maths.free › Algebra › 1. Foundations › Use the Language of Algebra

Use the Language of Algebra

Find factors, prime factorizations, and least common multiples

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 2 and a counting number.

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

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 15 by 3, we get a counting number. In fact, $15\div 3$ is 5, so 15 is $5\cdot 3.$

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

Example

Try it.

Is 5,625 divisible by ⓐ 2? ⓑ 3? ⓒ 5 or 10? ⓓ 6?

Solution

ⓐ

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

ⓒ

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.

ⓓ

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.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 $x,y,a,b,c.$

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.

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 “>”.

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,

\[\begin{array}{l}aa.\ \text{For example,}\ 7<11\ \text{is equivalent to}\ 11>7. \\ a>b\ \text{is equivalent to}\ b4\ \text{is equivalent to}\ 4<17.\end{array}\]

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.

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

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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:

\[\begin{array}{l}4\cdot 2+1 \\ 8+1 \\ 9\end{array}\]

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

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.

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”.

\[\begin{array}{llll}\text{P}\text{arentheses} & & & \ \text{P}\text{lease} \\ \text{E}\text{xponents} & & & \ \text{E}\text{xcuse} \\ \text{M}\text{ultiplication}\ \text{D}\text{ivision} & & & \ \text{M}\text{y}\ \text{D}\text{ear} \\ \text{A}\text{ddition}\ \text{S}\text{ubtraction} & & & \ \text{A}\text{unt}\ \text{S}\text{ally}\end{array}\]

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.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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.

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

Example

Try it.

Evaluate when $x=4:$ ⓐ ${x}^{2}$ ⓑ ${3}^{x}$ ⓒ $2{x}^{2}+3x+8.$

Solution

ⓐ



Use definition of exponent.
Simplify.

ⓑ



Use definition of exponent.
Simplify.

ⓒ



Follow the order of operations.

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.

Examples of terms are $7,y,5{x}^{2},9a,$ and ${b}^{5}.$

The constant that multiplies the variable is called the coefficient.

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?

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

We say,

\[\begin{array}{llll}\text{Simplify.} & & & \ 4x+7x+x \\ \text{Add the coefficients.} & & & \ 12x\end{array}\]

How To Combine Like Terms

Try it.

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

Solution

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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. 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	$a\div b,a\text{/}b,\frac{a}{b},ba$

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

Try it.

Translate each English phrase into an algebraic expression:

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

Solution

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

\[\begin{array}{l}\text{twelve more than}\ y \\ \text{twelve added to}\ y \\ y+12\end{array}\]

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

\[\begin{array}{l}\text{seven less than}\ 49{x}^{2} \\ \text{seven subtracted from}\ 49{x}^{2} \\ 49{x}^{2}-7\end{array}\]

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

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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
Inequality Symbols

Inequality Symbols Words
$a\ne b$ a is not equal to b.
$a a is less than b.
\(a\le b$ a is less than or equal to b.
$a>b$ a is greater than b.
$a\ge b$ a is greater than or equal to b.
Grouping Symbols
$\begin{array}{llllll}\text{Parentheses} & & & & & (\ ) \\ \text{Brackets} & & & & & [\ ] \\ \text{Braces} & & & & & \{\ \}\end{array}$
Exponential Notation
${a}^{n}$ means multiply a by itself, n times.
The expression ${a}^{n}$ is read a to the ${n}^{th}$ 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.

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.

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	$a\div b,a\text{/}b,\frac{a}{b},ba$

Use the Language of Algebra

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.

Try it.

Solution

Divisible by 2, 3, 6

Try it.

896

Solution

Divisible by 2

Try it.

942

Try it.

22,335

Solution

Divisible by 3, 5

Try it.

39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

Try it.

Solution

$2\cdot 43$

Try it.

455

Solution

$5\cdot 7\cdot 13$

Try it.

400

Try it.

432

Solution

$2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3$

Try it.

627

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

Try it.

