Section 1.2 Fractions

Objectives Section 1.2 Fractions Factor and prime factor natural numbers Recognize special fraction forms Multiply and divide fractions Build equivalent fractions Simplify fractions Add and subtract fractions Simplify answers Compute with mixed numbers Objective 1: Factor and Prime Factor Natural Numbers To compute with fractions, we need to know how to factor natural numbers. To factor a number means to express it as a product of two or more numbers. For example, some ways to factor 8 are 1 8, 4 2, and 2 2 2. The numbers 1, 2, 4, and 8 that were used to write the products are called factors of 8. In general, a factor is a number being multiplied. Sometimes a number has only two factors, itself and 1. We call such numbers prime numbers. Objective 1: Factor and Prime Factor Natural Numbers A prime number is a natural number greater than 1 that has only itself and 1 as factors. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. A composite number is a natural number, greater than 1, that is not prime. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18. Every composite number can be factored into the product of two or more prime numbers. This product of these prime numbers is called its prime factorization. 1

EXAMPLE 1 Find the prime factorization of 210. EXAMPLE 1 Find the prime factorization of 210. Strategy We will use a series of steps to express 210 as a product of only prime numbers. First, write 210 as the product of two natural numbers other than 1. Why To prime factor a number means to write it as a product of prime numbers. Neither 10 nor 21 are prime numbers, so we factor each of them. EXAMPLE 1 Find the prime factorization of 210. Writing the factors in order, from least to greatest, the prime-factored form of 210 is 2 3 5 7. Two other methods for prime factoring 210 are shown below. Objective 2: Recognize Special Fraction Forms Fractions can describe the number of equal parts of a whole. Consider the circle with 5 of 6 equal parts colored red. We say that 5/6 (five-sixths) of the circle is shaded. In a fraction, the number above the fraction bar is called the numerator, and the number below is called the denominator. 2

Objective 2: Recognize Special Fraction Forms Fractions are also used to indicate division. For example, 8/2 indicates that the numerator, 8, is to be divided by the denominator, 2: If the numerator and denominator of a fraction are the same nonzero number, the fraction indicates division of a number by itself, and the result is 1. If a denominator is 1, the fraction indicates division by 1, and the result is simply the numerator. Special fraction forms: For any nonzero number a, a/a = 1 and a/ 1 = a. Objective 3: Multiply and Divide Fractions Multiplying fractions: To multiply two fractions, multiply the numerators and multiply the denominators. For any two fractions a/b and c/d, (a/b)(c/d) = ac/bd. One number is called the reciprocal of another if their product is 1. To find the reciprocal of a fraction, we invert its numerator and denominator. Dividing fractions: To divide two fractions, multiply the first fraction by the reciprocal of the second. For any two fractions a/b and c/d, where c 0, a/b c/d = (a/b)(d/c). EXAMPLE 2 Strategy To find the product, we will multiply the numerators, 7 and 3, and multiply the denominators, 8 and 5. Why This is the rule for multiplying two fractions. : Multiply: 7/8 3/5. Objective 4: Build Equivalent Fractions The two rectangular regions on the right are identical. The first one is divided into 10 equal parts. Since 6 of those parts are red, 6/10 of the figure is shaded. The second figure is divided into 5 equal parts. Since 3 of those parts are red, 3/5 of the figure is shaded. We can conclude that 6/10 = 3/5 because 6/10 and 3/5 represent the same shaded portion of the figure. We say that 6/10 and 3/5 are equivalent fractions. Two fractions are equivalent if they represent the same number. 3

Objective 4: Build Equivalent Fractions Equivalent Fractions: Two fractions are equivalent if they represent the same number. Equivalent fractions represent the same portion of a whole. Writing a fraction as an equivalent fraction with a larger denominator is called building the fraction. Since any number multiplied by 1 remains the same (identical), 1 is called the multiplicative identity element. Multiplication property of 1: The product of 1 and any number is that number. For any number, 1 a = a and a 1=a EXAMPLE 4 Write 3/5 as an equivalent fraction with a denominator of 35. Strategy We will compare the given denominator to the required denominator and ask, By what must we multiply 5 to get 35? Why The answer to that question helps us determine the form of 1 to be used to build an equivalent fraction. EXAMPLE 4 Write 3/5 as an equivalent fraction with a denominator of 35. We need to multiply the denominator of 3/5 by 7 to obtain a denominator of 35. It follows that 7/7 should be the form of 1 that is used to build 3/5. Multiplying 3/5 by 7/7 changes its appearance but does not change its value, because we are multiplying it by 1. Objective 4: Build Equivalent Fractions Building Fractions : To build a fraction, multiply it by 1 in the form of c/c, where c is any nonzero number. The Fundamental Property of Fractions: If the numerator and denominator of a fraction are multiplied by the same nonzero number, the resulting fraction is equivalent to the original fraction. 4

Objective 5: Simplify Fractions Every fraction can be written in infinitely many equivalent forms. For example, some equivalent forms of 10/15 are: Objective 5: Simplify Fractions To simplify a fraction, we write it in simplest form by removing a factor equal to 1. For example, to simplify 10/15, we note that the greatest factor common to the numerator and denominator is 5 and proceed as follows: Simplest form of a fraction: A fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factors other than 1. To simplify 10/15, we removed a factor equal to 1 in the form of 5/5. The result,2/3, is equivalent to 10/15. We can easily identify the greatest common factor of the numerator and the denominator of a fraction if we write them in prime-factored form. EXAMPLE 5 Simplify each fraction, if possible: a. 63/42; b. 33/40 EXAMPLE 5 Simplify each fraction, if possible: a. 63/42; b. 33/40 Strategy We will begin by prime factoring the numerator and denominator of the fraction. Then, to simplify it, we will remove a factor equal to 1. Why We need to make sure that the numerator and denominator have no common factors other than 1. If that is the case, then the fraction is in simplest form. a. After prime factoring 63 and 42, we see that the greatest common factor of the numerator and the denominator is 3 7 = 21. b. To attempt to simplify the fraction, Prime factor 33 and 40. Since the numerator and the denominator have no common factors other than 1, the fraction is in simplest form (lowest terms). 5

