maths.free › Algebra › The Language of Algebra › Use the Language of Algebra

Use the Language of Algebra

Use variables and algebraic symbols

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.

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

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

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.

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.
$3\cdot 9$
Try it.
$28\div 4$
Solution
28 divided by 4, the quotient of twenty-eight and four
Try it.
$36\ge 19$
Solution
thirty-six is greater than or equal to nineteen
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
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
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.
$5{r}^{2}$
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.
$7u+2+3u+1$
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.

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $20$ years old and Alex is $23,$ so Alex is $3$ years older than Greg. When Greg was $12,$ Alex was $15.$ When Greg is $35,$ Alex will be $38.$ No matter what Greg’s age is, Alex’s age will always be $3$ years more, right?
In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The $3$ years between them always stays the same, so the age difference is the constant.
In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age $g.$ Then we could use $g+3$ to represent Alex’s age.
Greg’s age Alex’s age
$12$ $15$
$20$ $23$
$35$ $38$
$g$ $g+3$
Letters are used to represent variables. Letters often used for variables are $x,y,a,b,\text{and}\ c.$
Variables and Constants
A variable is a letter that represents a number or quantity whose value may change.
A constant is a number whose value always stays the same.
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
Operation Notation Say: The result is…
Addition $a+b$ $a\ \text{plus}\ b$ the sum of $a$ and $b$
Subtraction $a-b$ $a\ \text{minus}\ b$ the difference of $a$ and $b$
Multiplication $a\cdot b,(a)(b),(a)b,a(b)$ $a\ \text{times}\ b$ The product of $a$ and $b$
Division $a\div b,a/b,\ \frac{a}{b},ba$ $a$ divided by $b$ The quotient of $a$ and $b$
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 $x\ \text{times}\ y$)? To make it clear, use • or parentheses for multiplication.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
The sum of $5$ and $3$ means add $5$ plus $3,$ which we write as $5+3.$
The difference of $9$ and $2$ means subtract $9$ minus $2,$ which we write as $9-2.$
The product of $4$ and $8$ means multiply $4$ times $8,$ which we can write as $4\cdot 8.$
The quotient of $20$ and $5$ means divide $20$ by $5,$ which we can write as $20\div 5.$
Equality Symbol
$a=b\ \text{is read}\ a\ \text{is equal to}\ b$
The symbol $=$ is called the equal sign.
Common Grouping Symbols
parentheses $(\ )$
brackets $[\ ]$
braces $\{\ \}$
Condensed: the full section is in OpenStax Prealgebra 2e.

Identify Expressions and Equations

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. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
Expression Words Phrase
$3+5$ $3\ \text{plus}\ 5$ the sum of three and five
$n-1$ $n$ minus one the difference of $n$ and one
$6\cdot 7$ $6\ \text{times}\ 7$ the product of six and seven
$\frac{x}{y}$ $x$ divided by $y$ the quotient of $x$ and $y$
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with 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. Here are some examples of equations:
Equation Sentence
$3+5=8$ 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.
Expressions and Equations
An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
An equation is made up of two expressions connected by an equal sign.
Example
Try it.
Determine if each is an expression or an equation:
ⓐ $\ 16-6=10$
ⓑ $\ 4\cdot 2+1$
ⓒ $\ x\div 25$
ⓓ $\ y+8=40$
Solution
ⓐ $\ 16-6=10$ This is an equation, two expressions are connected with an equal sign.
ⓑ $\ 4\cdot 2+1$ This is an expression, no equal sign.
ⓒ $\ x\div 25$ This is an expression, no equal sign.
ⓓ $\ y+8=40$ This is an equation, two expressions are connected with an equal sign.
Note
Try it.
Determine if each is an expression or an equation:
ⓐ $\ 23+6=29$
ⓑ $\ 7\cdot 3-7$
Solution
ⓐ equation
ⓑ expression
Note
Try it.
Determine if each is an expression or an equation:
$\ y\div 14$
$\ x-6=21$
Solution
ⓐ expression
ⓑ equation

