**1.1 Use the Language of Algebra**

The topics covered in this section are:

- Find factors, prime factorizations, and least common multiples
- Use variables and algebraic symbols
- Simplify expressions using the order of operations
- Evaluate an expression
- Identify and combine like terms
- Translate an English phrase to an algebraic expression

**1.1.1 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 $2$.

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

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, $15 \div 3$ is $5$, so $15$ is $5 \cdot 3$.

**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**

Is $5,625$ divisible by…

- $2$?
- $3$?
- $5$ or $10$?
- $6$?

**Solution**

**Part 1.**

Is $5,625$ divisible by $2$? | |

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

**Part 2.**

Is $5,625$ divisible by $3$? | |

What is the sum of the digits? | $5+6+2+5=18$ |

Is the sum divisible by $3$? | Yes. $5,625$ is divisible by $3$. |

**Part 3.**

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 $10$. |

**Part 4.**

Is $5,625$ divisible by $6$? | |

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

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 $8 \cdot 9 = 72$, we say that $8$ and $9$ are **factors** of $72$. When we write $72=8 \cdot 9$, we say we have factored $72$.

Other ways to factor $72$ are $1 \cdot 72$, $2 \cdot36$, $3 \cdot 24$, $4 \cdot 18$, and $6 \cdot 12$. The number $72$ has many factors:

$1,2,3,4,6,8,9,12,18,24,36,$ and $72$.

**Factors**

In the expression $a \cdot b$, both $a$ and $b$ are called **factors**. If $a \cdot b = m$, and both $a$ and $b$ are integers, 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 greater than $1$ 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!

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 2**

**How to Find the Prime Factorization of a Composite Number**

Factor $48$.

**Solution**

We say $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$ 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 $6 \cdot 8$, the result would still be the same. Finish the prime factorization and verify this for yourself.

**HOW TO: Find the prime factorization of a composite number.**

- Find two factors whose product is the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree.
- If a factor is not prime, write it as the product of two factors and continue the process.
- 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 3**

**How to Find the Least Common Multiple Using the Prime Factors Method**

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

**Solution**

Notice that the prime factors of $12 (2 \cdot 2 \cdot 3)$ and the prime factors of $18 (2 \cdot 3 \cdot 3)$ are included in the LCM $(2 \cdot 2 \cdot 3 \cdot 3)$. 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.

**HOW TO: Find the least common multiple using the Prime Factors Method.**

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

**1.1.2 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**.

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 | $a-b$ | $a$ minus $b$ | the difference of $a$ and $b$ |

Multiplication | $a \cdot b$, $ab$, $(a)(b)$, $(a)b$, $a(b)$ | $a$ timess $b$ | the product of $a$ and $b$ |

Division | $a \div b$, $a/b$, $\frac{a}{b}$, $b \overline{)a}$ | $a$ divided by $b$ | the quotient of $a$ and $b$; $a$ s 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**

The expressions $a<b$ 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 |
---|---|

$a \neq b$ | $a$ is not equal to $b$. |

$a<b$ | $a$ is less than $b$ |

$a \leq b$ | $a$ is less than or equal to $b$. |

$a>b$ | $a$ is greater than $b$. |

$a \geq b$ | $a$ 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.

**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.

$8(14-8)$ | $21-3[2+4(9-8)]$ | $24 \div \{ 13-2[1(6-5)+4] \}$ |

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 | Words | English Phrase |
---|---|---|

$3+5$ | $3$ plus $5$ | the sum of three and five |

$n-1$ | $n$ minus one | the difference of $n$ and one |

$6 \cdot 7$ | $6$ times $7$ | the product of six and seven |

$\frac{x}{y}$ | $x$ divided by $y$ | the 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.

Equation | English Sentence |
---|---|

$3+5=18$ | The sum of three and five is equal to eight. |

$n-1=14$ | $n$ minus one equals fourteen. |

$6 \cdot 7 =42$ | The product of six and seven is equal to forty-two. |

$x=53$ | $x$ is equal to fifty-three. |

$y+9=2y-3$ | $y$ plus nine is equal to two $y$ minus three. |

Suppose we need to multiply $2$ nine times. We could write this as $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ . This is tedious and it can be hard to keep track of all those $2$s, so we use exponents. 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 times we need to multiply the base.

**Exponential Notation**

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

$a^{n}$ means multiply $a$ by itself, $n$ times.

The expression $a^{n}$ is read $a$ to the $n^{ \mathrm{th}}$ power.

While we read $a^{n}$ as “$a$ to the $n^{ \mathrm{th}}$ power, we usually read:

$a^{2}$ as “$a$ squared”

$a^{3}$ as “$a$ cubed”

We’ll see later why $a^{2}$ and $a^{3}$ have special names.

Table 1.1 shows how we read some expressions with exponents.

Expression | In 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 |

**Table 1.1**

**1.1.3 Simplify Expressions Using the Order of Operations**

To **simplify an expression** means to do all the math possible. For example, to simplify $4 \cdot 2 + 1$ we would first multiply $4 \cdot 2$ 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:

$4 \cdot 2 + 1$

$8+1$

$9$

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 $4+3 \cdot 7$. Some students simplify this getting $49$, by adding $4+3$ and then multiplying that result by $7$. Others get $25$, by multiplying $3 \cdot 7$ first and then adding $4$.