Objective 5: Simplify Fractions Simplifying fractions: Factor (or prime factor) the numerator and denominator to determine their common factors. Remove factors equal to 1 by replacing each pair of factors common to the numerator and denominator with the equivalent fraction 1/1. Multiply the remaining factors in the numerator and in the denominator. Objective 6: Add and Subtract Fractions In algebra as in everyday life, we can only add or subtract objects that are similar. For example, we can add dollars to dollars, but we cannot add dollars to oranges. This concept is important when adding or subtracting fractions. Consider the problem 2/5 + 1/5. When we write it in words, it is apparent we are adding similar objects. Because the denominators of 2/5 and 1/5 are the same, we say that they have a common denominator. Objective 6: Add and Subtract Fractions To add (or subtract) fractions that have the same denominator, add (or subtract) their numerators and write the sum (or difference) over the common denominator. For any fractions a/d and b/d, a/d + b/d = (a+b)/d and a/d b/d = (a b)/d. Objective 6: Add and Subtract Fractions Now we consider the problem 2/5 + 1/3. Since the denominators are not the same, we cannot add these fractions in their present form. To add (or subtract) fractions with different denominators, we express them as equivalent fractions that have a common denominator. The smallest common denominator, called the least or lowest common denominator, is usually the easiest common denominator to use. 6

Objective 6: Add and Subtract Fractions Least Common Denominator (LCD): The least or lowest common denominator (LCD) for a set of fractions is the smallest number each denominator will divide exactly (divide with no remainder). Objective 6: Add and Subtract Fractions The denominators of 2/5 and 1/3 are 5 and 3. The numbers 5 and 3 divide many numbers exactly (30, 45, and 60, to name a few), but the smallest number that they divide exactly is 15. Thus, 15 is the LCD for 2/5 and 1/3.To find 2/5 + 1/3, we find equivalent fractions that have denominators of 15 and we use the rule for adding fractions. Objective 6: Add and Subtract Fractions When adding (or subtracting) fractions with unlike denominators, the least common denominator is not always obvious. Prime factorization is helpful in determining the LCD. Finding the LCD Using Prime Factorization: Prime factor each denominator. The LCD is a product of prime factors, where each factor is used the greatest number of times it appears in any one factorization found in step 1. EXAMPLE 7 Subtract: 3/10 5/28. Strategy We will begin by expressing each fraction as an equivalent fraction that has the LCD for its denominator. Then we will use the rule for subtracting fractions with like denominators. Why To add or subtract fractions, the fractions must have like denominators. 7

EXAMPLE 7 Subtract: 3/10 5/28. To find the LCD, we find the prime factorization of both denominators and use each prime factor the greatest number of times it appears in any one factorization. EXAMPLE 7 Subtract: 3/10 5/28. Since 140 is the smallest number that 10 and 28 divide exactly, we write 3/10 and 5/28 as fractions with the LCD 140. Objective 6: Add and Subtract Fractions Adding and subtracting fractions that have different denominators: Find the LCD. Rewrite each fraction as an equivalent fraction with the LCD as the denominator. To do so, build each fraction using a form of 1 that involves any factors needed to obtain the LCD. Add or subtract the numerators and write the sum or difference over the LCD. Simplify the result, if possible. Objective 7: Simplify Answers When adding, subtracting, multiplying, or dividing fractions, remember to express the answer in simplest form. 8

EXAMPLE 8 Perform the operations and simplify: a. 45(4/9); b. 5/12 + 3/2 1/4. Strategy We will perform the indicated operations and then make sure that the answer is in simplest form. EXAMPLE 8 (a) Perform the operations and simplify: a. 45(4/9); b. 5/12 + 3/2 1/4. Why Fractional answers should always be given in simplest form. EXAMPLE 8 (b) Perform the operations and simplify: a. 45(4/9); b. 5/12 + 3/2 1/4. Since the smallest number that 12, 2, and 4 divide exactly is 12, the LCD is 12. Objective 8: Compute with Mixed Numbers A mixed number represents the sum of a whole number and a fraction. For example, 5¾ means 5 + ¾ and 179 15/16 means 179+15/16. 9

EXAMPLE 9 Divide: 5¾ 2. Strategy We begin by writing the mixed number 5¾ and the whole number 2 as fractions. Then we use the rule for dividing two fractions. EXAMPLE 9 Why To multiply (or divide) with mixed numbers, we first write them as fractions, and then multiply (or divide) as usual. Objectives Section 1.3 The Real Numbers Define the set of integers. Define the set of rational numbers. Define the set of irrational numbers. Classify real numbers. Graph sets of real numbers on the number line. Find the absolute value of a real number. 10

Objective 1: Define the Set of Integers A set is a collection of objects, such as a set of golf clubs or a set of dishes. Natural numbers are the numbers that we use for counting. To write this set, we list its members (or elements) within braces { }. The set of natural numbers is {1, 2, 3, 4, 5,...} Objective 1: Define the Set of Integers The natural numbers, together with 0, form the set of whole numbers. The set of whole numbers is {0, 1, 2, 3, 4, 5,...} Read as the set containing one, two, three, four, five, and so on. Objective 1: Define the Set of Integers Whole numbers are not adequate for describing many real-life situations. For example, if you write a check for more than what s in your account, the account balance will be less than zero. We can use the number line below to visualize numbers less than zero. A number line is straight and has uniform markings. The arrowheads indicate that it extends forever in both directions. For each natural number on the number line, there is a corresponding number, called its opposite, to the left of 0. In the diagram, we see that 3 and (negative three) are opposites, as are (negative five) and 5. Note that 0 is its own opposite. Objective 1: Define the Set of Integers Two numbers that are the same distance from 0 on the number line, but on opposite sides of it, are called opposites. The whole numbers, together with their opposites, form the set of integers. The set of integers is {..., 4, 3, 2, 1, 0, 1, 2, 3, 4,... } On the number line, numbers greater than 0 are to the right of 0. They are called positive numbers. 11

