1.1 Using the Language of Algebra - Intermediate Algebra

learning goals

At the end of this section you can:

Find factors, prime factorization, and least common multiples
Use of variables and algebraic symbols
Simplify expressions using order of operations
evaluate an expression
Identify and combine similar terms.
Translate an English sentence into an algebraic expression

To be prepared 1.1

This chapter is intended as a brief overview of the concepts required in an advanced algebra course. For a more detailed introduction to the topics covered in this chapter, see theElementary algebrachapter "Basics".

Find factors, prime factorization, and least common multiples

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

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (1)

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

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (2)

We can find multiples of any number if we continue this process.

meteraflezing	1	2	3	4	5	6	7	8	9	10	11	12
multiples of 2	2	4	6	8	10	12	14	sixteen	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	sixteen	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	sixteen	24	32	40	48	56	64	72	80	88	96
multiples of 9	9	18	27	36	45	54	63	72	81	90	99	108

multiple of a number

a number is onemultiplevon $norte$ if is the product of a countable number and $norte .$

Another way to say 15 is a multiple of 3 is to say 15 is a multiple of 3divisibletimes 3. That is, if we divide 3 by 15, we get a countable number. Actually, $15 \div 3$ is 5, then 15 $5 \cdot 3 .$

divisible by a number

yes a number $Metro$ is a multiple ofnorte, SoMetroesdivisiblevonnorte.

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

Divisibility tests

A number is divisible by:

2 if the last digit is 0, 2, 4, 6 or 8.

3 if the sum of the digits is divisible by $3 .$

5 if the last digit is a 5 or $0 .$

6 if it is divisible by 2 and $3 .$

10 if it ends with $0 .$

Example 1.1

5625 is divisible byⓐ2?ⓑ3?ⓒ5 of 10?ⓓ6?

Solution

ⓐ $\begin{matrix} Is 5625 divisible by 2? \\ Does it end in 0, 2, 4, 6 or 8? & NEE. \\ 5,625 is not divisible by 2. \end{matrix}$
ⓑ $\begin{matrix} Is 5625 divisible by 3? \\ What is the sum of the digits? & 5 + 6 + 2 + 5 = 18 \\ Is the sum divisible by 3? & In. \\ 5,625 is divisible by 3. \end{matrix}$
ⓒ $\begin{matrix} Is 5625 divisible by 5 or 10? \\ What is the last digit? they are 5 & 5625 is divisible by 5 but not by 10. \end{matrix}$
ⓓ $\begin{matrix} Is 5625 divisible by 6? \\ Is it divisible by 2 and by 3? & No, 5625 is not divisible by 2, so 5625 \\ not divisible by 6 \end{matrix}$

try it 1.1

4,962 is divisible byⓐ2?ⓑ3?ⓒ5?ⓓ6?ⓔ10?

try it 1.2

3,765 is divisible byⓐ2?ⓑ3?ⓒ5?ⓓ6?ⓔ10?

In math, there are often multiple ways to talk about the same ideas. So far we've seen that asMetrois a multiple ofnorteWe can state thatMetrois divisible bynorte. For example, since 72 is a multiple of 8, we say that 72 is divisible by 8. Since 72 is a multiple of 9, we say that 72 is divisible by 9. We can also express this differently.

Out $8 \cdot 9 = 72,$ we say that 8 and 9 arefactorsof 72. When we wrote $72 = 8 \cdot 9,$ We say we decomposed 72.

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (3)

Other ways to factor 72 are $1 \cdot 72, 2 \cdot 36, 3 \cdot 24, 4 \cdot 18,$ J $6 \cdot 12 .$ Number 72 has many factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,$ J $72 .$

factors

In $A \cdot B = Metro,$ SoAJBSonfactorsvonMetro.

Some numbers, like 72, have many factors. Other numbers have only two factors. NASTYPrime numberis a count greater than 1 whose only factors are 1 and itself.

Prime number and composite number

APrime numberis a counting number greater than 1 whose only factors are 1 and the number itself.

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

The table contains the numbers from 2 to 20 with their factors. Make sure you at least agree on the term "main" or "composition"!

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (4)

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

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

factorization into prime factors

Isfactorization into prime factorsa number is the product of primes equal to the number.

To find the prime factorization of a composite number, find any two factors of the number and make two branches of them. If a factor is prime, this branch is complete. Circle this cousin. Otherwise, it's easy to lose track of prime numbers.

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

Example 1.2

How to find the prime factorization of a composite number

factorization 48.

Solution

we say $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$ is the prime factorization of 48. We generally write prime numbers in ascending order. Multiply the factors to check your answer.

If we first factor 48 differently, for example if $6 \cdot 8,$ The result would remain the same. Complete the prime factorization and find out for yourself.

try it 1.3

Find the prime factorization of $80 .$

try it 1.4

Find the prime factorization of $60 .$

If

Find the prime factorization of a composite number.

