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Intermediate Algebra

1.1 Use the Language of Algebra

Intermediate Algebra1.1 Use the Language of Algebra
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Find factors, prime factorizations, and least common multiples
  • Use variables and algebraic symbols
  • Simplify expressions using the order of operations
  • Evaluate an expression
  • Identify and combine like terms
  • Translate an English phrase to an algebraic expression
Be Prepared 1.1

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 chapter, Foundations.

Find Factors, Prime Factorizations, and Least Common Multiples

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

Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.

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

Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.

We could find the multiples of any number by continuing this process.

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

Multiple of a Number

A number is a multiple of nn 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 3 into 15, we get a counting number. In fact, 15÷315÷3 is 5, so 15 is 5·3.5·3.

Divisible by a Number

If a number mm is a multiple of n, then m is divisible by n.

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

Divisibility Tests

A number is divisible by:

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

   3 if the sum of the digits is divisible by 3.3.

   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.

Example 1.1

Is 5,625 divisible by 2? 3? 5 or 10? 6?

Try It 1.1

Is 4,962 divisible by 2? 3? 5? 6? 10?

Try It 1.2

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.

8 times 9 is 72. 8 and 9 are factors. 72 is the product.

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.

Factors

If a·b=m,a·b=m, then a and b are factors of m.

Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.

Prime number and Composite number

A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.

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

The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!

This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.

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

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

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.

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

Example 1.2

How to Find the Prime Factorization of a Composite Number

Factor 48.

Try It 1.3

Find the prime factorization of 80.80.

Try It 1.4

Find the prime factorization of 60.60.

How To

Find the prime factorization of a composite number.

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

One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.

Example 1.3

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

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.

Try It 1.5

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

Try It 1.6

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

How To

Find the least common multiple using the Prime Factors Method.

  1. Step 1. Write each number as a product of primes.
  2. Step 2. List the primes of each number. Match primes vertically when possible.
  3. Step 3. Bring down the columns.
  4. Step 4. Multiply the factors.

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.x,y,a,b,c.

Variable

A variable is a letter that represents a number whose value may change.

A number whose value always remains the same is called a constant.

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 Symbols

Operation Notation Say: The result is…
Addition a+ba+b aa plus bb the sum of aa and bb
Subtraction abab aa minus bb the difference of aa and bb
Multiplication a·b,ab,(a)(b),a·b,ab,(a)(b), (a)b,a(b)(a)b,a(b) aa times bb the product of aa and bb
Division a÷b,a/b,ab,baa÷b,a/b,ab,ba aa divided by bb the quotient of aa and b;b;
aa is called the dividend, and bb is called the divisor

When two quantities have the same value, we say they are equal and connect them with an equal sign.

Equality Symbol

a=ba=b is read “a is equal to b.”

The symbol “=” is called the equal sign.

On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>”.

Inequality

For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.

The expressions a<ba<b or a>ba>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<bis equivalent tob>a.For example,7<11is equivalent to11>7.a>bis equivalent tob<a.For example,17>4is equivalent to4<17.a<bis equivalent tob>a.For example,7<11is equivalent to11>7.a>bis equivalent tob<a.For example,17>4is equivalent to4<17.

Inequality Symbols

Inequality Symbols Words
abab a is not equal to b.
a<ba<b a is less than b.
abab a is less than or equal to b.
a>ba>b a is greater than b.
abab a is greater than or equal to b.

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.

Grouping Symbols

Parentheses()Brackets[]Braces{}Parentheses()Brackets[]Braces{}

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

8(148)213[2+4(98)]24÷{132[1(65)+4]}8(148)213[2+4(98)]24÷{132[1(65)+4]}

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations.

Expression

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

ExpressionWordsEnglish Phrase3+53 plus 5the sum of three and fiven1nminus onethe difference ofnand one6·76 times 7the product of six and sevenxyxdivided byythe quotient ofxandyExpressionWordsEnglish Phrase3+53 plus 5the sum of three and fiven1nminus onethe difference ofnand one6·76 times 7the product of six and sevenxyxdivided byythe quotient ofxandy

Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.

An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.

Equation

An equation is two expressions connected by an equal sign.

EquationEnglish Sentence3+5=8The sum of three and five is equal to eight.n1=14nminus one equals fourteen.6·7=42The product of six and seven is equal to forty-two.x=53xis equal to fifty-three.y+9=2y3yplus nine is equal to twoyminus three.EquationEnglish Sentence3+5=8The sum of three and five is equal to eight.n1=14nminus one equals fourteen.6·7=42The product of six and seven is equal to forty-two.x=53xis equal to fifty-three.y+9=2y3yplus nine is equal to twoyminus 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 2·2·22·2·2 as 2323 and 2·2·2·2·2·2·2·2·22·2·2·2·2·2·2·2·2 as 29.29. In expressions such as 23,23, 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.