8, 12

Solution

Try it.

12, 16

Try it.

28, 40

Solution

280

Try it.

84, 90

Try it.

55, 88

Solution

440

Try it.

60, 72

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

Try it.

${2}^{3}-12\div (9-5)$

Solution

Try it.

${3}^{2}-18\div (11-5)$

Try it.

$2+8(6+1)$

Solution

Try it.

$4+6(3+6)$

Try it.

$20\div 4+6(5-1)$

Solution

Try it.

$33\div 3+4(7-2)$

Try it.

$3(1+9\cdot 6)-{4}^{2}$

Solution

149

Try it.

$5(2+8\cdot 4)-{7}^{2}$

Try it.

$2[1+3(10-2)]$

Solution

Try it.

$5[2+4(3-2)]$

Try it.

$8+2[7-2(5-3)]-{3}^{2}$

Solution

Try it.

$10+3[6-2(4-2)]-{2}^{4}$

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Try it.

When $x=2,$
ⓐ ${x}^{6}$
ⓑ ${4}^{x}$
ⓒ $2{x}^{2}+3x-7$

Solution

ⓐ 64 ⓑ 16 ⓒ 7

Try it.

When $x=3,$
ⓐ ${x}^{5}$
ⓑ ${5}^{x}$
ⓒ $3{x}^{2}-4x-8$

Try it.

When $x=4,y=1$
${x}^{2}+3xy-7{y}^{2}$

Solution

Try it.

When $x=3,y=2$
$6{x}^{2}+3xy-9{y}^{2}$

Try it.

When $x=10,y=7$
${(x-y)}^{2}$

Solution

Try it.

When $a=3,b=8$
${a}^{2}+{b}^{2}$

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

Try it.

$7x+2+3x+4$

Solution

$10x+6$

Try it.

$8y+5+2y-4$

Try it.

$10a+7+5a-2+7a-4$

Solution

$22a+1$

Try it.

$7c+4+6c-3+9c-1$

Try it.

$3{x}^{2}+12x+11+14{x}^{2}+8x+5$

Solution

$17{x}^{2}+20x+16$

Try it.

$5{b}^{2}+9b+10+2{b}^{2}+3b-4$

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

Try it.

ⓐ the difference of $5{x}^{2}$ and $6xy$
ⓑ the quotient of $6{y}^{2}$ and $5x$
ⓒ Twenty-one more than ${y}^{2}$
ⓓ $6x$ less than $81{x}^{2}$

Solution

ⓐ $5{x}^{2}-6xy$ ⓑ $\frac{6{y}^{2}}{5x}$
ⓒ ${y}^{2}+21$ ⓓ $81{x}^{2}-6x$

Try it.

ⓐ the difference of $17{x}^{2}$ and $5xy$
ⓑ the quotient of $8{y}^{3}$ and $3x$
ⓒ Eighteen more than ${a}^{2}$;
ⓓ $11b$ less than $100{b}^{2}$

Try it.

ⓐ the sum of $4a{b}^{2}$ and $3{a}^{2}b$
ⓑ the product of $4{y}^{2}$ and $5x$
ⓒ Fifteen more than $m$
ⓓ $9x$ less than $121{x}^{2}$

Solution

ⓐ $4a{b}^{2}+3{a}^{2}b$ ⓑ $20x{y}^{2}$
ⓒ $m+15$ ⓓ $121{x}^{2}-9x$

Try it.

ⓐ the sum of $3{x}^{2}y$ and $7x{y}^{2}$
ⓑ the product of $6x{y}^{2}$ and $4z$
ⓒ Twelve more than $3{x}^{2}$
ⓓ $7{x}^{2}$ less than $63{x}^{3}$

Try it.

ⓐ eight times the difference of $y$ and nine
ⓑ the difference of eight times $y$ and 9

Solution

ⓐ $8(y-9)$ ⓑ $8y-9$

Try it.

ⓐ seven times the difference of $y$ and one
ⓑ the difference of seven times $y$ and 1

Try it.

ⓐ five times the sum of $3x$ and $y$
ⓑ the sum of five times $3x$ and $y$

Solution

ⓐ $5(3x+y)$ ⓑ $15x+y$

Try it.

ⓐ eleven times the sum of $4{x}^{2}$ and $5x$
ⓑ the sum of eleven times $4{x}^{2}$ and $5x$

Try it.

Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let c represent the number of country songs. Write an expression for the number of rock songs.

Solution

$2c+14$

Try it.

The number of women in a Statistics class is 8 more than twice the number of men. Let $m$ represent the number of men. Write an expression for the number of women.

Try it.

Greg has nickels and pennies in his pocket. The number of pennies is seven less than three times the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.

Solution

$3n-7$

Try it.

Jeannette has $\text{\$}5$ and $\text{\$}10$ bills in her wallet. The number of fives is three more than six times the number of tens. Let $t$ represent the number of tens. Write an expression for the number of fives.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 variables 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 .

Greg’s age	Alex’s age
$12$	$15$
$20$	$23$
$35$	$38$
$\text{g}$	$\text{g}+3$

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 in the table below. You’ll probably recognize some of them.

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 times b	the product of a and b
Division	$a\div b,a\text{/}b,\frac{a}{b},ba$	a divided by b	the quotient of a and b, a is called the dividend, and b is called the divisor

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 $4\cdot 8.$

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

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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’d 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:

\[\begin{array}{l}4\cdot 2+1 \\ 8+1 \\ 9\end{array}\]

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

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:

\[4+3\cdot 7\]

If you simplify this expression, what do you get?

Some students say 49,

\[\begin{array}{llllll} & & & & & \ 4+3\cdot 7 \\ \text{Since}\ 4+3\ \text{gives}\ 7. & & & & & \ 7\cdot 7 \\ \text{And}\ 7\cdot 7\ \text{is}\ 49. & & & & & \ 49\end{array}\]

Others say 25,

\[\begin{array}{llllll} & & & & & 4+3\cdot 7 \\ \text{Since}\ 3\cdot 7\ \text{is}\ 21. & & & & & 4+21 \\ \text{And}\ 21+4\ \text{makes}\ 25. & & & & & 25\end{array}\]

Imagine the confusion in our banking system if every problem had several different correct answers!

\[\begin{array}{llll}\text{P}\text{arentheses} & & & \text{P}\text{lease} \\ \text{E}\text{xponents} & & & \text{E}\text{xcuse} \\ \text{M}\text{ultiplication}\ \text{D}\text{ivision} & & & \text{M}\text{y}\ \text{D}\text{ear} \\ \text{A}\text{ddition}\ \text{S}\text{ubtraction} & & & \text{A}\text{unt}\ \text{S}\text{ally}\end{array}\]

Example

Try it.

Simplify: ⓐ $4+3\cdot 7$ ⓑ $(4+3)\cdot 7.$

Solution

ⓐ


Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.
Add.

ⓑ


Are there any parentheses? Yes.
Simplify inside the parentheses.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Evaluate an Expression

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

Example

Try it.

Evaluate $7x-4,$ when ⓐ $x=5$ and ⓑ $x=1.$

Solution

ⓐ



Multiply.
Subtract.

ⓑ



Multiply.
Subtract.

Example

Try it.

Evaluate the following for $x=4,$ when ⓐ ${x}^{2}$ ⓑ ${3}^{x}.$

Solution

ⓐ

	${x}^{2}$

Use definition of exponent.	$4\cdot 4$
Simplify.	$16$

ⓑ

	${3}^{\text{x}}$

Use definition of exponent.	$3\cdot 3\cdot 3\cdot 3$
Simplify.	$81$

Example

Try it.

Evaluate $2{x}^{2}+3x+8$ when $x=4.$

Solution

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

Follow the order of operations.	$2(16)+3(4)+8$
	$32+12+8$
	$52$

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.

Examples of terms are $7,y,5{x}^{2},9a,\text{and}\ {b}^{5}.$

The constant that multiplies the variable is called the coefficient.

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 $x=1\cdot x.$

Example

Try it.

Solution

ⓐ The coefficient of 14y is 14.

ⓑ The coefficient of $15{x}^{2}$ is 15.

ⓒ The coefficient of a is 1 since $a=1\ a.$

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

\[\begin{array}{llllllllllllllll}5x & & & 7 & & & {n}^{2} & & & 4 & & & 3x & & & 9{n}^{2}\end{array}\]

The 7 and the 4 are both constant terms.

7 and 4 are like terms.
5x and 3x are like terms.
${x}^{2}$ and $9{x}^{2}$ are like terms.

Example

Try it.

Identify the like terms: ${y}^{3},$ $7{x}^{2},$ 14, 23, $4{y}^{3},$ 9x, $5{x}^{2}.$

Solution

${y}^{3}$ and $4{y}^{3}$ are like terms because both have ${y}^{3};$ the variable and the exponent match.

$7{x}^{2}$ and $5{x}^{2}$ are like terms because both have ${x}^{2};$ the variable and the exponent match.

14 and 23 are like terms because both are constants.

There is no other term like 9x.

Example

Try it.

Identify the terms in each expression.

ⓐ $9{x}^{2}+7x+12$
ⓑ $8x+3y$

Solution

ⓐ The terms of $9{x}^{2}+7x+12$ are $9{x}^{2},$ 7x, and 12.
ⓑ The terms of $8x+3y$ are 8x and 3y.

\[\begin{array}{l}4x+7x+x \\ x+x+x+x\ +x+x+x+x+x+x+x\ +x \\ 12x\end{array}\]

How To Combine Like Terms

Try it.

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

Solution

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Translate an English Phrase to an Algebraic Expression

In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. 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 $a$	$a\div b,a\text{/}b,\frac{a}{b},ba$

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

Try it.

Translate each English phrase into an algebraic expression: ⓐ the difference of $17x$ and $5$ ⓑ the quotient of $10{x}^{2}$ and $7.$

Solution

ⓐ 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 key word is “quotient,” which tells us the operation is division.

This can also be written $10{x}^{2}\text{/}7\ \text{or}\ \frac{10{x}^{2}}{7}.$

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.

Example

Try it.

Translate the English phrase into an algebraic expression: ⓐ Seventeen more than y ⓑ Nine less than $9{x}^{2}.$

Solution

ⓐ The key words are more than. They tell us the operation is addition. More than means “added to.”
\[\begin{array}{l}\text{Seventeen more than}\ y \\ \text{Seventeen added to}\ y \\ y+17\end{array}\]
ⓑ The key words are less than. They tell us to subtract. Less than means “subtracted from.”
\[\begin{array}{l} \\ \\ \\ \text{Nine less than}\ 9{x}^{2} \\ \text{Nine subtracted from}\ 9{x}^{2} \\ 9{x}^{2}-9\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Notation The result is…

$\begin{array}{llll}\ ∘\ a+b & & & \ \text{the sum of}\ a\ \text{and}\ b \\ ∘\ a-b & & & \ \text{the difference of}\ a\ \text{and}\ b \\ ∘\ a\cdot b,ab,(a)(b)\ (a)b,a(b) & & & \ \text{the product of}\ a\ \text{and}\ b \\ ∘\ a\div b,a\text{/}b,\frac{a}{b},ba & & & \ \text{the quotient of}\ a\ \text{and}\ b\end{array}$
Inequality

$\begin{array}{llll}\ ∘\ ab\ \text{is read}\ \text{“}a\ \text{is greater than}\ b\text{”} & & & a\ \text{is to the right of}\ b\ \text{on the number line}\end{array}$
Inequality Symbols Words

$\begin{array}{llll}\ ∘\ a\ne b & & & \ a\ \text{is}\ \text{not equal to}\ b \\ ∘\ ab & & & \ a\ \text{is}\ \text{greater than}\ b \\ ∘\ a\ge b & & & \ a\ \text{is}\ \text{greater than or equal to}\ b\end{array}$
Grouping Symbols
Parentheses $(\ )$
Brackets $[\ ]$
Braces $\{\ \}$
Exponential Notation
${a}^{n}$ means multiply $a$ by itself, $n$ times. The expression ${a}^{n}$ is read $a$ to the ${n}^{th}$ power.
Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
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.
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.

Use the Language of Algebra

Use Variables and Algebraic Symbols
In the following exercises, translate from algebra to English.
Try it.
$16-9$
Solution
16 minus 9, the difference of sixteen and nine
Try it.
$3\cdot 9$
Try it.
$28\div 4$
Solution
28 divided by 4, the quotient of twenty-eight and four
Try it.
$x+11$
Try it.
$(2)(7)$
Solution
2 times 7, the product of two and seven
Try it.
$(4)(8)$
Try it.
$14<21$
Solution
fourteen is less than twenty-one
Try it.
$17<35$
Try it.
$36\ge 19$
Solution
thirty-six is greater than or equal to nineteen
Try it.
$6n=36$
Try it.
$y-1>6$
Solution
y minus 1 is greater than 6, the difference of y and one is greater than six
Try it.
$y-4>8$
Try it.
$2\le 18\div 6$
Solution
2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six
Try it.
$a\ne 1\cdot 12$
In the following exercises, determine if each is an expression or an equation.
Try it.
$9\cdot 6=54$
Solution
equation
Try it.
$7\cdot 9=63$
Try it.
$5\cdot 4+3$
Solution
expression
Try it.
$x+7$
Try it.
$x+9$
Solution
expression
Try it.
$y-5=25$
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
Try it.
${5}^{3}$
Solution
125
Try it.
${8}^{3}$
Try it.
${2}^{8}$
Solution
256
Try it.
${10}^{5}$
In the following exercises, simplify using the order of operations.
Try it.
ⓐ $3+8\cdot 5$ ⓑ $(3+8)\cdot 5$
Solution
ⓐ 43 ⓑ 55
Try it.
ⓐ $2+6\cdot 3$ ⓑ $(2+6)\cdot 3$
Try it.
${2}^{3}-12\div (9-5)$
Solution
5
Try it.
${3}^{2}-18\div (11-5)$
Try it.
$3\cdot 8+5\cdot 2$
Solution
34
Try it.
$4\cdot 7+3\cdot 5$
Try it.
$2+8(6+1)$
Solution
58
Try it.
$4+6(3+6)$
Try it.
$4\cdot 12\text{/}8$
Solution
6
Try it.
$2\cdot 36\text{/}6$
Try it.
$(6+10)\div (2+2)$
Solution
4
Try it.
$(9+12)\div (3+4)$
Try it.
$20\div 4+6\cdot 5$
Solution
35
Try it.
$33\div 3+8\cdot 2$
Try it.
${3}^{2}+{7}^{2}$
Solution
58
Try it.
${(3+7)}^{2}$
Try it.
$3(1+9\cdot 6)-{4}^{2}$
Solution
149
Try it.
$5(2+8\cdot 4)-{7}^{2}$
Try it.
$2[1+3(10-2)]$
Solution
50
Try it.
$5[2+4(3-2)]$
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Try it.
$7x+8$ when $x=2$
Solution
22
Try it.
$8x-6$ when $x=7$
Try it.
${x}^{2}$ when $x=12$
Solution
144
Try it.
${x}^{3}$ when $x=5$
Try it.
${x}^{5}$ when $x=2$
Solution
32
Try it.
${4}^{x}$ when $x=2$
Try it.
${x}^{2}+3x-7$ when $x=4$
Solution
21
Try it.
$6x+3y-9$ when
$x=6,y=9$
Try it.
${(x-y)}^{2}$ when
$x=10,y=7$