Step 1.Find two factors whose product is the given number and use those numbers to make two branches.
Step 2.If a factor is prime, this branch is complete. Circle the prime number like a leaf on a tree.
Step 3.If a factor is not prime, write it as the product of two factors and move on.
Level 4.Write the composite number as the product of all the primes circled.

One of the reasons we look for primes is to use these techniques to find themleast common multiplefrom two digits. This is useful when we add and subtract fractions with different denominators.

least common multiple

Isleast common multiple (mcm)of two numbers is the smallest number that is a multiple of both numbers.

To find the least common multiple of two numbers, we use the prime factor method. Let's find the LCM of 12 and 18 using their prime factors.

Example 1.3

How to find the least common multiple using the prime factor method

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

Solution

Note that the prime factors of 12 $(2 \cdot 2 \cdot 3)$ and the prime factors of 18 $(2 \cdot 3 \cdot 3)$ are included in the LCM $(2 \cdot 2 \cdot 3 \cdot 3) .$ So 36 is the least common multiple of 12 and 18.

When assigning common prime numbers, each common prime factor is used only once. This way you know for sure that 36 is the right oneat leastcommon multiple.

try it 1.5

Find the lcm of 9 and 12 using the prime factor method.

try it 1.6

Find the lcm of 18 and 24 using the prime factor method.

If

Find the least common multiple using the prime factor method.

Step 1.Write each number as a product of primes.
Step 2.List the primes of each number. Fit primes vertically whenever possible.
Step 3.Lower the pillars.
Level 4.Multiply the factors.

Use of variables and algebraic symbols

In algebra, we use a letter of the alphabet to represent a number whose value can change. We call that oneVariableand are the most commonly used letters for variables $X, J, A, B, C .$

Variable

AVariableis a letter representing a number whose value can change.

A number whose value always remains the same is calledContinual.

Continual

AContinualis a number whose value always remains the same.

To write algebraically we need some operation symbols, as well as numbers and variables. There are different types of symbols that we will use. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. The symbols used to identify these operations are listed below.

operational icons

Operation	Notation	Participation:	The result is…
additive	$A + B$	$A$ further $B$	the sum of $A$ J $B$
Subtract	$A - B$	$A$ not less $B$	The difference of $A$ J $B$
multiplication	$A \cdot B, A B, (A) (B),$ $(A) B, A (B)$	$A$ mal $B$	the product of $A$ J $B$
division	$A \div B, A / B, \frac{A}{B}, B A$	$A$ divided by $B$	the quotient of $A$ J $B;$ $A$ is called dividend and $B$ it's called the divider

If two quantities have the same value, we say they are equal and connect them with aevenSign.

equality symbol

$A = B$ It's for reading"Ais equal toB.“

The symbol "=" is called the equals sign.

Over hemLine number, the numbers increase from left to right. The number line can be used to explain the symbols "<" and ">".

inequality

The expressions $A < B$ O $A > B$ It can be read left to right or right to left, although in English we usually read left to right. Generally,

$\begin{matrix} A < B It is equal to B > A . For example, 7 < 11 It is equal to 11 > 7 . \\ A > B It is equal to B < A . For example, 17 > 4 It is equal to 4 < 17 . \end{matrix}$

inequality symbols

inequality symbols	Words
$A \neq B$	Aesis not the same b.
$A < B$	Aesinferior and b.
$A \leq B$	Aesless than or equal to b.
$A > B$	Aesgreater than b.
$A \geq B$	Aesgreater than or equal to b.

(Video) Intermediate Algebra 1.1 - Understanding the Language of Algebra

Grouping symbols in algebra are very similar to commas, colons, and other punctuation marks in English. They help identify oneExpression, which can be a number, a variable, or a combination of numbers and variables using editing symbols. We will now introduce three types of grouping symbols.

grouping of symbols

$\begin{matrix} bracket & () \\ supports & [] \\ braces & {} \end{matrix}$

Here are some examples of expressions that contain grouping symbols. We will simplify such expressions later in this section.

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

What is the difference between a phrase and a sentence in English? A sentence expresses a single thought which is incomplete in itself, but a sentence represents a complete statement. A sentence consists of a subject and a verb, in algebra we have thatexpressionsJequations.

Expression

InExpressionis a number, a variable, or a combination of numbers and variables that use operator symbols.

$\begin{matrix} Expression & Words & English sentence \\ 3 + 5 & 3 plus 5 & the sum of three and five \\ norte - 1 & norte minus one & The difference of norte and a \\ 6 \cdot 7 & 6 in painting 7 & the product of six and seven \\ \frac{X}{J} & X divided by J & the quotient of X J J \end{matrix}$

Note that the English sentences do not form a complete sentence as the sentence does not contain a verb.