Objective 1: Define the Set of Integers Positive numbers can be written with or without a positive sign +. Numbers less than 0 are to the left of 0 on the number line. They are called negative numbers. Negative numbers are always written with a negative sign. Positive and negative numbers are called signed numbers. Objective 2: Define the Set of Rational Numbers Fractions such as 1/4 and 2/3, that are quotients of two integers, are called rational numbers. A rational number is any number that can be expressed as a fraction (ratio) with an integer numerator and a nonzero integer denominator. Negative fractions are also rational numbers. For any numbers a and b, where b is not 0, (a/b) = ( a/b) = (a/ b). Positive and negative mixed numbers are also rational numbers because they can be expressed as fractions. For example, 7 5 / 8 = 61/8. Any natural number, whole number, or integer can be expressed as a fraction with a denominator of 1. For example, 5 = 5/1, 0 = 0/1, and 3 = 3/1. Therefore, every natural number, whole number, and integer is also a rational number. Objective 2: Define the Set of Rational Numbers Many numerical quantities are written in decimal notation. For instance, a candy bar might cost $0.89, a dragster might travel at 203.156 mph, or a business loss might be $4.7 million. These decimals are called terminating decimals because their representations terminate (stop). Terminating decimals can be expressed as fractions: 0.89 = 89/100, 203.156 = 203 156 / 1000 = 203,156/1,000. Therefore, terminating decimals are rational numbers. Decimals such as 0.3333... and 2.8167167167..., which have a digit (or block of digits) that repeats, are called repeating decimals. Since any repeating decimal can be expressed as a fraction, repeating decimals are rational numbers. The set of rational numbers cannot be listed as we listed other sets in this section. Instead, we use set-builder notation. Objective 2: Define the Set of Rational Numbers To find the decimal equivalent for a fraction, we divide its numerator by its denominator. For example, to write 1/4 and 5/22 as decimals, we proceed as follows: 12

Objective 2: Define the Set of Rational Numbers The decimal equivalent of 1/4 is 0.25 and the decimal equivalent of 5/22 is 0.2272727... We can use an overbar to write repeating decimals in more compact form. 0.2272727...=0.227. Objective 3: Define the Set of Irrational Numbers Not all numbers are rational numbers. One example is the square root of 2, written 2. It is the number that, when multiplied by itself, gives 2: 2 2 = 2. It can be shown that 2 cannot be written as a fraction with an integer numerator and an integer denominator. Therefore, it is not rational; it is an irrational number. It is interesting to note that a square with sides of length 1 inch has a diagonal that is 2 inches long. The number represented by the Greek letter π (pi) is another example of an irrational number. A circle, with a 1-inch diameter, has a circumference of π inches. Expressed in decimal form, 2 = 1.414213562... and π = 3.141592654... These decimals neither terminate nor repeat. An irrational number is a nonterminating, nonrepeating decimal. An irrational number cannot be expressed as a fraction with an integer numerator and an integer denominator. Objective 4: Classify Real Numbers The set of real numbers is formed by combining the set of rational numbers and the set of irrational numbers. Every real number has a decimal representation. If it is rational, its corresponding decimal terminates or repeats. If it is irrational, its decimal representation is nonterminating and nonrepeating. A real number is any number that is a rational number or an irrational number. Objective 4: Classify Real Numbers The following diagram shows how various sets of numbers are related. Note that a number can belong to more than one set. For example, 6 is an integer, a rational number, and a real number. 13

EXAMPLE 1 Which numbers in the following set are natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers? { 3.4, 2/5, 0, 6, 1¾, π, 16} EXAMPLE 1 Strategy We begin by scanning the given set, looking for any natural numbers. Then we scan it five more times, looking for whole numbers, for integers, for rational numbers, for irrational numbers, and finally, for real numbers. Why We need to scan the given set of numbers six times, because numbers in that set can belong to more than one classification. Objective 5: Graph Sets of Numbers on the Number Line Every real number corresponds to a point on the number line, and every point on the number line corresponds to exactly one real number. As we move right on the number line, the values of the numbers increase. As we move left, the values decrease. On the number line below, we see that 5 is greater than 3, because 5 lies to the right of 3. Similarly, 3 is less than 5, because it lies to the left of 5. Objective 5: Graph Sets of Numbers on the Number Line The inequality symbol > means is greater than. It is used to show that one number is greater than another. The inequality symbol < means is less than. It is used to show that one number is less than another. For example, To distinguish between these inequality symbols, remember that each one points to the smaller of the two numbers involved. 14

EXAMPLE 2 Use one of the symbols > or < to make each statement true: a. 4? 4, b. 2? 3, c. 4.47? 12.5, d. 3/4? 5/8 Strategy To pick the correct inequality symbol to place between a given pair of numbers, we need to determine the position of each number on a number line. Why For any two numbers on a number line, the number to the left is the smaller number and the number to the right is the larger number. EXAMPLE 2 a. Since 4 is to the left of 4 on the number line, we have 4 < 4. b. Since 2 is to the right of 3 on the number line, we have 2 > 3. c. Since 4.47 is to the left of 12.5 on the number line, we have 4.47 < 12.5. d. To compare fractions, express them in terms of the same denominator, preferably the LCD. If we write 3/4 as an equivalent fraction with denominator 8, we see that 3/4 = (3 2)/(4 2) = 6/8. Therefore, 3/4 > 5/8. To compare the fractions, we could also convert each to its decimal equivalent. Since 3/4 = 0.75 and 5/8 = 0.625, we know that 3/4 > 5/8. Objective 6: Find the Absolute Value of a Real Number A number line can be used to measure the distance from one number to another. For example, in the following figure we see that the distance from 0 to 4 is 4 units and the distance from 0 to 3 is 3 units. EXAMPLE 4 Find each absolute value: a. 18, b. 7/8, c. 98.6, d. 0 To express the distance that a number is from 0 on a number line, we can use absolute values. The absolute value of a number is its distance from 0 on the number line. To indicate the absolute value of a number, we write the number between two vertical bars. From the figure above, we see that 4 = 4. It also follows from the figure that 3 = 3. Strategy We need to determine the distance that the number within the vertical absolute value bars is from 0. Why The absolute value of a number is the distance between 0 and the number on a number line. 15