The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.

**HOW TO: Use the order of operations.**

- Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

- Exponents
- Simplify all expressions with exponents.

- Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.

- 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”.

Parentheses | Please |

Exponents | Excuse |

Multiplication Division | My Dear |

Addition Subtraction | Aunt Sally |

It’s good that “**M**y **D**ear” goes together, as this reminds us that **m**ultiplication and **d**ivision 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, “**A**unt **S**ally” goes together and so reminds us that **a**ddition and **s**ubtraction also have equal priority and we do them in order from left to right.

**Example 4**

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 of division? No. | |

Any addition or subtraction? Yes. | |

Add. | $15$ |

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

**Example 5**

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 symbols)? Yes. | $5+2^{3} +3[6-3(4-2)]$ |

Focus on the parentheses that are inside the brackets. 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. | $5+8 +3[0]$ |

Is there any multiplication or division? Yes. | |

Multiply. | $5+8+0$ |

Is there any addition of subtraction? Yes. | |

Add. | $13+0$ |

Add. | $13$ |

**1.1.4 Evaluate an 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 6**

Evaluate when $x=4$:

- $x^2$
- $3^{x}$
- $2x^{2} +3x+8$

**Solution**

**Part 1.**

$x^{2}$ | |

Replace $x$ whith $4$ | $4^{2}$ |

Use definition of exponent. | $4 \cdot 4$ |

Simplify. | $16$ |

**Part 2.**

$3^{x}$ | |

Replace $x$ whith $4$ | $3^{4}$ |

Use definition of exponent. | $3 \cdot 3 \cdot 3 \cdot 3$ |

Simplify. | $81$ |

**Part 3.**

$2x^{2} +3x+8$ | |

Replace $x$ whith $4$ | $2(4)^{2} + 3(4)+8$ |

Follow the order of operations. | $2(16)+3(4)+8$ |

$32+12+8$ | |

$52$ |

**1.1.5 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 $7,y,5x^{2},9a,$ and $b^{5}$.

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 $x=1 \cdot x$.

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?

$5x \ \ \ \ 7 \ \ \ \ n^{2} \ \ \ \ 4 \ \ \ \ 3x \ \ \ \ 9n^{2}$

We say,

$7$ and $4$ are like terms.

$5x$ and $3x$ are like terms.

$n^{2}$ and $9n^{2}$ 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. | $4x+7x+x$ |

Add the coefficients. | $12x$ |

**Example 7**

**How To Combine LIke Terms**

Simplify: $2x^{2} + 3x + 7 +x^{2} + 4x + 5$.

**Solution**

**HOW TO: Combine like terms.**

- Identify like terms.
- Rearrange the expression so like terms are together.
- Add or subtract the coefficients and keep the same variable for each group of like terms.

**1.1.6 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 1.2 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 $a$ | $a-b$ |

Multiplication | $a$ times $b$ the product of $a$ and $b$ twice $a$ | $a \cdot b, ab, a(b),(a)(b)$ $2a$ |

Division | $a$ divided by $b$ the quotient of $a$ and $b$ the ratio of $a$ and $b$ $b$ divided into $a$ | $a \div b, a/b, \frac{a}{b} , b \overline{)a}$ |

**Table 1.2**

Look closely at these phrases using the four operations:

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

**Example 8**

Translate each English phrase into an algebraic expression:

- the difference of $14x$ and $9$
- the quotient of $8y^{2}$ and $3$
- twelve more than $y$
- seven less than $49x^{2}$

**Solution**

**Part 1.**

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

**Part 2.**

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

**Part 3.**

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

twelve more than $y$

twelve added to $y$

$y+12$

**Part 4.**

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

seven less than $49x^{2}$

seven subtracted from $49x^{2}$

$49x^{2} -7$

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

**Example 9**

Translate the English phrase into an algebraic expression:

- eight times the sum of $x$ and $y$
- the sum of eight times $x$ and $y$

**Solution**

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

**Part 1. **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 $y$

$8(x+y)$

**Part 2.** 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 eight times $x$ and $y$

$8x+y$

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 10**

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.

**Solution**

Write a phrase about the length of the rectangle. | $14$ less than the width |

Substitute $w$ for “the width” | $w$ |

Rewrite less than as subtracted from. | $14$ subtracted from $w$ |

Translate the phrase into algebra. | $w-14$ |

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

**Example 11**

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.

**Solution**

Write a phrase about the number of dimes. | seven less than four times the number of quarters |

Substitute $q$ for the number of quarters. | $7$ less than $4$ times $q$ |

Translate $4$ times $q$. | $7$ less than $4q$ |

Translate the phrase into algebra. | $4q-7$ |

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*Marecek, L., & Mathis, A. H. (2020). Use the Language of Algebra. In Intermediate Algebra 2e. OpenStax. https://openstax.org/books/intermediate-algebra-2e/pages/1-1-use-the-language-of-algebra. License: CC BY 4.0. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction*