This chapter is intended to be a brief review of concepts that will be needed in an Intermediate Algebra course. A more thorough introduction to the topics covered in this chapter can be found in the Elementary Algebra 2e chapter, Foundations.
In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are a, b, c, m, n, x, and y. Further discussion of constants and variables appears later in this section.
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 |
A number is a multiple of n n if it is the product of a counting number and n . n .
Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact, 15 ÷ 3 15 ÷ 3 is 5, so 15 is 5 · 3 . 5 · 3 .
If a number m m is a multiple of n, then m is divisible by n.
If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:
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 . 3 .
5 if the last digit is 5 or 0 . 0 .
6 if it is divisible by both 2 and 3 . 3 .
10 if it ends with 0 . 0 .
Is 5,625 divisible by ⓐ 2? ⓑ 3? ⓒ 5 or 10? ⓓ 6?
| 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 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?
Is 3,765 divisible by ⓐ 2? ⓑ 3? ⓒ 5? ⓓ 6? ⓔ 10?
In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.
Since 8 · 9 = 72 , 8 · 9 = 72 , we say that 8 and 9 are factors of 72. When we write 72 = 8 · 9 , 72 = 8 · 9 , we say we have factored 72.
Other ways to factor 72 are 1 · 72 , 2 · 36 , 3 · 24 , 4 · 18 , 1 · 72 , 2 · 36 , 3 · 24 , 4 · 18 , and 6 · 12 . 6 · 12 . The number 72 has many factors: 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 18 , 24 , 36 , 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 18 , 24 , 36 , and 72 . 72 .
In the expression a · b a · b , both a and b are called factors . If a · b = m , a · b = m , and both a and b are integers, then a and b are factors of m.
Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.
A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
A composite number is a counting number greater than 1 that is not prime. A composite number has factors other than 1 and the number itself.
The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.
A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.
The prime factorization of a number is the product of prime numbers that equals the number. These prime numbers are called the prime factors.
To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.
If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.
We say 2 · 2 · 2 · 2 · 3 2 · 2 · 2 · 2 · 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 · 8 , 6 · 8 , the result would still be the same. Finish the prime factorization and verify this for yourself.
Find the prime factorization of 80 . 80 .
Find the prime factorization of 60 . 60 .
One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.
Find the least common multiple (LCM) of 12 and 18 using the prime factors method.
Notice that the prime factors of 12 ( 2 · 2 · 3 ) ( 2 · 2 · 3 ) and the prime factors of 18 ( 2 · 3 · 3 ) ( 2 · 3 · 3 ) are included in the LCM ( 2 · 2 · 3 · 3 ) . ( 2 · 2 · 3 · 3 ) . So 36 is the least common multiple of 12 and 18.
By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.
Find the LCM of 9 and 12 using the Prime Factors Method.
Find the LCM of 18 and 24 using the Prime Factors Method.
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 . x , y , a , b , c .
A variable is a letter that represents a number whose value may change.
A number whose value always remains the same is called a constant .
A constant is a number whose value always stays the same.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | a + b a + b | a a plus b b | the sum of a a and b b |
| Subtraction | a − b a − b | a a minus b b | the difference of a a and b b |
| Multiplication | a · b , a b , ( a ) ( b ) , a · b , a b , ( a ) ( b ) , ( a ) b , a ( b ) ( a ) b , a ( b ) | a a times b b | the product of a a and b b |
| Division | a ÷ b , a / b , a b , b a a ÷ b , a / b , a b , b a | a a divided by b b | the quotient of a a and b ; b ; a a is called the dividend, and b b is called the divisor |
When two quantities have the same value, we say they are equal and connect them with an equal sign.
a = b a = b is read “a is equal to b.”
The symbol “=” is called the equal sign.
On the number line , the numbers get larger as they go from left to right. The number line can be used to explain the symbols “”.
The expressions a < b a < b or a >b a > b can be read from left to right or right to left, though in English we usually read from left to right. In general,
a < b is equivalent to b >a . For example, 7 < 11 is equivalent to 11 >7 . a > b is equivalent to b < a . For example, 17 >4 is equivalent to 4 < 17 . a < b is equivalent to b >a . For example, 7 < 11 is equivalent to 11 >7 . a > b is equivalent to b < a . For example, 17 >4 is equivalent to 4 < 17 .
| Inequality Symbols | Words |
|---|---|
| a ≠ b a ≠ b | a is not equal to b. |
| a < b a < b | a is less than b. |
| a ≤ b a ≤ b | a is less than or equal to b. |
| a > b a > b | a is greater than b. |
| a ≥ b a ≥ b | a is greater than or equal to b. |
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression , which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
8 ( 14 − 8 ) 21 − 3 [ 2 + 4 ( 9 − 8 ) ] 24 ÷ < 13 − 2 [ 1 ( 6 − 5 ) + 4 ] >8 ( 14 − 8 ) 21 − 3 [ 2 + 4 ( 9 − 8 ) ] 24 ÷ < 13 − 2 [ 1 ( 6 − 5 ) + 4 ] >What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations.
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
Expression Words English Phrase 3 + 5 3 plus 5 the sum of three and five n − 1 n minus one the difference of n and one 6 · 7 6 times 7 the product of six and seven x y x divided by y the quotient of x and y Expression Words English Phrase 3 + 5 3 plus 5 the sum of three and five n − 1 n minus one the difference of n and one 6 · 7 6 times 7 the product of six and seven x y x divided by y the quotient of x and y
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.
An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
An equation is two expressions connected by an equal sign.
Equation English Sentence 3 + 5 = 8 The sum of three and five is equal to eight. n − 1 = 14 n minus one equals fourteen. 6 · 7 = 42 The product of six and seven is equal to forty-two. x = 53 x is equal to fifty-three. y + 9 = 2 y − 3 y plus nine is equal to two y minus three. Equation English Sentence 3 + 5 = 8 The sum of three and five is equal to eight. n − 1 = 14 n minus one equals fourteen. 6 · 7 = 42 The product of six and seven is equal to forty-two. x = 53 x is equal to fifty-three. y + 9 = 2 y − 3 y plus nine is equal to two y minus three.
Suppose we need to multiply 2 nine times. We could write this as 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 . 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 . This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write 2 · 2 · 2 2 · 2 · 2 as 2 3 2 3 and 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 as 2 9 . 2 9 . In expressions such as 2 3 , 2 3 , the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base .
We say 2 3 2 3 is in exponential notation and 2 · 2 · 2 2 · 2 · 2 is in expanded notation.
a n a n means multiply a by itself, n times.
The expression a n a n is read a to the n t h n t h power.
While we read a n a n as “ a “ a to the n t h n t h power”, we usually read:
a 2 “ a squared” a 3 “ a cubed” a 2 “ a squared” a 3 “ a cubed”We’ll see later why a 2 a 2 and a 3 a 3 have special names.
Table 1.1 shows how we read some expressions with exponents.
| Expression | In Words | |
|---|---|---|
| 7 2 | 7 to the second power or | 7 squared |
| 5 3 | 5 to the third power or | 5 cubed |
| 9 4 | 9 to the fourth power | |
| 12 5 | 12 to the fifth power |
To simplify an expression means to do all the math possible. For example, to simplify 4 · 2 + 1 4 · 2 + 1 we would first multiply 4 · 2 4 · 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 · 2 + 1 8 + 1 9 4 · 2 + 1 8 + 1 9By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
To simplify an expression, do all operations in the expression.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations . Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression 4 + 3 · 7 . 4 + 3 · 7 . Some students simplify this getting 49, by adding 4 + 3 4 + 3 and then multiplying that result by 7. Others get 25, by multiplying 3 · 7 3 · 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”.
P arentheses P lease E xponents E xcuse M ultiplication D ivision M y D ear A ddition S ubtraction A unt S ally P arentheses P lease E xponents E xcuse M ultiplication D ivision M y D ear A ddition S ubtraction A unt S ally
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.
Simplify: 18 ÷ 6 + 4 ( 5 − 2 ) . 18 ÷ 6 + 4 ( 5 − 2 ) .