EXAMPLE 4 a. Since 18 is a distance of 18 from 0 on the number line, 18 = 18. b. Since 7/8 is a distance of 7/8 from 0 on the number line, 7/8 = 7/8. c. Since 98.6 is a distance of 98.6 from 0 on the number line, 98.6 = 98.6. Section 1.4 Adding Real Numbers; Properties of Addition d. Since 0 is a distance of 0 from 0 on the number line, 0 = 0. Objectives Add two numbers that have the same sign Add two numbers that have different signs Use properties of addition Identify opposites (additive inverses) Objective 1: Add Two Numbers That Have the Same Sign A number line can be used to explain the addition of signed numbers. For example, to compute 5 + 2, we begin at 0 and draw an arrow five units long that points right. It represents 5. From the tip of that arrow, we draw a second arrow two units long that points right. It represents 2. Since we end up at 7, it follows that 5 + 2 = 7. The numbers that we added, 5 and 2, are called addends, and the result, 7, is called the sum. 16

Objective 1: Add Two Numbers That Have the Same Sign To compute 5 + ( 2), we begin at 0 and draw an arrow five units long that points left. It represents 5. From the tip of that arrow, we draw a second arrow two units long that points left. It represents 2. Since we end up at 7, it follows that 5 + ( 2) = 7. Objective 1: Add Two Numbers That Have the Same Sign When we use a number line to add numbers with the same sign, the arrows point in the same direction and they build upon each other. Furthermore, the answer has the same sign as the numbers that we added. These observations suggest the following rules. Adding Two Numbers That Have the Same (Like) Signs: 1. To add two positive numbers, add them as usual. The final answer is positive. 2. To add two negative numbers, add their absolute values and make the final answer negative. EXAMPLE 1 Add: a. 20 + ( 15), b. 7.89 + ( 0.6), c. 1/3 + ( 1/2) EXAMPLE 1 Strategy We will use the rule for adding two numbers that have the same sign. Why In each case, we are asked to add two negative numbers. 17

Objective 2: Add Two Numbers That Have Different Signs To compute 5 + ( 2), we begin at 0 and draw an arrow five units long that points right. From the tip of that arrow, we draw a second arrow two units long that points left. Since we end up at 3, it follows that 5 + ( 2) = 3. In terms of money, if you won $5 and then lost $2, you would have $3 left. Objective 2: Add Two Numbers That Have Different Signs To compute 5 + 2, we begin at 0 and draw an arrow five units long that points left. From the tip of that arrow, we draw a second arrow two units long that points right. Since we end up at 3, it follows that 5 + ( 2) = 3. In terms of money, if you lost $5 and then won $2, you have lost $3. Objective 2: Add Two Numbers That Have Different Signs When we use a number line to add numbers with different signs, the arrows point in opposite directions and the longer arrow determines the sign of the answer. If the longer arrow represents a positive number, the sum is positive. If it represents a negative number, the sum is negative. These observations suggest the following rules. Adding Two Numbers That Have Different (Unlike) Signs: To add a positive number and a negative number, subtract the smaller absolute value from the larger. 1. If the positive number has the larger absolute value, the final answer is positive. 2. If the negative number has the larger absolute value, make the final answer negative. EXAMPLE 2 Add: a. 20 + 32, b. 5.7 + ( 7.4), c. 19/25 + 2/5 Strategy We will use the rule for adding two numbers that have different (unlike) signs. Why In each case, we are asked to add a positive number and a negative number. 18

EXAMPLE 2 EXAMPLE 2 a. c. Since 2/5 = 10/25, the fraction 19/25 has the larger absolute value. To find 19/25 +2/5, we subtract the smaller absolute value from the larger: b. To find 5.7 + ( 7.4), subtract the smaller absolute value, 5.7, from the larger, 7.4. Since the negative decimal, 7.4, has the larger absolute value, make the final answer negative: 5.7 + ( 7.4) = 1.7. Since the negative fraction 19/25 has the larger absolute value, make the final answer negative: 19/25 + 10/25 = 9/25. Objective 3: Use Properties of Addition The addition of two numbers can be done in any order and the result is the same. For example, 8 + ( 1) = 7 and 1 + 8 = 7. This example illustrates that addition is commutative. The Commutative Property of Addition: Changing the order when adding does not affect the answer. For any real numbers a and b, a + b = b + a. Objective 3: Use Properties of Addition In the following example, we add 3 + 7 + 5 in two ways. We will use grouping symbols ( ), called parentheses, to show this. Standard practice requires that the operation within the parentheses be performed first. It doesn t matter how we group the numbers in this addition; the result is 9. This example illustrates that addition is associative. The Associative Property of Addition: Changing the grouping when adding does not affect the answer. For any real numbers a, b, and c, (a + b) + c = a + (b + c). 19

EXAMPLE 4 Strategy We will use the associative property to group 2 with 98. Then, we evaluate the expression by following the rules for the order of operations. Why It is helpful to regroup because 98 and 2 are a pair of numbers that are easily added. Find the sum: 98 + (2 + 17) Objective 4: Identify Opposites (Additive Inverses) Recall that two numbers that are the same distance from 0 on a number line, but on opposite sides of it, are called opposites. To develop a property for adding opposites, we will find 4 + 4 using a number line. We begin at 0 and draw an arrow four units long that points left, to represent 4. From the tip of that arrow, we draw a second arrow, four units long that points right, to represent 4. We end up at 0; therefore, 4 + 4 = 0. Objective 4: Identify Opposites (Additive Inverses) This example illustrates that when we add opposites, the result is 0. Therefore, 1.6 + ( 1.6) = 0 and 3/4 + 3/4 = 0. Also, whenever the sum of two numbers is 0, those numbers are opposites. For these reasons, opposites are also called additive inverses. Addition Property of Opposites (Inverse Property of Addition): The sum of a number and its opposite (additive inverse) is 0. For any real number a and its opposite or additive inverse a, a + ( a) = 0. EXAMPLE 6 Strategy Instead of working from left to right, we will use the commutative and associative properties of addition to add pairs of opposites. Why Since the sum of a number and its opposite is 0, it is helpful to identify such pairs in an addition. Add: 12 + ( 5) + 6 + 5 + ( 12) 20