Simplify Expressions with Exponents

To simplify a numerical 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:
\[4\cdot 2+1\]\[8+1\]\[9\]
Suppose we have the expression $2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2.$ We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. 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 factors of the base we have to multiply.
\[\text{means multiply three factors of 2}\]
We say ${2}^{3}$ is in exponential notation and $2\cdot 2\cdot 2$ is in expanded notation.
Exponential Notation
For any expression ${a}^{n},a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.
\[{a}^{n}\text{means multiply}\ n\ \text{factors of}\ a\]
The expression ${a}^{n}$ is read $a$ to the ${n}^{th}$ power.
For powers of $n=2$ and $n=3,$ we have special names.
\[\begin{array}{l}{a}^{2}\ \text{is read as}\ \text{"}a\ \text{squared"} \\ {a}^{3}\ \text{is read as}\ \text{"}a\ \text{cubed"}\end{array}\]
lists some examples of expressions written in exponential notation.
Exponential Notation 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
Example
Try it.
Write each expression in exponential form:
ⓐ $\ 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16$
ⓑ $\ 9\cdot 9\cdot 9\cdot 9\cdot 9$
ⓒ $\ x\cdot x\cdot x\cdot x$
ⓓ $\ a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a$
Solution
ⓐ The base 16 is a factor 7 times. ${16}^{7}$
ⓑ The base 9 is a factor 5 times. ${9}^{5}$
ⓒ The base $x$ is a factor 4 times. ${x}^{4}$
ⓓ The base $a$ is a factor 8 times. ${a}^{8}$
Note
Try it.
Write each expression in exponential form:
$41\cdot 41\cdot 41\cdot 41\cdot 41$
Solution
41⁵
Note
Try it.
Write each expression in exponential form:
$7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7$
Solution
7⁹
Example
Try it.
Write each exponential expression in expanded form:
ⓐ $\ {8}^{6}$
ⓑ $\ {x}^{5}$
Solution
ⓐ The base is $8$ and the exponent is $6,$ so ${8}^{6}$ means $8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8$
ⓑ The base is $x$ and the exponent is $5,$ so ${x}^{5}$ means $x\cdot x\cdot x\cdot x\cdot x$
Note
Try it.
Simplify:
ⓐ $\ {5}^{3}$
ⓑ $\ {1}^{7}$
Solution
ⓐ 125
ⓑ 1
Condensed: the full section is in OpenStax Prealgebra 2e.

Simplify Expressions Using the Order of Operations

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:
\[4+3\cdot 7\]\[\begin{array}{llllllllllllllll}\text{Some students say it simplifies to 49.} & \ & & \text{Some students say it simplifies to 25.} \\ \begin{array}{lll} & & 4+3\cdot 7 \\ \text{Since}\ 4+3\ \text{gives 7.} & \ & 7\cdot 7 \\ \text{And}\ 7\cdot 7\ \text{is 49.} & \ & 49\end{array} & & & \begin{array}{lll} & & 4+3\cdot 7 \\ \text{Since}\ 3\cdot 7\ \text{is 21.} & & 4+21 \\ \text{And}\ 21+4\ \text{makes 25.} & & 25\end{array}\end{array}\]
Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.
Order of Operations
When simplifying mathematical expressions perform the operations in the following order:
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.
Order of Operations
Please Parentheses
Excuse Exponents
My Dear Multiplication and Division
Aunt Sally Addition and Subtraction
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.
Note
Note
Try it.
Simplify the expressions:
ⓐ $\ 12-5\cdot 2$
ⓑ $\ (12-5)\cdot 2$
Solution
ⓐ 2
ⓑ 14
Note
Try it.
Simplify the expressions:
ⓐ $\ 8+3\cdot 9$
ⓑ $(8+3)\cdot 9$
Solution
ⓐ 35
ⓑ 99
Note
Try it.
Simplify:
$42\div 7\cdot 3$
Solution
18
Note
Try it.
Simplify:
$12\cdot 3\div 4$
Solution
9
Note
Try it.
Simplify:
$30\div 5+10(3-2)$
Solution
16
Condensed: the full section is in OpenStax Prealgebra 2e.