Incomparisonare two expressions connected by an equals sign. If you read the words that represent the symbols in an equation, you will have a complete sentence in English. The equal sign indicates the verb.

comparison

Incomparisonare two expressions connected by an equals sign.

$\begin{matrix} comparison & sentence in English \\ 3 + 5 = 8 & The sum of three and five equals eight. \\ norte - 1 = 14 & norte minus one is fourteen. \\ 6 \cdot 7 = 42 & The product of six and seven is forty-two. \\ X = 53 & X equals fifty-three. \\ J + 9 = 2 J - 3 & J plus nine is two J minus three. \end{matrix}$

Suppose we need to multiply 2 nine times. We could write this as $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 .$ This is annoying and it can be hard to keep track of all those twos. That's why we use exponents. We write $2 \cdot 2 \cdot 2$ if $2^{3}$ J $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ if $2^{9} .$ In expressions like $2^{3},$ the 2 is calledbaseand the 3 is calledExponent. IsExponenttells us how many times to multiplybase.

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (14)

exponential notation

we say $2^{3}$ is inexponential notationJ $2 \cdot 2 \cdot 2$ is inExtensive list.

$A^{norte}$ means multiplyAyourself,nortemal.

The expression $A^{norte}$ It is readAUnpleasant ${norte}^{T H}$ Power.

while we read $A^{norte}$ if $„ A$ Unpleasant ${norte}^{T H}$ Power', we usually read:

$\begin{matrix} A^{2} & „ A checked" \\ A^{3} & „ A cut in pieces" \end{matrix}$

We'll see why later $A^{2}$ J $A^{3}$ they have special names.

Table 1.1shows how we read some expressions with exponents.

Expression	In words
7²	7 raised to the second power or	7 squared
5³	5 cubed or	5 gerold
9⁴	9 to the fourth power
12⁵	12 to the power of five

Mesa 1.1

Simplify expressions using order of operations

Asimplify an expressionit means calculating as much as possible. For example, to simplify $4 \cdot 2 + 1$ First we would multiply $4 \cdot 2$ to get 8, then add 1 to get 9. A good practice to develop is to work your way down the page and write each step of the process below the previous step. The example just described looks like this:

$\begin{matrix} 4 \cdot 2 + 1 \\ 8 + 1 \\ 9 \end{matrix}$

By not using an equals sign when simplifying an expression, you can avoid confusing expressions with equations.

simplify an expression

Asimplify an expression, perform all operations on the expression.

We've introduced most of the symbols and notations used in algebra, but now we need to clarify themorder of operations. Otherwise, the terms may have different meanings and result in different values.

For example, look at the expression $4 + 3 \cdot 7 .$ Some students simplify this by taking 49 and adding $4 + 3$ and multiply the result by 7. Others get 25 by multiplication $3 \cdot 7$ first and then add 4.

The same expression should produce the same result. That's why mathematicians have created guidelines called the order of operations.

If

Use the order of operations.

Step 1.
Brackets and other grouping symbols
- Simplify any expressions inside parentheses or other grouping symbols by working on the inner parentheses first.
Step 2.
exponents
- Simplify all expressions with exponents.
Step 3.
multiplication and division
- Do all multiplications and divisions in order from left to right. These operations have the same priority.
Level 4.
addition and subtraction
- Do all additions and subtractions from left to right. These operations have the same priority.

Students often ask, "How do I remember the order?" To help you remember, take the first letter of each keyword and replace it with the silly phrase "Excuse me dear Aunt Sally."

$\begin{matrix} PAG dor & PAG rent \\ mi xponents & mi forgiveness \\ METRO multiplication D Vision & METRO J D Ohr \\ A additive S Subtract & A In S allies \end{matrix}$

It's good that"METROJDear” goes hand in hand, because that reminds usMetromultiplication andDivision have the same priority. We don't always do multiplication before division, or we always do division before multiplication. We do them in order from left to right.

Also "AInS"Ally" goes hand in hand, so remind us of thatAsupplement andSSubtractions also have the same priority and are done from left to right.

Example 1.4

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

Solution


bracket? Yes, subtract first.
exponents? NO.
Multiply or divide? Yes.
Divide first because we multiply and divide from left to right.
Another multiplication or division? Yes.
Multiply.
Another division multiplication? NO.
Add or subtract? Yes.
Add to.

try it 1.7

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

try it 1.8

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

If there are multiple grouping symbols, let's simplify and expand on the inner parentheses first.

Example 1.5

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

Solution


Are there brackets (or any other group symbols)? Yes.
Focus on the brackets within the supports. Subtract.
Continue inside the parentheses and multiply.
Continue inside the parentheses and subtract.
The expression in parentheses requires without further simplifications.
Is there an exponent? Yes. Simplify exponents.
Is there multiplication or division? Yes.
Multiply.
Is there addition to subtraction? Yes.
Add to.
Add to.

try it 1.9

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

try it 1.10

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

evaluate an expression

In the previous examples, we simplified expressions using order of operations. We will now evaluate some expressions, again in order of operations. NASTYevaluate an expressionmeans to get the value of the expression when the variable is replaced with a specific number.

evaluate an expression

Aevaluate an expressionmeans to get the value of the expression when the variable is replaced with a specific number.