Objectives Section 1.5 Subtracting Real Numbers Use the definition of subtraction. Solve application problems using subtraction. Objective 1: Use the Definition of Subtraction A minus symbol is used to indicate subtraction. However, this symbol is also used in two other ways, depending on where it appears in an expression. Objective 1: Use the Definition of Subtraction This observation illustrates the following rule. Opposite of an Opposite: The opposite of the opposite of a number is that number. For any real number a, ( a) = a. Read as the opposite of the opposite of a is a. In ( 5), parentheses are used to write the opposite of a negative number. When such expressions are encountered in computations, we simplify them by finding the opposite of the number within the parentheses. 21

EXAMPLE 1 Simplify each expression: a. ( 45), b. ( h), c. 10 Strategy To simplify each expression, we will use the concept of opposite. Why In each case, the outermost symbol is read as the opposite. EXAMPLE 1 a. The number within the parentheses is 45. Its opposite is 45. Therefore, ( 45) = 45. b. The opposite of the opposite of h is h. Therefore, ( h) = h. c. The notation 10 means the opposite of the absolute value of negative ten. Since 10 = 10, we have: Objective 1: Use the Definition of Subtraction (continued) To develop a rule for subtraction, we consider the following illustration. It represents the subtraction 5 2 = 3. The illustration above also represents the addition 5 + ( 2) = 3. We see that: Objective 1: Use the Definition of Subtraction (continued) Subtraction of Real Numbers: To subtract two real numbers, add the first number to the opposite (additive inverse) of the number to be subtracted. For any real numbers a and b, a b = a + ( b). Read as a minus b equals a plus the opposite of b. 22

Objective 2: Solve Application Problems Using Subtraction Subtraction finds the difference between two numbers. When we find the difference between the maximum value and the minimum value of a collection of measurements, we are finding the range of the values. EXAMPLE 5 The record high temperature in the United States was 134 F in Death Valley, California, on July 10, 1913. The record low was 80 F at Prospect Creek, Alaska, on January 23, 1971. Find the temperature range for these extremes. Strategy We will subtract the lowest temperature from the highest temperature. Why The range of a collection of data indicates the spread of the data. It is the difference between the largest and smallest values. : U.S. Temperatures The temperature range for these extremes is 214 F. Objectives Section 1.6 Multiplying and Dividing Real Numbers; Multiplication and Division Properties Multiply signed numbers Use properties of multiplication Divide signed numbers Use properties of division 23

Objective 1: Multiply Signed Numbers Multiplication represents repeated addition. For example, 4(3) is equal to the sum of four 3 s. This example illustrates that the product of two positive numbers is positive. To develop a rule for multiplying a positive number and a negative number, we will find 4( 3), which is equal to the sum of four 3 s. 4( 3) = 3 + ( 3) + ( 3) + ( 3) = 12. We see that the result is negative. This example illustrates that the product of a positive number and a negative number is negative. Multiplying Two Numbers That Have Different (Unlike) Signs: To multiply a positive real number and a negative real number, multiply their absolute values. Then make the final answer negative. Multiplying Two Numbers That Have the Same (Like) Signs: To multiply two real numbers that have the same sign, multiply their absolute values. The final answer is positive. EXAMPLE 1 Multiply: a. 8( 12), b. 151 5, c. ( 0.6)(1.2), d. 3/4( 4/15) Strategy We will use the rule for multiplying two numbers that have different signs. Why In each case, we are asked to multiply a positive number and a negative number. EXAMPLE 1 Objective 2: Use Properties of Multiplication The multiplication of two numbers can be done in any order; the result is the same. For example, 9(4) = 36 and 4( 9) = 36. This illustrates that multiplication is commutative. The Commutative Property of Multiplication: Changing the order when multiplying does not affect the answer. For any real numbers a and b, ab = ba. 24

Objective 2: Use Properties of Multiplication In the following example, we multiply 3 7 5 in two ways. Recall that the operation within the parentheses should be performed first. It doesn t matter how we group the numbers in this multiplication; the result is 105. This example illustrates that multiplication is associative. The Associative Property of Multiplication: Changing the grouping when multiplying does not affect the answer. For any real numbers a, b, and c, (ab)c = a(bc). EXAMPLE 3 Multiply: a. 5( 37)( 2), b. 4( 3)( 2)( 1) Strategy First, we will use the commutative and associative properties of multiplication to reorder and regroup the factors. Then we will perform the multiplications. Why Applying of one or both of these properties before multiplying can simplify the computations and lessen the chance of a sign error. EXAMPLE 3 Using the commutative and associative properties of multiplication, we can reorder and regroup the factors to simplify computations. Objective 2: Use Properties of Multiplication (continued) Some more properties of multiplication: Multiplying Negative Numbers: The product of an even number of negative numbers is positive. The product of an odd number of negative numbers is negative. Multiplication Property of 0: The product of 0 and any real number is 0. For any real number a, 0 a = 0 and a 0 = 0. Multiplication Property of 1(Identity Property of Multiplication): The product of 1 and any number is that number. For any real number a, 1 a = a and a 1 = a. Since any number multiplied by 1 remains the same (is identical), the number 1 is called the identity element for multiplication. Multiplicative Inverses (Inverse Property of Multiplication): The product of any number and its multiplicative inverse (reciprocal) is 1. For any nonzero real number a, a(1/a) = 1. Two numbers whose product is 1 are reciprocals or multiplicative inverses of each other. For example, 8 is the multiplicative inverse of 1/8, and 1/8 is the multiplicative inverse of 8. 25

