.  
Multiply. CNX_IntAlg_Figure_01_01_010h_img.jpg
Is there any addition of subtraction? Yes.  
Add. CNX_IntAlg_Figure_01_01_010i_img.jpg
Add. CNX_IntAlg_Figure_01_01_010j_img.jpg

example 1.2.14

Simplify: 9+53[4(9+3)].

Answer

86

example 1.2.15

Simplify: 722[4(5+1)].

Answer

1

EXAMPLE 1.2.22

Simplify: ⓐ (2)424.

Solution

Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the −2 to the 4thpower. In part (b), the exponent means to raise just the 2 to the 4th power and then take the opposite.

(2)4Write in expanded form.(2)(2)(2)(2)Multiply.4(2)(2)Multiply.8(2)Multiply.16

24Write in expanded form.(2·2·2·2)We are asked to findthe opposite of 24.Multiply.(4·2·2)Multiply.(8·2)Multiply.16

EXAMPLE 1.2.23

Simplify: ⓐ (3)434.

Answer

ⓐ 81 ⓑ −81

The last example showed us the difference between (2)4 and 24. This distinction is important to prevent future errors.  

Evaluate a Variable Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

EVALUATE AN EXPRESSION

To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

Example 1.2.16

Evaluate when x=4: ⓐ x23x2x2+3x+8.

Answer

  CNX_IntAlg_Figure_01_01_011b_img.jpg
. CNX_IntAlg_Figure_01_01_011c_img.jpg
Use definition of exponent. CNX_IntAlg_Figure_01_01_011d_img.jpg
Simplify. CNX_IntAlg_Figure_01_01_011e_img.jpg
  CNX_IntAlg_Figure_01_01_012b_img.jpg
. CNX_IntAlg_Figure_01_01_012c_img.jpg
Use definition of exponent. CNX_IntAlg_Figure_01_01_012d_img.jpg
Simplify. CNX_IntAlg_Figure_01_01_012e_img.jpg
  CNX_IntAlg_Figure_01_01_013b_img.jpg
. CNX_IntAlg_Figure_01_01_013c_img.jpg
Follow the order of operations. CNX_IntAlg_Figure_01_01_013d_img (1).jpg
  CNX_IntAlg_Figure_01_01_013e_img.jpg
  CNX_IntAlg_Figure_01_01_013f_img.jpg

example 1.2.17

Evaluate when x=3, ⓐ x24x3x2+4x+1.

Answer

ⓐ 9 ⓑ 64 ⓒ40

example 1.2.18

Evaluate when x=6, ⓐ x32x6x24x7.

Answer

ⓐ 216 ⓑ 64 ⓒ 185

EXAMPLE 1.2.28

Evaluate 4x22xy+3y2 when x=2,y=1.

Answer
  CNX_IntAlg_Figure_01_02_015b_img.jpg
CNX_IntAlg_Figure_01_02_015a_img.jpg CNX_IntAlg_Figure_01_02_015c_img.jpg
Simplify exponents. CNX_IntAlg_Figure_01_02_015d_img.jpg
Multiply. CNX_IntAlg_Figure_01_02_015e_img.jpg
Subtract. CNX_IntAlg_Figure_01_02_015f_img.jpg
Add. CNX_IntAlg_Figure_01_02_015g_img.jpg

EXAMPLE 1.2.29

Evaluate: 3x22xy+6y2 when x=1,y=2.

Answer

31

EXAMPLE 1.2.30

Evaluate: 4x2xy+5y2 when x=2,y=3.

Answer

67 

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

TERM

A term is a constant or the product of a constant and one or more variables.

Examples of terms are 7,y,5x2,9a, and b5.

The constant that multiplies the variable is called the coefficient.

COEFFICIENT

The coefficient of a term is the constant that multiplies the variable in a term.

Think of the coefficient as the number in front of the variable. The coefficient of the term 3x is 3. When we write x, the coefficient is 1, since x=1x.

Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.

Look at the following 6 terms. Which ones seem to have traits in common?