To evaluate an expression, replace the variable in the expression with this number and simplify the expression.

Example 1.6

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

Solution

ⓐ



Use the definition of the exponent.
Simplify.

ⓑ



Use the definition of the exponent.
Simplify.

ⓒ



Follow the order of operations.

try it 1.11

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

try it 1.12

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

Identify and combine similar terms

Algebraic expressions are made up of terms. NASTYExpressionis a constant or the product of a constant and one or more variables.

Expression

AExpressionis a constant or the product of a constant and one or more variables.

Examples of terms are $7, J, 5 X^{2}, 9 A,$ J $B^{5} .$

The constant that multiplies the variable is calledcoefficient.

coefficient

Iscoefficientof a term is the constant that multiplies the variable in a term.

Think of the coefficient as the number for the variable. The coefficient of the term $3 X$ is 3. If we write $X,$ the coefficient is 1, since $X = 1 \cdot X .$

Some terms have common features. If two terms are constants or have the same variable and exponent, we say they aresimilar terms.

Check out the next 6 terms. Which ones seem to have common features?

$5 X 7 {norte}^{2} 4 3 X 9 {norte}^{2}$

we say,

$7$ J $4$ they are just concepts.

(Video) Intermediate Algebra 1.1 Real Numbers and Algebraic Expressions

$5 X$ J $3 X$ they are just concepts.

${norte}^{2}$ J $9 {norte}^{2}$ they are just concepts.

Similar terms

Terms are mentioned that are constants or have the same variables raised to the same powersimilar terms.

If an expression contains similar terms, you can simplify the expression by combining the same terms. We add the coefficients, keeping the same variable.

$\begin{matrix} Simplify. & 4 X + 7 X + X \\ Add up the coefficients. & 12 X \end{matrix}$

Example 1.7

Combine similar terms

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

Solution

try it 1.13

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

try it 1.14

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

If

Combine similar terms.

Step 1.Identify similar terms.
Step 2.Rearrange the expression so that the same terms go together.
Step 3.Add or subtract the coefficients, keeping the same variable for each group of like terms.

Translate an English sentence into an algebraic expression

We list many operation symbols used in algebra. Now we will use them to convert English sentences into algebraic expressions. The symbols and variables we talked about will help us with this.Table 1.2summarize them.

Operation	Zin	Expression
additive	AfurtherB the sum of $A$ JB Aincreased byB Bmore thanA the sum ofAJB Badded toA	$A + B$
Subtract	Anot less $B$ The difference ofAJB Areduced withinB Bless thanA Bwithdrawn fromA	$A - B$
multiplication	AmalB the product of $A$ J $B$ twiceA	$A \cdot B, A B, A (B), (A) (B)$ $2 A$
division	Adivided byB the quotient ofAJB The relationship ofAJB BassignedA	$A \div B, A / B, \frac{A}{B}, B A$

Mesa 1.2

Take a closer look at these sentences using the four operations:

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (49)

Each sentence asks us to operate on two numbers. find the wordsvonJJto find the numbers.

Example 1.8

Translate an English sentence into an algebraic expression:

ⓐThe difference of $14 X$ year 9ⓑthe quotient of $8 J^{2}$ year 3ⓒsweet but what $J$ ⓓseven less than $49 X^{2}$

Solution

ⓐThe keyword isDifference, which tells us that the operation is a subtraction. find the wordsvonJand Tor find the numbers you want to subtract.

ⓑThe keyword isQuotient, which tells us that the operation is a division.

ⓒThe keywords aremore than.They tell us that the operation is a sum.More thanmeans "added".

$\begin{matrix} sweet but what J \\ twelve added J \\ J + 12 \end{matrix}$

ⓓThe keywords areless than. They tell us to jerk off.Less thanmeans "taken".

$\begin{matrix} seven less than 49 X^{2} \\ seven subtracted 49 X^{2} \\ 49 X^{2} - 7 \end{matrix}$

try it 1.15

Translate the English expression into an algebraic expression:

ⓐThe difference of $14 X^{2}$ year 13ⓑthe quotient of $12 X$ year 2ⓒ13 but what $z$
ⓓ18 less than $8 X$

try it 1.16

Translate the English expression into an algebraic expression:

ⓐthe sum of $17 J^{2}$ J 19ⓑthe product of $7$ JJ ⓒeleven more thanX ⓓfourteen less than 11A

We look closely at the words to distinguish between multiplying a sum and adding a product.