Objective 3: Divide Signed Numbers Every division fact can be written as an equivalent multiplication fact. For any real numbers a, b, and c, where b 0, a/b = c provided that c b = a. Quotient divisor = dividend We can use this relationship between multiplication and division to develop rules for dividing signed numbers. For example, 15/5 = 3 because 3(5) = 15. From this example, we see that the quotient of two positive numbers is positive. Objective 3: Divide Signed Numbers We summarize this and other such rules and note that they are similar to the rules for multiplication. Dividing Two Real Numbers: To divide two real numbers, divide their absolute values. 1. The quotient of two numbers that have the same (like) signs is positive. 2. The quotient of two numbers that have different (unlike) signs is negative. EXAMPLE 5 Divide and check the result: a. 81/ 9, b. 45/ 9, c. 2.87 0.7, d. 5/16 ( 1/2) EXAMPLE 5 Strategy We will use the rules for dividing signed numbers. In each case, we need to ask, Is it a quotient of two numbers with the same sign or different signs? Quotient divisor = dividend Quotient divisor = dividend Why The signs of the numbers that we are dividing determine the sign of the result. Quotient divisor = dividend 26

Objective 4: Use Properties of Division Whenever we divide a number by 1, the quotient is that number. Therefore, 12/1 = 12, 80/1 = 80, and 7.75/1 = 7.75. Furthermore, whenever we divide a nonzero number by itself, the quotient is 1. Therefore, 35/35 = 1, 4/ 4 = 1, and 0.9/0.9 = 1. These observations suggest the following properties of division. Any number divided by 1 is the number itself. Any number (except 0) divided by itself is 1. For any real number a, a/1 = a and a/a = 1 (where a 0). Objective 4: Use Properties of Division Division is not commutative. For example, and 6/3 3/6. In general, a/b b/a. We will now consider division that involves zero. First, we examine division of zero. Let s look at two examples. 0/2 = 0 because 0(2) = 0. 0/ 5 = 0 because 0( 5) = 0. These examples illustrate that 0 divided by a nonzero number is 0. Objective 4: Use Properties of Division To examine division by zero, let s look at 2/0 and its related multiplication statement. 2/0 =? because?(0) = 2 There is no number that can make 0(?) = 2 true because any number multiplied by 0 is equal to 0, not 2. Therefore, 2/0 does not have an answer. We say that such a division is undefined. EXAMPLE 7 Find each quotient, if possible: a. 0/8, b. 24/0 Strategy In each case, we need to determine if we have division of 0 or division by 0. Why Division of 0 by a nonzero number is defined, and the result is 0. However, division by 0 is undefined; there is no result. 27

EXAMPLE 7 Find each quotient, if possible: a. 0/8, b. 24/0 Section 1.7 Exponents and Order of Operations Objectives Objective 1: Evaluate Exponential Expressions Evaluate exponential expressions Use the order of operations rules Evaluate expressions containing grouping symbols Find the mean (average) In the expression 3 3 3 3 3, the number 3 repeats as a factor five times. We can use exponential notation to write this product in a more compact form. Exponent and Base: An exponent is used to indicate repeated multiplication. It is how many times the base is used as a factor. In the exponential expression 3 5, the base is 3, and 5 is the exponent. The expression is called a power of 3. Some other examples of exponential expressions are: 28

Objective 1: Evaluate Exponential Expressions To evaluate (find the value of) an exponential expression, we write the base as a factor the number of times indicated by the exponent. Then we multiply the factors. EXAMPLE 2 Evaluate each expression: a. 5 3, b. ( 2/3) 3, c. 10 1, d. (0.6) 2, e. ( 3) 4, f. ( 3) 5 Strategy We will rewrite each exponential expression as a product of repeated factors, and then perform the multiplication. This requires that we identify the base and the exponent. Why The exponent tells the number of times the base is to be written as a factor. EXAMPLE 2 EXAMPLE 2 29

Objective 1: Evaluate Exponential Expressions Even and Odd Powers of a Negative Number: When a negative number is raised to an even power, the result is positive. When a negative number is raised to an odd power, the result is negative. Although the expressions ( 4) 2 and 4 2 look alike, they are not. When we find the value of each expression, it becomes clear that they are not equivalent. Objective 2: Use the Order of Operations Rules Suppose you have been asked to contact a friend if you see a Rolex watch for sale when you are traveling in Europe. While in Switzerland, you find the watch and send the text message shown on the left. The next day, you get the response shown on the right. Something is wrong. The first part of the response (No price too high!) says to buy the watch at any price. The second part (No! Price too high.) says not to buy it, because it s too expensive. The placement of the exclamation point makes us read the two parts of the response differently, resulting in different meanings. When reading a mathematical statement, the same kind of confusion is possible. Objective 2: Use the Order of Operations Rules For example, consider the expression 2 + 3 6. We can evaluate this expression in two ways. We can add first, and then multiply. Or we can multiply first, and then add. However, the results are different. If we don t establish a uniform order of operations, the expression has two different values. To avoid this possibility, we will always use the following set of priority rules. Objective 2: Use the Order of Operations Rules Order of Operations: 1. Perform all calculations within parentheses and other grouping symbols following the order listed in Steps 2 4 below, working from the innermost pair of grouping symbols to the outermost pair. 2. Evaluate all exponential expressions. 3. Perform all multiplications and divisions as they occur from left to right. 4. Perform all additions and subtractions as they occur from left to right. When grouping symbols have been removed, repeat Steps 2 4 to complete the calculation. If a fraction is present, evaluate the expression above and the expression below the bar separately. Then simplify the fraction, if possible. 30