5x7n243x9n2

We say,

7 and 4 are like terms.

5x and 3x are like terms.

n2 and 9n2 are like terms.

LIKE TERMS

Terms that are either constants or have the same variables raised to the same powers are called like terms.

If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

Simplify.4x+7x+xAdd the coefficients.12x

ExAMPLE 1.2.19: How To Combine Like Terms

Simplify: 2x2+3x+7+x2+4x+5.

Answer
Step 1 is to identify the like terms in 2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5. The like terms are 2 x squared and x squared, then 3 x and 4 x, then 7 and 5.
Figure_01_01_014b_img.jpg" />

Step 3 is to combine the like terms to get 3 x squared plus 7 x plus 12.

ExAMPLE 1.2.20

Simplify: 3x2+7x+9+7x2+9x+8.

Answer

10x2+16x+17

ExAMPLE 1.2.21

Simplify: 4y2+5y+2+8y2+4y+5.

Answer

12y2+9y+7

COMBINE LIKE TERMS.

  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add or subtract the coefficients and keep the same variable for each group of like terms.
Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table summarizes them.

Operation Phrase Expression
Addition a plus b

the sum of a and b

a increased by b

b more than a

the total of a and b

b added to a

a+b
Subtraction a minus b

the difference of a and b

a decreased by b

b less than a

b subtracted from a

ab
Multiplication a times b

the product of aa and bb

twice a

a·b,ab,a(b),(a)(b)

2a

Division a divided by b

the quotient of a and b

the ratio of a and b

b divided into a

a÷b,a/b,ab,b¯)a

Look closely at these phrases using the four operations:

The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.

Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.

Example 1.2.22

Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.

Translate each English phrase into an algebraic expression:

ⓐ the difference of 14x and 9

ⓑ the quotient of 8y2 and 3

ⓒ twelve more than y

ⓓ seven less than 49x2

Answer

ⓐ The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

The difference of 14 x and 9, 14 x minus 9.

ⓑ The key word is quotient, which tells us the operation is division.


The quotient of 8 y squared and 3, divide 8 y squared by 3, 8 y squared divided by 3. This can also be written as 8 y squared slash 3 or 8 y squared upon 3.

ⓒ The key words are more than. They tell us the operation is addition. More than means “added to.”

twelve more than ytwelve added to yy+12

ⓓ The key words are less than. They tell us to subtract. Less than means “subtracted from.”

seven less than 49x2seven subtracted from 49x249x27

Exercise 1.2.23

Translate the English phrase into an algebraic expression:

ⓐ the difference of 14x2 and 13

ⓑ the quotient of 12x and 2

ⓒ 13 more than z

ⓓ 18 less than 8x

Answer

14x21312x÷2

z+138x18

Example 1.2.24

Translate the English phrase into an algebraic expression:

ⓐ the sum of 17y2 and 19

ⓑ the product of 7 and y

ⓒ Eleven more than x

ⓓ Fourteen less than 11a

Answer

17y2+197y

x+1111a14

We look carefully at the words to help us distinguish between multiplying a sum and adding a product.

Example 1.2.25

Translate the English phrase into an algebraic expression:

ⓐ eight times the sum of x and y

ⓑ the sum of eight times x and y

Answer

There are two operation words—times tells us to multiply and sum tells us to add.

ⓐ Because we are multiplying 8 times the sum, we need parentheses around the sum of x and y, (x+y). This forces us to determine the sum first. (Remember the order of operations.)

eight times the sum of x and y8(x+y)

ⓑ To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of eight times x and y.

The sum of 8 times x and y is 8 x plus y.

Example 1.2.26

Translate the English phrase into an algebraic expression:

ⓐ four times the sum of p and q

ⓑ the sum of four times p and q

Answer

4(p+q)4p+q

Example 1.2.27

Translate the English phrase into an algebraic expression:

ⓐ the difference of two times x and 8

ⓑ two times the difference of x and 8

Answer

2x82(x8)

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.