Example 1.9

Translate the English expression into an algebraic expression:

ⓐeight times the sum ofXJJ ⓑthe sum of eight timesXJJ

Solution

There are two editing words:maltells us to multiply andadditiveinvites us to add something.

ⓐSince we're multiplying 8 by the sum, we need parentheses around the sum ofXJJ, $(X + J) .$ This forces us to determine the sum first. (Note the order of operations.)

$\begin{matrix} eight times the sum of X J J \\ 8 (X + J) \end{matrix}$

ⓑTo get a sum, we look for the wordsvonJJto see what's being added. Here we take the sumvoneight timesXJJ.

try it 1.17

Translate the English expression into an algebraic expression:

ⓐfour times the sum ofPageJQ
ⓑthe sum of four timesPageJQ

try it 1.18

Translate the English expression into an algebraic expression:

ⓐthe difference of two timesXyear 8
ⓑtwice as much differenceXyear 8

Later in the course we will use our algebraic skills to solve applications. The first step is to translate an English sentence into an algebraic expression. In the following two examples we will see how you can do this.

Example 1.10

The length of a rectangle is 14 less than its width. To leavewrepresent the width of the rectangle. Write an expression for the length of the rectangle.

Solution

$\begin{matrix} Write a sentence about the length of the rectangle. & 14 less than the width \\ Ersatz w for "the width". & w \\ Write again less than if withdrawn from . & 14 subtracted from w \\ Translate the sentence into algebra. & w - 14 \end{matrix}$

try it 1.19

The length of a rectangle is 7 less than its width. To leavewrepresent the width of the rectangle. Write an expression for the length of the rectangle.

try it 1,20

The width of a rectangle is 6 less than its length. To leavejorepresent the length of the rectangle. Write an expression for the width of the rectangle.

The expressions in the example below are used in the typical coin shuffling problems we will see later.

Example 1.11

June has tens and quarters in her bag. The number of dimes is seven less than four times the number of quarters. To leaveQrepresent the number of dimes Write an expression for the number of dimes.

Solution

$\begin{matrix} Write a sentence about the number of cents. & seven less than four times as many rooms \\ Ersatz Q by the number of quarters. & 7 less than 4 times Q \\ translate 4 times Q . & 7 less than 4 Q \\ Translate the sentence into algebra. & 4 Q - 7 \end{matrix}$

try it 1.21

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

try it 1.22

Lauren has dimes and nickels in her pocket. The number of dimes is three times more than seven times the number of nickels. To leavenorterepresent the number of nickels Write an expression for the number of dimes.

Section 1.1 Exercises

Practice creates masters

Identify multiples and factors

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

(Video) 1.1 ALGEBRA - Expressions and Variables

896

942

22.335

39.075

Find prime factors and least common multiples

Find the prime factorization in the following exercises.

455

10.

400

11.

432

12.

627

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

13.

8, 12

14.

12, 16

15.

28, 40

sixteen.

84, 90

17.

55, 88

18.

60, 72

Simplify expressions using order of operations

Simplify each expression in the following exercises.

19.

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

20.

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

21.

$2 + 8 (6 + 1)$

22.

$4 + 6 (3 + 6)$

23.

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

24.

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

25.

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

26.

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

27.

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

28.

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

29.

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

30.

(Video) Intermediate Algebra: 1.1: Simplifying Radicals

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

evaluate an expression

In the following exercises, evaluate the following expressions.

31.

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

32.

If $X = 3,$
ⓐ $X^{5}$
ⓑ $5^{X}$
ⓒ $3 X^{2} - 4 X - 8$

33.

If $X = 4, J = 1$
$X^{2} + 3 X J - 7 J^{2}$

34.

If $X = 3, J = 2$
$6 X^{2} + 3 X J - 9 J^{2}$

35.

If $X = 10, J = 7$
${(X - J)}^{2}$

36.

If $A = 3, B = 8$
$A^{2} + B^{2}$

Simplify expressions by combining similar terms

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

37.

$7 X + 2 + 3 X + 4$

38.

$8 J + 5 + 2 J - 4$

39.

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

40.

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

41.

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

42.

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

Translate an English sentence into an algebraic expression

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

43.

ⓐThe difference of $5 X^{2}$ J $6 X J$
ⓑthe quotient of $6 J^{2}$ J $5 X$
ⓒtwenty-one more than $J^{2}$
ⓓ $6 X$ less than $81 X^{2}$

44.

ⓐThe difference of $17 X^{2}$ J $5 X J$
ⓑthe quotient of $8 J^{3}$ J $3 X$
ⓒeighteen more than $A^{2}$ ;
ⓓ $11 B$ less than $100 B^{2}$

45.

ⓐthe sum of $4 A B^{2}$ J $3 A^{2} B$
ⓑthe product of $4 J^{2}$ J $5 X$
ⓒfifteen more than $Metro$
ⓓ $9 X$ less than $121 X^{2}$

46.

ⓐthe sum of $3 X^{2} J$ J $7 X J^{2}$
ⓑthe product of $6 X J^{2}$ J $4 z$
ⓒtwelve more than $3 X^{2}$
ⓓ $7 X^{2}$ less than $63 X^{3}$

47.

ⓐeight times the difference of $J$ and nine
ⓑthe eightfold difference $J$ year 9

48.

ⓐseven times the difference of $J$ and a
ⓑthe difference of seven times $J$ year 1

49.

ⓐfive times the sum of $3 X$ J $J$
ⓑthe sum of five times $3 X$ J $J$

50.

ⓐeleven times the sum of $4 X^{2}$ J $5 X$
ⓑthe sum of eleven times $4 X^{2}$ J $5 X$

51.

Eric has rock and country songs on his playlist. At 14, the number of rock songs is more than twice that of country songs. To leaveCrepresent the number of country songs. Write an expression for the number of rock songs.

52.

At the age of 8, the number of women taking a statistics course is more than twice as high as that of men. To leave $Metro$ represent the number of males Write an expression for the number of females.

53.

Greg has nickels and pennies in his pocket. The number of pennies is seven less than three the number of nickels. To leavenorterepresent the number of nickels Write an expression for the number of cents.

54.

Jeannette Tiene $PS 5$ J $PS 10$ bills in your wallet. The number of fives is three more than six times the number of tens. To leave $T$ represent the number of tens Write an expression for the number of fives.

writing exercises

55.

Explain in your own words how to find the prime factorization of a composite number.

56.

Why is it important to use order of operations to simplify an expression?

57.

Explain how to identify the similar terms in the expression $8 A^{2} + 4 A + 9 - A^{2} - 1 .$

58.

Explain the difference between the sentences "4 times the sum ofXJJ' and 'the sum of 4 timesXJJ“.

Autocheque

ⓐUse this checklist to assess whether you have mastered the objectives of this section.

1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (53)

ⓑIf most of your checks:

...with trust.Congratulations! You have completed the objectives of this section. Think about the study techniques you've been using so you can keep using them. What have you done to make sure you can do these things? Be specific.

...with a little help.This needs to be addressed quickly, because problems you don't master will become holes in your path to success. In mathematics, each topic builds on previous work. It's important to make sure you have a solid foundation before moving forward. Who can you ask for help? Your classmates and teacher are good resources. Are there math teachers anywhere on campus? Can your study skills be improved?

…No, I do not understand!This is a warning signal and you should not ignore it. You must get help immediately or you will quickly become overwhelmed. See your teacher as soon as possible to discuss your situation. Together you can make a plan to get the help you need.

(Video) Algebra 1: 1.1 Variables and Expressions

FAQs

Is algebra 1 intermediate algebra? ›

Algebra I is equivalent to Elementary Algebra. Algebra II is equivalent to Intermediate Algebra. College Algebra follows both of the above.

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What is intermediate algebra? ›

Intermediate Algebra is a branch of mathematics that substitutes letters for numbers and uses simplification techniques to solve equations. Algebraic equations: A scale, what is done on one side of the scale with a number is also done to the other side of the scale.

Is intermediate algebra before geometry? ›

Geometry is typically taken before algebra 2 and after algebra 1. Whether or not a student can take algebra 2 before Geometry depends on each student's school policies. However, I would recommend taking the traditional order of math classes.

What do you need to know for intermediate algebra? ›

Table of Contents

Review of Real Numbers and Absolute Value.
Operations with Real Numbers.
Square and Cube Roots of Real Numbers.
Algebraic Expressions and Formulas.
Rules of Exponents and Scientific Notation.
Polynomials and Their Operations.
Solving Linear Equations.
Solving Linear Inequalities with One Variable.

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Does it matter what letters are used in algebra? ›

You can use any letter you like, although and are commonly used to represent the unknown elements of equations. A letter used to substitute for a number in algebra is called a variable, because it stands for different numbers each time you use it.

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What is math with letters called? ›

A basic distinction between algebra and arithmetic is the use of symbols. In algebra, symbols (usually letters) are used to represent numbers. To solve math problems, you should know what variables and constants are.

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Where is algebra used in real life? ›

utilizing linear algebra, and this uniqueness starts to expose a lot of applications. Other real-world applications of linear algebra include ranking in search engines, decision tree induction, testing software code in software engineering, graphics, facial recognition, prediction and so on.

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Is intermediate algebra harder than geometry? ›

Geometry has less math in it than algebra, and the math that is required is less complicated. However, Geometry also requires you to memorize a lot of rules and formulas, which can be more difficult than basic algebra for some people. If you need help in a math class, you should ask your teacher.