Solution
9
Try it.
${(x+y)}^{2}$ when $x=6,y=9$
Try it.
${a}^{2}+{b}^{2}$ when $a=3,b=8$
Solution
73
Try it.
${r}^{2}-{s}^{2}$ when $r=12,s=5$
Try it.
$2l+2w$ when
$l=15,w=12$
Solution
54
Try it.
$2l+2w$ when
$l=18,w=14$
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
Try it.
8a
Solution
8
Try it.
13m
Try it.
$5{r}^{2}$
Solution
5
Try it.
$6{x}^{3}$
In the following exercises, identify the like terms.
Try it.
${x}^{3},8x,14,8y,5,8{x}^{3}$
Solution
${x}^{3}\ \text{and}\ 8{x}^{3},14\ \text{and}\ 5$
Try it.
$6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}$
Try it.
$9a,{a}^{2},16,16{b}^{2},4,9{b}^{2}$
Solution
$16\ \text{and}\ 4,16{b}^{2}\ \text{and}\ 9{b}^{2}$
Try it.
$3,25{r}^{2},10s,10r,4{r}^{2},3s$
In the following exercises, identify the terms in each expression.
Try it.
$15{x}^{2}+6x+2$
Solution
$15{x}^{2},6x,2$
Try it.
$11{x}^{2}+8x+5$
Try it.
$10{y}^{3}+y+2$
Solution
$10{y}^{3},y,2$
Try it.
$9{y}^{3}+y+5$
In the following exercises, simplify the following expressions by combining like terms.
Try it.
$10x+3x$
Solution
13x
Try it.
$15x+4x$
Try it.
$4c+2c+c$
Solution
7c
Try it.
$6y+4y+y$
Try it.
$7u+2+3u+1$
Solution
$10u+3$
Try it.
$8d+6+2d+5$
Try it.
$10a+7+5a-2+7a-4$
Solution
$22a+1$
Try it.
$7c+4+6c-3+9c-1$
Try it.
$3{x}^{2}+12x+11+14{x}^{2}+8x+5$
Solution
$17{x}^{2}+20x+16$
Try it.
$5{b}^{2}+9b+10+2{b}^{2}+3b-4$
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.
Try it.
the difference of 14 and 9
Solution
$14-9$
Try it.
the difference of 19 and 8
Try it.
the product of 9 and 7
Solution
$9\cdot 7$
Try it.
the product of 8 and 7
Try it.
the quotient of 36 and 9
Solution
$36\div 9$
Try it.
the quotient of 42 and 7
Try it.
the sum of 8x and 3x
Solution
$8x+3x$
Try it.
the sum of 13x and 3x
Try it.
the quotient of y and 3
Solution
$\frac{y}{3}$
Try it.
the quotient of y and 8
Try it.
eight times the difference of y and nine
Solution
$8(y-9)$
Try it.
seven times the difference of y and one
Try it.
Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let c represent the number of classical CDs. Write an expression for the number of rock CDs.
Solution
$c+3$
Try it.
The number of girls in a second-grade class is 4 less than the number of boys. Let b represent the number of boys. Write an expression for the number of girls.
Try it.
Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.
Solution
$2n-7$
Try it.
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Is 5,625 divisible by ⓐ 2? ⓑ 3? ⓒ 5 or 10? ⓓ 6?

Atbildēt uz šo jautājumu

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

ⓓ
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.

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

Atbildēt uz šo jautājumu

ⓐ yes ⓑ yes ⓒ no ⓓ yes
ⓔ no

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

Atbildēt uz šo jautājumu

ⓐ no ⓑ yes ⓒ yes ⓓ no
ⓔ no

Factor 48.

Atbildēt uz šo jautājumu

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.