Key Concepts

Operation Notation Say: The result is…
Addition $a+b$ $a\ \text{plus}\ b$ the sum of $a$ and $b$
Multiplication $a\cdot b,(a)(b),(a)b,a(b)$ $a\ \text{times}\ b$ The product of $a$ and $b$
Subtraction $a-b$ $a\ \text{minus}\ b$ the difference of $a$ and $b$
Division $a\div b,a/b,\ \frac{a}{b},ba$ $a$ divided by $b$ The quotient of $a$ and $b$
Equality Symbol
$a=b$ is read as $a$ is equal to $b$
The symbol $=$ is called the equal sign.
Inequality
a
$a$ is to the left of $b$ on the number line
$a>b$ is read $a$ is greater than $b$
$a$ is to the right of $b$ on the number line
Algebraic Notation Say
$a=b$ $a$ is equal to $b$
$a\ne b$ $a$ is not equal to $b$
a $a$ is less than $b$
$a>b$ $a$ is greater than $b$
$a\le b$ $a$ is less than or equal to $b$
$a\ge b$ $a$ is greater than or equal to $b$
Exponential Notation
For any expression ${a}^{n}$ is a factor multiplied by itself $n$ times, if $n$ is a positive integer.
${a}^{n}$ means multiply $n$ factors of $a$
The expression of ${a}^{n}$ is read $a$ to the $n\text{th}$ 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.

Use the Language of Algebra

Use Variables and Algebraic Symbols
In the following exercises, translate from algebraic notation to words.
Try it.
$5\cdot 6$
Solution
5 times 6, the product of five and six
Try it.
$3\cdot 9$
Try it.
$28\div 4$
Solution
28 divided by 4, the quotient of twenty-eight and four
Try it.
$45\div 5$
Try it.
$36\ge 19$
Solution
thirty-six is greater than or equal to nineteen
Try it.
$42\ge 27$
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.
$3\le 20\div 4$
Try it.
$a\ne 7\cdot 4$
Solution
a is not equal to 7 times 4, a is not equal to the product of seven and four
Try it.
$a\ne 1\cdot 12$
Identify Expressions and Equations
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.
$6\cdot 3+5$
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
Try it.
$3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3$
Solution
3⁷
Try it.
$4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4$
Try it.
$x\cdot x\cdot x\cdot x\cdot x$
Solution
x⁵
Try it.
$y\cdot y\cdot y\cdot y\cdot y\cdot y$
In the following exercises, write in expanded form.
Try it.
${10}^{5}$
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.
Try it.
ⓐ $\ 3+8\cdot 5$
ⓑ $\ \text{(3}+\text{8)}\cdot 5$
Solution
ⓐ 43
ⓑ 55
Try it.
ⓐ $\ 2+6\cdot 3$
ⓑ $\ \text{(2}+\text{6)}\cdot 3$
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.
$4\cdot 12/8$
Solution
6
Try it.
$2\cdot 36/6$
Try it.
$6+10/2+2$
Solution
13
Try it.
$9+12/3+4$
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.
$20\div (4+6)\cdot 5$
Solution
10
Try it.
$33\div (3+8)\cdot 2$
Try it.
${4}^{2}+{5}^{2}$
Solution
41
Try it.
${3}^{2}+{7}^{2}$
Try it.
${(4+5)}^{2}$
Solution
81
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.
$5[2+4(3-2)]$
Try it.
Explain the difference between an expression and an equation.
Try it.
Why is it important to use the order of operations to simplify an expression?
Condensed: the full section is in OpenStax Prealgebra 2e.

Guhitamo iyi Rimwe, i Bya. Rimwe munsi, Cyangwa Ubwoko:.

Imibare

Gukoresha (40)

Rimwe ku Urupapuro Itangira. i Kuri Kugenzura; Gufungura in i ya: Intera.

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

i

ⓐ
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?

i

ⓐ yes ⓑ yes ⓒ no ⓓ yes
ⓔ no

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

i

ⓐ no ⓑ yes ⓒ yes ⓓ no
ⓔ no

Factor 48.

i

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 $80.$

i

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

Find the prime factorization of $60.$

i

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

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.

i

$36$

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

i

$72$

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

i

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

i

16

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

i

23

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

i

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

i

86

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

i

1

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

i

ⓐ

Use definition of exponent.
Simplify.