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Is intermediate algebra more advanced than college algebra? ›

"College Algebra" is more advanced than Intermediate Algebra. Find and study from an Intermediate Algebra book. You could, if you are strong at Algebra, study College Algebra, since it contains almost everything that is in Intermediate. Most sure learning would be to first study Intermediate Algebra.

Which is harder algebra or calculus? ›

Calculus is the hardest mathematics subject and only a small percentage of students reach Calculus in high school or anywhere else. Linear algebra is a part of abstract algebra in vector space. However, it is more concrete with matrices, hence less abstract and easier to understand.

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What is the lowest level of math in college? ›

What is college-level math? Entry-level math in college is considered the stepping stone to more advanced math. Algebra 1, trigonometry, geometry, and calculus 1 are the basic math classes. Once you have successfully navigated through these courses, you can trail blazed through more advanced courses.

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Can you go from algebra 1 to algebra 2? ›

Yes, depending on the student. Taking algebra 2 directly after algebra 1, and then taking geometry is just fine for some students, but I do not recommend this if the student struggles with math (or is somewhat math resistant). It can be a tougher road to go.

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Is algebra 2 high school algebra? ›

Algebra 2 is the third math course in high school and will guide you through among other things linear equations, inequalities, graphs, matrices, polynomials and radical expressions, quadratic equations, functions, exponential and logarithmic expressions, sequences and series, probability and trigonometry.

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Is intermediate algebra lower than algebra 2? ›

Algebra II is equivalent to Intermediate Algebra. College Algebra follows both of the above.

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What is 12th grade math called? ›

By 12th grade, most students will have completed Algebra I, Algebra II, and Geometry, so high school seniors may want to focus on a higher level mathematics course such as Precalculus or Trigonometry. Students taking an advanced mathematics course will learn concepts like: Graphing exponential and logarithmic functions.

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Is college algebra algebra 1 or algebra 2? ›

In fact, the standard CA course in American colleges and universities is identical to high school Algebra II. Many students will have completed that course by the end of their junior year in high school.

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What is easier college algebra or intermediate algebra? ›

Intermediate algebra is a U.S. college course for which you usually do not earn college credit. It is an easier more basic treatment of algebra, without a lot of the material considered as college algebra.

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Is intermediate algebra harder than statistics? ›

Statistics requires a lot more memorization and a deeper level of analysis/inference skills while algebra requires little memorization and very little analysis outside of algebraic applications.

Is Algebra 1 a freshman math? ›

What grade is Algebra 1? Algebra 1 is typically taught late in middle school or early in high school. In the United States, 9th grade (freshman year) seems to be the most common grade for students to take an Algebra 1 class. Some high schools also offer Algebra 1 to 10th graders.

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What is the highest level of math in college? ›

A doctoral degree is the highest level of education available in mathematics, often taking 4-7 years to complete. Like a master's degree, these programs offer specializations in many areas, including computer algebra, mathematical theory analysis, and differential geometry.

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What is the hardest math in college? ›

Advanced Calculus is the hardest math subject, according to college professors. One of the main reasons students struggle to understand the concepts in Advanced Calculus is because they do not have a good mathematical foundation. Calculus builds on the algebraic concepts learned in previous classes.

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Is algebra 1 hard for a 7th grader? ›

In math, concepts begin to jump from concrete to more abstract, making 7th grade algebra a challenging course for many students. It's important that you set your child up for success when they move up to 8th grade.

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What grade do you need to pass algebra 1? ›

Students who begin Algebra 1 before 9th grade must earn a minimum score 80% on the Algebra 1 (High School) Readiness Test, and earn a minimum score of 80% on the Geometry Readiness test (or other CWCS-approved Algebra proficiency exam) upon completion of Algebra 1.

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Should I take algebra 1 in 8th grade? ›

Spielhagen found that those who took Algebra 1 in eighth grade did as well as similar students who took it in ninth grade. Those taking the course in eighth grade “stayed in the mathematics pipeline longer and attended college at greater rates” than similar students who took it in ninth grade, Spielhagen said.

What grade do you learn algebra basics? ›

Typically, algebra is taught to strong math students in 8th grade and to mainstream math students in 9th grade. In fact, some students are ready for algebra earlier.

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How hard is college algebra? ›

College Algebra is not difficult if you've taken Pre-Algebra and Algebra in the past and done well. However, if you haven't done well, or it's a been a while since you've taken Pre-Algebra and Algebra, College Algebra will be difficult.

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What is the most common letter in algebra? ›

The letters x, y, z, a, b, c, m, and n are probably the most commonly used variables. The letters e and i have special values in algebra and are usually not used as variables. The letter o is usually not used because it can be mistaken for 0 (zero).

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What does m mean in algebra? ›

In algebra, the letter "m" refers to the slope of a line. The slope of a line determines both its steepness and direction. The greater the magnitude of the slope, the steeper it is. A positive slope will result in an upward trending line, and a negative slope will result in a downward trending line.