The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.

Exponential Notation

We say 2323 is in exponential notation and 2·2·22·2·2 is in expanded notation.

anan means multiply a by itself, n times.

The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.

The expression anan is read a to the nthnth power.

While we read anan as aa to the nthnth power”, we usually read:

a2asquared”a3acubed”a2asquared”a3acubed”

We’ll see later why a2a2 and a3a3 have special names.

Table 1.1 shows how we read some expressions with exponents.

Expression In Words
72 7 to the second power or 7 squared
53 5 to the third power or 5 cubed
94 9 to the fourth power
125 12 to the fifth power
Table 1.1

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify 4·2+14·2+1 we would first multiply 4·24·2 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+18+194·2+18+19

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

Simplify an Expression

To simplify an expression, do all operations in the expression.

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression 4+3·7.4+3·7. Some students simplify this getting 49, by adding 4+34+3 and then multiplying that result by 7. Others get 25, by multiplying 3·73·7 first and then adding 4.

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

How To

Use the order of operations.

  1. Step 1. Parentheses and Other Grouping Symbols
    • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  2. Step 2. Exponents
    • Simplify all expressions with exponents.
  3. Step 3. Multiplication and Division
    • Perform all multiplication and division in order from left to right. These operations have equal priority.
  4. Step 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”.

ParenthesesPleaseExponentsExcuseMultiplicationDivisionMyDearAdditionSubtractionAuntSallyParenthesesPleaseExponentsExcuseMultiplicationDivisionMyDearAdditionSubtractionAuntSally

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.

Example 1.4

Simplify: 18÷6+4(52).18÷6+4(52).

Try It 1.7

Simplify: 30÷5+10(32).30÷5+10(32).

Try It 1.8

Simplify: 70÷10+4(62).70÷10+4(62).

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

Example 1.5

Simplify: 5+23+3[63(42)].5+23+3[63(42)].

Try It 1.9

Simplify: 9+53[4(9+3)].9+53[4(9+3)].

Try It 1.10

Simplify: 722[4(5+1)].722[4(5+1)].

Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

Evaluate an Expression

To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

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

Example 1.6

Evaluate when x=4:x=4: x2x2 3x3x 2x2+3x+8.2x2+3x+8.

Try It 1.11

Evaluate when x=3,x=3, x2x2 4x4x 3x2+4x+1.3x2+4x+1.

Try It 1.12

Evaluate when x=6,x=6, x3x3 2x2x 6x24x7.6x24x7.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

Term

A term is a constant or the product of a constant and one or more variables.

Examples of terms are 7,y,5x2,9a,7,y,5x2,9a, and b5.b5.

The constant that multiplies the variable is called the coefficient.

Coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

Think of the coefficient as the number in front of the variable. The coefficient of the term 3x3x is 3. When we write x,x, the coefficient is 1, since x=1·x.x=1·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?

5x7n243x9n25x7n243x9n2

We say,

77 and 44 are like terms.

5x5x and 3x3x are like terms.

n2n2 and 9n29n2 are like terms.

Like Terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

Simplify.4x+7x+xAdd the coefficients.12xSimplify.4x+7x+xAdd the coefficients.12x

Example 1.7

How To Combine Like Terms

Simplify: 2x2+3x+7+x2+4x+5.2x2+3x+7+x2+4x+5.

Try It 1.13

Simplify: 3x2+7x+9+7x2+9x+8.3x2+7x+9+7x2+9x+8.

Try It 1.14

Simplify: 4y2+5y+2+8y2+4y+5.4y2+5y+2+8y2+4y+5.

How To

Combine like terms.

  1. Step 1. Identify like terms.
  2. Step 2. Rearrange the expression so like terms are together.
  3. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table 1.2 summarizes them.

Operation Phrase Expression
Addition a plus b
the sum of aa and b
a increased by b
b more than a
the total of a and b
b added to a
a+ba+b
Subtraction a minus bb
the difference of a and b
a decreased by b
b less than a
b subtracted from a
abab
Multiplication a times b
the product of aa and bb
twice a
a·b,ab,a(b),(a)(b)a·b,ab,a(b),(a)(b)

2a2a
Division a divided by b
the quotient of a and b
the ratio of a and b
b divided into a
a÷b,a/b,ab,baa÷b,a/b,ab,ba
Table 1.2

Look closely at these phrases using the four operations:

The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.

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

Example 1.8

Translate each English phrase into an algebraic expression:

the difference of 14x14x and 9 the quotient of 8y28y2 and 3 twelve more than yy seven less than 49x249x2

Try It 1.15

Translate the English phrase into an algebraic expression:

the difference of 14x214x2 and 13 the quotient of 12x12x and 2 13 more than zz
18 less than 8x8x

Try It 1.16

Translate the English phrase into an algebraic expression:

the sum of 17y217y2 and 19 the product of 77 and y Eleven more than x Fourteen less than 11a

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

Example 1.9

Translate the English phrase into an algebraic expression:

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

Try It 1.17

Translate the English phrase into an algebraic expression:

four times the sum of p and q
the sum of four times p and q

Try It 1.18

Translate the English phrase into an algebraic expression:

the difference of two times x and 8
two times the difference of x and 8

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.

Example 1.10

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

Try It 1.19

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.

Try It 1.20

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.

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

Example 1.11

June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

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. Let q represent the number of quarters. Write an expression for the number of dimes.

Try It 1.22

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.

Section 1.1 Exercises

Practice Makes Perfect

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.

1.

84

2.

96

3.

896

4.

942

5.

22,335

6.

39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

7.

86

8.

78

9.

455

10.

400

11.

432

12.

627

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

13.

8, 12

14.

12, 16

15.

28, 40

16.

84, 90

17.

55, 88

18.

60, 72

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

19.

2312÷(95)2312÷(95)

20.

3218÷(115)3218÷(115)

21.

2+8(6+1)2+8(6+1)

22.

4+6(3+6)4+6(3+6)

23.

20÷4+6(51)20÷4+6(51)

24.

33÷3+4(72)33÷3+4(72)

25.

3(1+9·6)423(1+9·6)42

26.

5(2+8·4)725(2+8·4)72

27.

2[1+3(102)]2[1+3(102)]

28.

5[2+4(32)]5[2+4(32)]

29.

8+2[72(53)]328+2[72(53)]32

30.

10+3[62(42)]2410+3[62(42)]24

Evaluate an Expression

In the following exercises, evaluate the following expressions.

31.

When x=2,x=2,
x6x6
4x4x
2x2+3x72x2+3x7

32.

When x=3,x=3,
x5x5
5x5x
3x24x83x24x8

33.

When x=4,y=1x=4,y=1
x2+3xy7y2x2+3xy7y2

34.

When x=3,y=2x=3,y=2
6x2+3xy9y26x2+3xy9y2

35.

When x=10,y=7x=10,y=7
(xy)2(xy)2

36.

When a=3,b=8a=3,b=8
a2+b2a2+b2

Simplify Expressions by Combining Like Terms

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

37.

7x+2+3x+47x+2+3x+4

38.

8y+5+2y48y+5+2y4

39.

10a+7+5a2+7a410a+7+5a2+7a4

40.

7c+4+6c3+9c17c+4+6c3+9c1

41.

3x2+12x+11+14x2+8x+53x2+12x+11+14x2+8x+5

42.

5b2+9b+10+2b2+3b45b2+9b+10+2b2+3b4

Translate an English Phrase to an Algebraic Expression

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

43.


the difference of 5x25x2 and 6xy6xy
the quotient of 6y26y2 and 5x5x
Twenty-one more than y2y2
6x6x less than 81x281x2

44.


the difference of 17x217x2 and 5xy5xy
the quotient of 8y38y3 and 3x3x
Eighteen more than a2a2;
11b11b less than 100b2100b2

45.


the sum of 4ab24ab2 and 3a2b3a2b
the product of 4y24y2 and 5x5x
Fifteen more than mm
9x9x less than 121x2121x2

46.


the sum of 3x2y3x2y and 7xy27xy2
the product of 6xy26xy2 and 4z4z
Twelve more than 3x23x2
7x27x2 less than 63x363x3

47.


eight times the difference of yy and nine
the difference of eight times yy and 9

48.


seven times the difference of yy and one
the difference of seven times yy and 1

49.


five times the sum of 3x3x and yy
the sum of five times 3x3x and yy

50.


eleven times the sum of 4x24x2 and 5x5x
the sum of eleven times 4x24x2 and 5x5x

51.

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.

52.

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

53.

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

54.

Jeannette has $5$5 and $10$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let tt 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 the order of operations to simplify an expression?

57.

Explain how you identify the like terms in the expression 8a2+4a+9a21.8a2+4a+9a21.

58.

Explain the difference between the phrases “4 times the sum of x and y” and “the sum of 4 times x and y”.

Self Check

Use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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