EXAMPLE 4 EXAMPLE 4 Evaluate: a. 3 2 3 4, b. 30 4 5 + 9, c. 24 6 2, d. 160 4 + 6( 2)( 3) Strategy We will scan the expression to determine what operations need to be performed. Then we will perform those operations, one-at-a-time, following the order of operations rules. a. Three operations need to be performed to evaluate this expression: multiplication, raising to a power, and subtraction. By the order of operations rules, we evaluate 2 3 first. b. This expression involves subtraction, multiplication, and addition. The order of operations rule tells us to multiply first. Why If we don t follow the correct order of operations, the expression can have more than one value. EXAMPLE 4 c. Since there are no calculations within parentheses nor are there exponents, we perform the multiplications and divisions as they occur from left to right. The division occurs before the multiplication, so it must be performed first. d. Although this expression contains parentheses, there are no operations to perform within them. Since there are no exponents, we will perform the multiplications as they occur from left to right. Objective 3: Evaluate Expressions Containing Grouping Symbols Grouping symbols serve as mathematical punctuation marks. They help determine the order in which an expression is to be evaluated. Examples of grouping symbols are parentheses ( ), brackets [ ], braces { }, absolute value symbols, and the fraction bar. Expressions can contain two or more pairs of grouping symbols. To evaluate the following expression, we begin within the innermost pair of grouping symbols, the parentheses. Then we work within the outermost pair, the brackets. 31

EXAMPLE 6 Evaluate: 4[2 + 3(4 8 2 )] 2 EXAMPLE 6 Evaluate: 4[2 + 3(4 8 2 )] 2 Strategy We will work within the parentheses first and then within the brackets. At each stage, we follow the order of operations rules. Why By the order of operations, we must work from the innermost pair of grouping symbols to the outermost. Objective 4: Find the Mean (Average) EXAMPLE 9 Hotel Reservations The arithmetic mean (or simply mean) of a set of numbers is a value around which the values of the numbers are grouped. The mean is also commonly called the average. Finding An Arithmetic Mean: To find the mean of a set of values, divide the sum of the values by the number of values. In an effort to improve customer service, a hotel electronically recorded the number of times the reservation desk telephone rang before it was answered by a receptionist. The results of the week-long survey are shown in the table. Find the average(mean) number of times the phone rang before a receptionist answered. Strategy First, we will determine the total number of times the reservation desk telephone rang during the week. Then we will divide that result by the total number of calls received. Why To find the average value of a set of values, we divide the sum of the values by the number of values. 32

EXAMPLE 9 Hotel Reservations To find the total number of rings, we multiply each number of rings (1, 2, 3, 4, and 5 rings) by the respective number of occurrences and add those subtotals. Total number of rings = 11(1) + 46(2) + 45(3) + 28(4) + 20(5) The total number of calls received was 11 + 46 + 45 + 28 + 20. To find the average, we divide the total number of rings by the total number of calls. Section 1.8 Algebraic Expressions Objectives Identify terms and coefficients of terms Translate word phrases to algebraic expressions Analyze problems to determine hidden operations Evaluate algebraic expressions Objective 1: Identify Terms and Coefficients of Terms Recall that variables and/or numbers can be combined with the operations of arithmetic to create algebraic expressions. Addition symbols separate expressions into parts called terms. For example, the expression x + 8 has two terms. Since subtraction can be written as addition of the opposite, the expression a 2 3a 9 has three terms. In general, a term is a product or quotient of numbers and/or variables. A single number or variable is also a term. Examples of terms are: 4, y, 6r, w 3, 3.7x 5, 3/n, 15ab 2. 33

Objective 1: Identify Terms and Coefficients of Terms The numerical factor of a term is called the coefficient of the term. For instance, the term 6r has a coefficient of 6 because 6r = 6 r. The coefficient of 15ab 2 is 15 because 15ab 2 = 15 ab 2. More examples are shown below. A term such as 4, that consists of a single number, is called a constant term. EXAMPLE 1 Identify the coefficient of each term in the expression: 7x 2 x + 6. Strategy We will begin by writing the subtraction as addition of the opposite. Then we will determine the numerical factor of each term. Why Addition symbols separate expressions into terms. EXAMPLE 1 If we write 7x 2 x + 6 as 7x 2 + ( x) + 6, we see that it has three terms: 7x 2, x, and 6. The numerical factor of each term is its coefficient. Objective 2: Translate Word Phrases to Algebraic Expressions The four tables show how key phrases can be translated into algebraic expressions. The coefficient of 7x 2 is 7 because 7x 2 means 7 x 2. The coefficient of x is 1 because x means 1 x. The coefficient of the constant 6 is 6. 34

Objective 2: Translate Word Phrases to Algebraic Expressions EXAMPLE 3 Write each phrase as an algebraic expression: a. one-half of the profit P b. 5 less than the capacity c c. the product of the weight w and 2,000, increased by 300 Strategy We will begin by identifying any key phrases. Why Key phrases can be translated to mathematical symbols. EXAMPLE 3 The algebraic expression is: ½P. Objective 3: Analyze Problems to Determine Hidden Operations Many applied problems require insight and analysis to determine which mathematical operations to use. Sometimes thinking in terms of specific numbers makes translating easier. Suppose the capacity was 100. Then 5 less than 100 would be 100 5. If the capacity is c, then we need to make it 5 less. The algebraic expression is: c 5. In the given wording, the comma after 2,000 means w is first multiplied by 2,000; then 300 is added to that product. The algebraic expression is: 2,000w + 300. 35

EXAMPLE 7 Vacations EXAMPLE 7 Vacations Disneyland, in California, was in operation 16 years before the opening of Disney World in Florida. Euro Disney, in France, was constructed 21 years after Disney World. Write algebraic expressions to represent the ages (in years) of each Disney attraction. In carefully reading the problem, we see that Disneyland was built 16 years before Disney World. That makes its age 16 years more than that of Disney World. The key phrase more than indicates addition. Strategy We will begin by letting x = the age of Disney World. Why The ages of Disneyland and Euro Disney are both related to the age of Disney World. x + 16 = the age of Disneyland Euro Disney was built 21 years after Disney World. That makes its age 21 years less than that of Disney World. The key phrase less than indicates subtraction. x 21 = the age of Euro Disney Objective 4: Evaluate Algebraic Expressions To evaluate an algebraic expression, we substitute given numbers for each variable and perform the necessary calculations in the proper order. EXAMPLE 10 Evaluate each expression for x = 3 and y = 4: a. y 3 + y 2, b. y x, c. 5xy 7, d. (y 0)/[x ( 1)] Strategy We will replace each x and y in the expression with the given value of the variable, and evaluate the expression using the order of operation rule. Why To evaluate an expression means to find its numerical value, once we know the value of its variable(s). 36