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What does C mean in math? ›

The Latin small letter c is used in math to represent a variable or coefficient.

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What is math dyslexia called? ›

Dyscalculia is a learning disorder that affects a person's ability to do math. Much like dyslexia disrupts areas of the brain related to reading, dyscalculia affects brain areas that handle math- and number-related skills and understanding.

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What does Y mean in algebra? ›

The letter y is commonly used as a variable in math. It is probably one of the first variables you will come across. It is usually used when referring to equations that you graph. It also is used to refer to functions.

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What does J stand for in math? ›

Similarly, the imaginary number i (sometimes written as j) is just a mathematical tool to represent the square root of –1, which has no other method of description.

What are the 4 types of algebra? ›

You will probably use the concept of algebra without realising it. Algebra is divided into different sub-branches such as elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra.

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Who is father of algebra? ›

Muhammad ibn Musa al-Khwarizmi was a 9th-century Muslim mathematician and astronomer. He is known as the “father of algebra”, a word derived from the title of his book, Kitab al-Jabr. His pioneering work offered practical answers for land distribution, rules on inheritance and distributing salaries.

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What is the main use of algebra? ›

algebra, Generalized version of arithmetic that uses variables to stand for unspecified numbers. Its purpose is to solve algebraic equations or systems of equations.

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What type of algebra is algebra 1? ›

Students' work with numbers and operations throughout elementary and middle school has led them to an understanding of the structure of the number system; in Algebra I, students explore the structure of algebraic expressions and polynomials.

What level class is algebra 1? ›

Some schools may offer Algebra I in either 9th/10th grade OR 11th/12th grade, but not both. Nonetheless, it is important that students have access to Algebra I sometime in their high school career.

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What is algebra 1 considered? ›

What is Algebra 1? Algebra 1 is a high school math course exploring how to use letters (called variables) and numbers with mathematical symbols to solve problems. Algebra 1 typically includes evaluating expressions, writing equations, graphing functions, solving quadratics, and understanding inequalities.

Is college intermediate algebra equivalent to algebra 2? ›

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Is 8th grade algebra algebra 1? ›

Grade 8 Algebra is a high school level Algebra 1 course, and is the first course on their growth in upper level mathematics. The fundamental purpose of this course is to formalize and extend the mathematics that students learned through mastery of the middle school standards.

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What is algebra 1 for 9th grade? ›

Algebra 1 formalizes and extends students' understanding and application of functions. Students primarily explore linear functions (as well as linear piecewise, absolute value, and step functions), quadratic functions, and exponential functions.

Read On ›

Is algebra 1 a high school level math? ›

Algebra 1 is the first high school credit math course that students take. After Algebra 1, students have options about what math course(s) they choose next.

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What grade is a passing grade in algebra 1? ›

What is algebra 1 called in college? ›

An introductory college algebra course, often referred to as "Algebra 1" or "College Algebra," is a requirement for many academic programs. Some college algebra courses list the intended audience, such as math, engineering or business students who need the class to further their academic goals.

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Is algebra 1 a pre-algebra? ›

Pre-algebra helps students to have the basic command of algebra topics. Algebra increases the complexity and understanding of the topics learned in pre-algebra. Pre-algebra is essential to understand algebra 1 and algebra 2. Algebra is a major branch that includes topics of pre-algebra, algebra 1, and algebra 2.

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Do colleges look at algebra 1? ›

Most colleges want students to have at least 3 years of high school math, though more selective colleges prefer 4 years. Prioritize taking several of the following courses: Algebra 1.

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Is college algebra 1 hard? ›

Is College Algebra difficult? College Algebra is not difficult if you've taken Pre-Algebra and Algebra in the past and done well. However, if you haven't done well, or it's a been a while since you've taken Pre-Algebra and Algebra, College Algebra will be difficult.

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1.1 Using the Language of Algebra - Intermediate Algebra | OpenStax (2023)

Find factors, prime factorization, and least common multiples

Solution

How to find the prime factorization of a composite number

Solution

Find the prime factorization of a composite number.

How to find the least common multiple using the prime factor method

Solution

Find the least common multiple using the prime factor method.

Use of variables and algebraic symbols

Simplify expressions using order of operations

Use the order of operations.

Solution

Solution

evaluate an expression

Solution

Identify and combine similar terms

Combine similar terms

Solution

Combine similar terms.

Translate an English sentence into an algebraic expression

Solution

Solution

Solution

Solution

Section 1.1 Exercises

Practice creates masters

writing exercises

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FAQs

Is algebra 1 intermediate algebra? ›

Is intermediate algebra before geometry? ›

Is algebra 2 high school algebra? ›

What grade do you need to pass algebra 1? ›

What are the 4 types of algebra? ›

What grade is a passing grade in algebra 1? ›

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