Find the prime factorization of $80.$

Atbildēt uz šo jautājumu

$2\cdot 2\cdot 2\cdot 2\cdot 5$

Find the prime factorization of $60.$

Atbildēt uz šo jautājumu

$2\cdot 2\cdot 3\cdot 5$

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

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

Atbildēt uz šo jautājumu

$36$

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

Atbildēt uz šo jautājumu

$72$

Simplify: $18\div 6+4(5-2).$

Atbildēt uz šo jautājumu

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

Simplify: $30\div 5+10(3-2).$

Atbildēt uz šo jautājumu

16

Simplify: $70\div 10+4(6-2).$

Atbildēt uz šo jautājumu

23

Simplify: $5+{2}^{3}+3[6-3(4-2)].$

Atbildēt uz šo jautājumu

Are there any parentheses (or other
grouping symbols)? Yes.
Focus on the parentheses that are inside the
brackets. Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires
no further simplification.
Are there any exponents? Yes. Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition of subtraction? Yes.
Add.
Add.

Simplify: $9+{5}^{3}-[4(9+3)].$

Atbildēt uz šo jautājumu

86

Simplify: ${7}^{2}-2[4(5+1)].$

Atbildēt uz šo jautājumu

1

Evaluate when $x=4:$ ⓐ ${x}^{2}$ ⓑ ${3}^{x}$ ⓒ $2{x}^{2}+3x+8.$

Atbildēt uz šo jautājumu

ⓐ

Use definition of exponent.
Simplify.

ⓑ

Use definition of exponent.
Simplify.

ⓒ

Follow the order of operations.

Evaluate when $x=3,$ ⓐ ${x}^{2}$ ⓑ ${4}^{x}$ ⓒ $3{x}^{2}+4x+1.$

Atbildēt uz šo jautājumu

ⓐ 9 ⓑ 64 ⓒ 40

Evaluate when $x=6,$ ⓐ ${x}^{3}$ ⓑ ${2}^{x}$ ⓒ $6{x}^{2}-4x-7.$

Atbildēt uz šo jautājumu

ⓐ 216 ⓑ 64 ⓒ 185

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

Simplify: $3{x}^{2}+7x+9+7{x}^{2}+9x+8.$

Atbildēt uz šo jautājumu

$10{x}^{2}+16x+17$

Simplify: $4{y}^{2}+5y+2+8{y}^{2}+4y+5.$

Atbildēt uz šo jautājumu

$12{y}^{2}+9y+7$

Translate each English phrase into an algebraic expression:
ⓐ the difference of $14x$ and 9 ⓑ the quotient of $8{y}^{2}$ and 3 ⓒ twelve more than $y$ ⓓ seven less than $49{x}^{2}$

Atbildēt uz šo jautājumu

ⓐ 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 key word is quotient, which tells us the operation is division.
ⓒ The key words are more than. They tell us the operation is addition. More than means “added to.”
\[\begin{array}{l}\text{twelve more than}\ y \\ \text{twelve added to}\ y \\ y+12\end{array}\]
ⓓ The key words are less than. They tell us to subtract. Less than means “subtracted from.”
\[\begin{array}{l}\text{seven less than}\ 49{x}^{2} \\ \text{seven subtracted from}\ 49{x}^{2} \\ 49{x}^{2}-7\end{array}\]

Translate the English phrase into an algebraic expression:
ⓐ the difference of $14{x}^{2}$ and 13 ⓑ the quotient of $12x$ and 2 ⓒ 13 more than $z$
ⓓ 18 less than $8x$

Atbildēt uz šo jautājumu

ⓐ $14{x}^{2}-13$ ⓑ $12x\div 2$
ⓒ $z+13$ ⓓ $8x-18$

Translate the English phrase into an algebraic expression:
ⓐ the sum of $17{y}^{2}$ and 19 ⓑ the product of $7$ and y ⓒ Eleven more than x ⓓ Fourteen less than 11a

Atbildēt uz šo jautājumu

ⓐ $17{y}^{2}+19$ ⓑ $7y$
ⓒ $x+11$ ⓓ $11a-14$

Translate the English phrase into an algebraic expression:
ⓐ eight times the sum of x and y ⓑ the sum of eight times x and y

Atbildēt uz šo jautājumu

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.)
\[\begin{array}{l}\text{eight times the sum of}\ x\ \text{and}\ y \\ 8(x+y)\end{array}\]
ⓑ 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.