EXAMPLE 10 EXAMPLE 10 Objectives Section 1.9 Simplifying Algebraic Expressions Using Properties of Real Numbers Use the commutative and associative properties. Simplify products Use the distributive property Identify like terms Combine like terms 37

Objective 1: Use the commutative and associative properties. Objective 1: Use the commutative and associative properties. The Commutative and Associative properties of multiplication can be explained as follows. Commutative Properties Changing the order when adding or multiplying does not affect the answer. a + b = b + a and ab = ba Associative Properties Changing the grouping when adding or multiplying does not affect the answer. (a + b) + c = a + (b + c) and (ab) = a (bc) These properties can be applied when working with algebraic expressions that involve addition and multiplication. EXAMPLE 1 Use the given property to complete each statement: EXAMPLE 1 a. To commute means to go back and forth. The commutative property of addition enables us to change the order of the terms, 9 and x. Strategy To fill in each blank, we will determine the way in which the property enables us to rewrite the given expression. b. We change the order of the factors, t and 5. Why We should memorize the properties by name because their names remind us how to use them. 38

EXAMPLE 1 c. To associate means to group together. The associative property of addition enables us to group the terms in a different way. Objective 2: Simplify Products The commutative and associative properties of multiplication can be used to simplify certain products. For example, let s simplify 8(4x). d. We change the grouping of the factors, 6, 10, and n. We have found that 8(4x) = 32x. We say that 8(4x) and 32x are equivalent expressions because for each value of x, they represent the same number. EXAMPLE 2 Multiply: a. 9(3b), b. 15a(6), c. 3(7p)( 5), d. ( 8 / 3 )( 3r / 8 ), e. 35(4/5)x EXAMPLE 2 Strategy We will use the commutative and associative properties of multiplication to reorder and regroup the factors in each expression. Why We want to group all of the numerical factors of an expression together so that we can find their product. 39

EXAMPLE 2 Objective 3: Use the Distributive Property Another property that is often used to simplify algebraic expressions is the distributive property. To introduce it, we will evaluate 4(5+3) in two ways. Each method gives a result of 32. This observation suggests the following property. The Distributive Property: For any real numbers a, b, and c, a(b + c) = ab + ac. Objective 3: Use the Distributive Property To illustrate one use of the distributive property, let s consider the expression 5(x + 3). Since we are not given the value of x, we cannot add x and 3 within the parentheses. However, we can distribute the multiplication by the factor of 5 that is outside the parentheses to x and to 3 and add those products. Objective 3: Use the Distributive Property Since subtraction is the same as adding the opposite, the distributive property also holds for subtraction. a(b c) = ab ac The distributive property can be extended to several other useful forms. Since multiplication is commutative, we have: For situations in which there are more than two terms within parentheses, we have: 40

EXAMPLE 5 Multiply: a. (6x + 4)½, b. 2(a 3b)8, c. 0.3(3a 4b + 7) EXAMPLE 5 Strategy We will multiply each term within the parentheses by the factor (or factors) outside the parentheses. Why In each case, we cannot simplify the expression within the parentheses. To multiply, we must use the distributive property. Objective 3: Use the Distributive Property We can use the distributive property to find the opposite of a sum. For example, to find (x + 10), we interpret the symbol as a factor of 1, and proceed as follows: Objective 4: Identify Like Terms Before we can discuss methods for simplifying algebraic expressions involving addition and subtraction, we need to introduce some new vocabulary. Like terms are terms containing exactly the same variables raised to exactly the same powers. Any constant terms in an expression are considered to be like terms. Terms that are not like terms are called unlike terms. In general, we have the following property of real numbers. The Opposite of a Sum: The opposite of a sum is the sum of the opposites. For any real numbers a and b, (a + b) = a + ( b). 41

Objective 4: Identify Like Terms EXAMPLE 7 List the like terms in each expression: a. 7r + 5 + 3r, b. 6x 4 6x 2 6x, c. 17m 3 + 3 2 + m 3 Strategy First, we will identify the terms of the expression. Then we will look for terms that contain the same variables raised to exactly the same powers. Why If two terms contain the same variables raised to the same powers, they are like terms. EXAMPLE 7 a. 7r + 5 + 3r contains the like terms 7r and 3r. b. Since the exponents on x are different, 6x 4 6x 2 6x contains no like terms. c. 17m 3 + 3 2 + m 3 contains two pairs of like terms: 17m 3 and m 3 are like terms, and the constant terms, 3 and 2, are like terms. Objective 5: Combine Like Terms To add or subtract objects, they must be similar. For example, fractions that are to be added must have a common denominator. When adding decimals, we align columns to be sure to add tenths to tenths, hundredths to hundredths, and so on. The same is true when working with terms of an algebraic expression. They can be added or subtracted only if they are like terms. 42

Objective 5: Combine Like Terms Recall that the distributive property can be written in the following forms: (b + c)a = ba + ca (b c)a = ba ca We can use these forms of the distributive property in reverse to simplify a sum or difference of like terms. For example, we can simplify 3x + 4x as follows: Objective 5: Combine Like Terms We can simplify 15m 2 9m 2 in a similar way: In each case, we say that we combined like terms. These examples suggest the following general rule. Combining Like Terms: Like terms can be combined by adding or subtracting the coefficients of the terms and keeping the same variables with the same exponents. EXAMPLE 8 Simplify by combining like terms, if possible: a. 2x + 9x, b. 8p + ( 2p) + 4p, c. 0.5s 3 0.3s 3, d. 4w + 6, e. (4/9)b + (7/9)b EXAMPLE 8 Strategy We will use the distributive property in reverse to add (or subtract) the coefficients of the like terms. We will keep the same variables raised to the same powers. Why To combine like terms means to add or subtract the like terms in an expression. 43