Example 1.2.28

The length of a rectangle is 14 less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

Answer

Write a phrase about the length of the rectangle.14 less than the widthSubstitute w for “the width.”wRewrite less than as subtracted from.14 subtracted from wTranslate the phrase into algebra.w14

Example 1.2.30

The width of a rectangle is 6 less than the length. Let l represent the length of the rectangle. Write an expression for the width of the rectangle.

Answer

l6

The expressions in the next example will be used in the typical coin mixture problems we will see soon.

Example 1.2.31

June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

Answer

Write a phrase about the number of dimes.7 less than 4 times qTranslate 4 times q.7 less than 4qTranslate the phrase into algebra.4q7

Example 1.2.32

Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

Answer

4q8

When we are working with integers, it can sometimes be difficult to translate words into a mathematical expression, even if there is no variable involved.  This next example looks at a situation in life when integers are used in calculations.

EXAMPLE 1.2.34: calculating with Integers

The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9 degrees. What was the difference in the morning and afternoon temperatures?

Figure shows a glass thermometer, with temperature markings ranging from minus 10 to 30. Two markings are highlighted, minus 9 degrees C and 11 degrees C.
Answer

Step 1 is to read the problem and make sure all the words and ideas are understood.
Step 2 is to identify what we are asked to find.  Here this is the difference of the morning and afternoon temperatures.
Step 3 is to write a phrase that gives the information to find it. In this case, the phrase is the difference of 11 and minus 9.
Step 4 is to translate the phrase to an expression.  Here this is eleven minus a negative nine.
In step 5, we simplify the expression to get 20.
Step 6 is to answer the question with a complete sentence.  The difference in temperatures was 20 degrees.

EXAMPLE 1.2.35

The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

Answer

The difference in temperatures was 45 degrees Fahrenheit.

Absolute Value

We saw in Section 0.2 that numbers such as 3 and −3 are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

Definition: ABSOLUTE VALUE

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number n is written as |n| and |n|0 for all numbers.

Absolute values are always greater than or equal to zero.

For example,

5 is 5 units away from 0, so |5|=5.5 is 5 units away from 0, so |5|=5.

Figure 1.2.7 illustrates this idea.

Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.
Figure 1.2.7: The numbers 5 and −5 are 5 units away from 0.

The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

In the next example, we’ll order expressions with absolute values.

EXAMPLE 1.2.5

Fill in <,>, or = for each of the following pairs of numbers:

  1. |5|___|5|
  2. 8___|8|
  3. 9___|9|
  4. (16)___|16|.
Solution

a.

 Simplify.Order. |5|55|5|______>>|5|55|5|

b.

 Simplify.Order. 8888______>>|8|88|8|

c.

 Simplify.[2pt]Order. 9999______==|9|[2pt]99|9|

d.

 Simplify.Order. (16)1616(16)______==|16|1616|16|

try-it.png 1.2.7

Fill in <,>, or = for each of the following pairs of numbers:

9___|9|2___|2|8___|8|(9)___|9|.

Answer

>  ⓑ >  ⓒ <=

We simplify the expressions inside absolute value bars first just like we do with parentheses.

EXAMPLE 1.2.6

Simplify: 24|193(62)|.

Solution

24|193(62)|Work inside parentheses first: subtract 2 from 6.24|193(4)|Work inside the absolute value bars: multiply 3(4)24|1912| subtract.24|7|Take the absolute value of 7.247Subtract.17

try-it.png 1.2.8

Simplify: 19|114(31)|.

Answer

16

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write n2 and read it “n squared.” The result is called the square of a number n. For example, 82 is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because 112 is 121. It will be helpful to learn to recognize the perfect square numbers.

SQUARE OF A NUMBER

If n2=m, then m is the square of n.

What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

(3)2=9(8)2=64(11)2=121(15)2=225

Because 102=100, we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of a number m.

SQUARE ROOT OF A NUMBER

If n2=m, then n is a square root of m.

Notice (10)2=100 also, so −10 is also a square root of 100. Therefore, both 10 and −10 are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, m, denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.

SQUARE ROOT NOTATION

m is read “the square root of m.”

Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.

If m=n2, then m=n, for n0.

The square root of m, m, is the positive number whose square is m.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100=10. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 100=10. We read 100 as “the opposite of the principal square root of 10,” or, "the negative square root of 10."

Exercise 1.2.22

Simplify: ⓐ 25121144.

Answer

25Since 52=255

121Since 112=12111

144The negative is in front of12the radical sign.

Exercise 1.2.23

Simplify: ⓐ 36169225

Answer

ⓐ 6 ⓑ 13 ⓒ −15

Exercise 1.2.24

Simplify: a) 16 b) 196 c) 100

Answer

a) 4 b) 14 c) −10

Can we simplify 25? Is there a number whose square is 25?

(__)2=25?

None of the numbers that we have dealt with so far has a square that is 25. Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to 25. The square root of a negative number is not a real number.

Key Concepts

  • Divisibility Tests
    A number is divisible by:
    2 if the last digit is 0, 2, 4, 6, or 8.
    3 if the sum of the digits is divisible by 3.
    5 if the last digit is 5 or 0.
    6 if it is divisible by both 2 and 3.
    10 if it ends with 0.
  • How to find the prime factorization of a composite number.
    1. Find two factors whose product is the given number, and use these numbers to create two branches.
    2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
    3. If a factor is not prime, write it as the product of two factors and continue the process.
    4. Write the composite number as the product of all the circled primes.
  • How To Find the least common multiple using the prime factors method.
    1. Write each number as a product of primes.
    2. List the primes of each number. Match primes vertically when possible.
    3. Bring down the columns.
    4. Multiply the factors.
  • Equality Symbol
    a=b is read “a is equal to b.” The symbol “=” is called the equal sign.
  • Inequality

    For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.

  • Inequality Symbols
    Inequality Symbols Words
    ab a is not equal to b.
    a<b a is less than b.
    ab a is less than or equal to b.
    a>b a is greater than b.
    ab a is greater than or equal to b.
  • Grouping Symbols Parentheses()Brackets[]Braces{}
  • Exponential Notation an means multiply a by itself, n times. The expression an is read a to the nth power.
  • Simplify an Expression
    To simplify an expression, do all operations in the expression.
  • How to use the order of operations.
    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.
  • How to combine like terms.
    1. Identify like terms.
    2. Rearrange the expression so like terms are together.
    3. Add or subtract the coefficients and keep the same variable for each group of like terms.
    Operation Phrase Expression
    Addition a plus b
    the sum of a and b
    a increased by b
    b more than a
    the total of a and b
    b added to a
    a+b
    Subtraction a minus b
    the difference of a and b
    a decreased by b
    b less than a
    b subtracted from a
    ab
    Multiplication a times b
    the product of a and b
    twice a

    a·b,ab,a(b),(a)(b)


    2a

    Division

    a divided by b

    the quotient of a and b

    the ratio of a and b

    b divided into a
    a÷b,a/b,ab,b¯)a

 

Glossary

coefficient
The coefficient of a term is the constant that multiplies the variable in a term.
composite number
A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.
constant
A constant is a number whose value always stays the same.
divisible by a number
If a number m is a multiple of n, then m is divisible by n.
equation
An equation is two expressions connected by an equal sign.
evaluate an expression
To evaluate an expression means to find the value of the expression when the variables are replaced by a given number.
expression
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
factors
If a·b=m, then a and b are factors of m.
least common multiple
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
like terms
Terms that are either constants or have the same variables raised to the same powers are called like terms.
multiple of a number
A number is a multiple of n if it is the product of a counting number and n.
order of operations
The order of operations are established guidelines for simplifying an expression.
prime factorization
The prime factorization of a number is the product of prime numbers that equals the number.
prime number
A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
simplify an expression
To simplify an expression means to do all the math possible.
term
A term is a constant, or the product of a constant and one or more variables.
variable
A variable is a letter that represents a number whose value may change.

This page titled 1.2: Use the Language of Algebra is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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