Translate the English phrase into an algebraic expression:
ⓐ four times the sum of p and q
ⓑ the sum of four times p and q

Atbildēt uz šo jautājumu

ⓐ $4(p+q)$ ⓑ $4p+q$

Translate the English phrase into an algebraic expression:
ⓐ the difference of two times x and 8
ⓑ two times the difference of x and 8

Atbildēt uz šo jautājumu

ⓐ $2x-8$ ⓑ $2(x-8)$

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

Atbildēt uz šo jautājumu

Write a phrase about the width of the rectangle. 14 less than the length
Substitute l for “the length.” l
Rewrite less than as subtracted from. 14 subtracted from l
Translate the phrase into algebra. l − 14

The length of a rectangle is 7 less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

Atbildēt uz šo jautājumu

$w-7$

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.

Atbildēt uz šo jautājumu

$l-6$

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.

Atbildēt uz šo jautājumu

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

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.

Atbildēt uz šo jautājumu

$4q-8$

Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let n represent the number of nickels. Write an expression for the number of dimes.

Atbildēt uz šo jautājumu

$7n+3$

${2}^{3}-12\div (9-5)$

Atbildēt uz šo jautājumu

5

${3}^{2}-18\div (11-5)$

$2+8(6+1)$

Atbildēt uz šo jautājumu

58

$4+6(3+6)$

$20\div 4+6(5-1)$

Atbildēt uz šo jautājumu

29

$33\div 3+4(7-2)$

$3(1+9\cdot 6)-{4}^{2}$

Atbildēt uz šo jautājumu

149

Symbols used here

\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Use the Language of Algebra

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
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.

Questions people ask

What does it mean to solve an equation?
To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.
Why do I sometimes get two answers?
A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.
How do I know whether to factor or use the quadratic formula?
Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Izmēģiniet savu

Jūsu matemātikas problēma

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Vairāk Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

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	\(a\cdot b,ab,(a)(b),\) \((a)b,a(b)\)	\(a\) times \(b\)	the product of \(a\) and \(b\)
Division	\(a\div b,a\text{/}b,\frac{a}{b},\ ba\)	\(a\) divided by \(b\)	the quotient of \(a\) and \(b;\) \(a\) is called the dividend, and \(b\) is called the divisor

Greg’s age	Alex’s age
\(12\)	\(15\)
\(20\)	\(23\)
\(35\)	\(38\)
\(\text{g}\)	\(\text{g}+3\)

	\({x}^{2}\)

Use definition of exponent.	\(4\cdot 4\)
Simplify.	\(16\)

	\({3}^{\text{x}}\)

Use definition of exponent.	\(3\cdot 3\cdot 3\cdot 3\)
Simplify.	\(81\)

	\(2{x}^{2}+3x+8\)

Follow the order of operations.	\(2(16)+3(4)+8\)
	\(32+12+8\)
	\(52\)

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.

Inequality Symbols	Words
\(a\ne b\)	a is not equal to b.
\(a	a is less than b.
\(a\le b\)	a is less than or equal to b.
\(a>b\)	a is greater than b.
\(a\ge b\)	a is greater than or equal to b.


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


Are there any parentheses (or other grouping symbols)? Yes.
Focus on the parentheses that are inside the brackets. Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes. Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition of subtraction? Yes.
Add.
Add.

Write a phrase about the width of the rectangle.	14 less than the length
Substitute l for “the length.”	l
Rewrite less than as subtracted from.	14 subtracted from l
Translate the phrase into algebra.	l − 14

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