ⓑ

Use definition of exponent.
Simplify.

ⓒ

Follow the order of operations.

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

i

ⓐ 9 ⓑ 64 ⓒ 40

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

i

ⓐ 216 ⓑ 64 ⓒ 185

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

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

i

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

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

i

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

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 $49{x}^{2}$

i

ⓐ 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 $8x$

i

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

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

i

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

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

i

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

i

ⓐ $4(p+q)$ ⓑ $4p+q$

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

i

ⓐ $2x-8$ ⓑ $2(x-8)$

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.

i

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.

i

$w-7$

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.

i

$l-6$

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.

i

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.

i

$4q-8$

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.

i

$7n+3$

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

i

5

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

$2+8(6+1)$

i

58

$4+6(3+6)$

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

i

29

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

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

i

149

← < entry > Ibanjirije Introduction to the Language of Algebra Komeza>> → Evaluate, Simplify, and Translate Expressions

Kuri Gushyiraho
A Kigenga Konti: & OYA & Ongera Ibisobanuro: ku, A Ibijyana Bya, in Rimwe, na A Kubaza Bigyanye iyi Ipaji:. Imibare ni Gufungura Kuri, in Cyangwa OYA.
Kwiyandikisha Ifashayinjira

Byakoreshejwe

Icyo ari cyo cyose Ibimenyetso ya: i Byuzuye Insobanuro:, A Ishusho, na buri Ibaruwa: in.

ku iyi Ipaji

Rimwe kugirango

Kuri: Use the Language of Algebra

Use variables and algebraic symbols
Identify expressions and equations
Simplify expressions with exponents
Simplify expressions using the order of operations
The
The
The
The

Abaturage

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.

Bya iyi Ipaji: Bivuye OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). na -; Ikosa. Inkomoko ni ya: - Ibicuruzwa - Koresha, iyi ni Kigenga,, na ni i.

in 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.

Operation	Notation	Say:	The result is…
Addition	\(a+b\)	\(a\ \text{plus}\ b\)	the sum of \(a\) and \(b\)
Subtraction	\(a-b\)	\(a\ \text{minus}\ b\)	the difference of \(a\) and \(b\)
Multiplication	\(a\cdot b,(a)(b),(a)b,a(b)\)	\(a\ \text{times}\ b\)	The product of \(a\) and \(b\)
Division	\(a\div b,a/b,\ \frac{a}{b},ba\)	\(a\) divided by \(b\)	The quotient of \(a\) and \(b\)

Common Grouping Symbols
parentheses	\((\ )\)
brackets	\([\ ]\)
braces	\(\{\ \}\)

Equation	Sentence
\(3+5=8\)	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.

ⓐ \(\ 16-6=10\)	This is an equation, two expressions are connected with an equal sign.
ⓑ \(\ 4\cdot 2+1\)	This is an expression, no equal sign.
ⓒ \(\ x\div 25\)	This is an expression, no equal sign.
ⓓ \(\ y+8=40\)	This is an equation, two expressions are connected with an equal sign.

Exponential Notation	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

ⓐ The base 16 is a factor 7 times.	\({16}^{7}\)
ⓑ The base 9 is a factor 5 times.	\({9}^{5}\)
ⓒ The base \(x\) is a factor 4 times.	\({x}^{4}\)
ⓓ The base \(a\) is a factor 8 times.	\({a}^{8}\)

Order of Operations
Please	Parentheses
Excuse	Exponents
My Dear	Multiplication and Division
Aunt Sally	Addition and Subtraction

Algebraic Notation	Say
\(a=b\)	\(a\) is equal to \(b\)
\(a\ne b\)	\(a\) is not equal to \(b\)
a	\(a\) is less than \(b\)
\(a>b\)	\(a\) is greater than \(b\)
\(a\le b\)	\(a\) is less than or equal to \(b\)
\(a\ge b\)	\(a\) 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