Review of Basic Algebraic Concepts

Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable. Applications of Linear Equations in One Variable.6 Literal Equations and Applications to Geometry.7 Linear Inequalities in One Variable.8 Properties of Integer Exponents and Scientific Notation In Chapter, we present operations on real numbers, solving equations, and applications. This puzzle will help you familiarize yourself with some basic terms and geometry formulas. You will use these terms and formulas when working the exercises in Sections. and.6. Across. Given that distance rate time, what is the distance from Oklahoma City to Boston if it takes 8 hr of driving 60 mph?. What is the sum of the measures of the angles of a triangle? 6. What number is twice 69? 8. If the measure of an angle is 8º, what is the measure of its supplement? 0. If a rectangle has an area of 8670 ft and a width of 8 ft, what is its length? Down. What is the next consecutive even integer after 08?. If the measure of an angle is º, what is the measure of its complement?. What is % of,6?. What is the next consecutive integer after 0? 7. What is 0% of 87,0? 9. What is the next consecutive odd integer after? 6 7 8 9 0

Chapter Review of Basic Algebraic Concepts Section. Concepts. Set of Real Numbers. Inequalities. Interval Notation. Union and Intersection of Sets. Translations Involving Inequalities Sets of Numbers and Interval Notation. Set of Real Numbers Algebra is a powerful mathematical tool that is used to solve real-world problems in science, business, and many other fields. We begin our study of algebra with a review of basic definitions and notations used to express algebraic relationships. In mathematics, a collection of elements is called a set, and the symbols { } are used to enclose the elements of the set. For example, the set {a, e, i, o, u} represents the vowels in the English alphabet. The set {,,, 7} represents the first four positive odd numbers. Another method to express a set is to describe the elements of the set by using set-builder notation. Consider the set {a, e, i, o, u} in set-builder notation. description of set Set-builder notation: {x x is a vowel in the English alphabet} the set of all x such that x is a vowel in the English alphabet Consider the set {,,, 7} in set-builder notation. description of set Set-builder notation: {x x is an odd number between 0 and 8} the set of all x such that x is an odd number between 0 and 8 Several sets of numbers are used extensively in algebra. The numbers you are familiar with in day-to-day calculations are elements of the set of real numbers. These numbers can be represented graphically on a horizontal number line with a point labeled as 0. Positive real numbers are graphed to the right of 0, and negative real numbers are graphed to the left. Each point on the number line corresponds to exactly one real number, and for this reason, the line is called the real number line (Figure -). 0 Negative numbers Positive numbers Figure - Several sets of numbers are subsets (or part) of the set of real numbers. These are The set of natural numbers The set of whole numbers The set of integers The set of rational numbers The set of irrational numbers

Section. Sets of Numbers and Interval Notation Definition of the Natural Numbers, Whole Numbers, and Integers The set of natural numbers is {,,,...}. The set of whole numbers is {0,,,,...}. The set of integers is {...,,,,0,,,,...}. The set of rational numbers consists of all the numbers that can be defined as a ratio of two integers. Definition of the Rational Numbers The set of rational numbers is p q 0 p and q are integers and q does not equal zero}. Example Identifying Rational Numbers Show that each number is a rational number by finding two integers whose ratio equals the given number. a. b. 8 c. 0.6 d. 0.87 7 Solution: a. 7 is a rational number because it can be expressed as the ratio of the integers and 7. b. 8 is a rational number because it can be expressed as the ratio of the integers 8 and 8 8. In this example we see that an integer is also a rational number. c. 0.6 represents the repeating decimal 0.6666666 and can be expressed as the ratio of and 0.6. In this example we see that a repeating decimal is a rational number. d. 0.87 is the ratio of 87 and 00 0.87 00. 87 In this example we see that a terminating decimal is a rational number. TIP: Any rational number can be represented by a terminating decimal or by a repeating decimal. Skill Practice Show that the numbers are rational by writing them as a ratio of integers.. 9 8. 0. 0.. 0. Some real numbers such as the number p (pi) cannot be represented by the ratio of two integers. In decimal form, an irrational number is a nonterminating, nonrepeating decimal. The value of p, for example, can be approximated as p.968979. However, the decimal digits continue indefinitely with no pattern. Other examples of irrational numbers are the square roots of nonperfect squares, such as and. Definition of the Irrational Numbers The set of irrational numbers is { x 0 x is a real number that is not rational}. Note: An irrational number cannot be written as a terminating decimal or as a repeating decimal. Skill Practice Answers. 9 0. 8.. 00

Chapter Review of Basic Algebraic Concepts The set of real numbers consists of both the rational numbers and the irrational numbers. The relationships among the sets of numbers discussed thus far are illustrated in Figure -. Real numbers 7 Integers,, Whole numbers 0 Rational numbers 0. 0. Irrational numbers 7 Natural numbers,,, Figure - Example Classifying Numbers by Set Check the set(s) to which each number belongs. The numbers may belong to more than one set. 6 Natural Whole Rational Irrational Real Numbers Numbers Integers Numbers Numbers Numbers 7. Solution: Natural Whole Rational Irrational Real Numbers Numbers Integers Numbers Numbers Numbers 6 7.

Section. Sets of Numbers and Interval Notation Skill Practice. Check the set(s) to which each number belongs. Natural Whole Integer Rational Irrational Real 0.7. Inequalities The relative size of two numbers can be compared by using the real number line. We say that a is less than b (written mathematically as a b) if a lies to the left of b on the number line. a a b b We say that a is greater than b (written mathematically as a b) if a lies to the right of b on the number line. b a b a Table - summarizes the relational operators that compare two real numbers a and b. Table - Mathematical Expression Translation Other Meanings a 6 b a is less than b b exceeds a b is greater than a a 7 b a is greater than b a exceeds b b is less than a a b a is less than or equal to b a is at most b a is no more than b a b a is greater than or equal to b a is no less than b a is at least b a b a is equal to b a b a is not equal to b a b a is approximately equal to b The symbols,,,, and are called inequality signs, and the expressions a 6 b, a 7 b, a b, a b, and a b are called inequalities. Skill Practice Answers. 0.7 Natural Whole Integer Rational Irrational Real

6 Chapter Review of Basic Algebraic Concepts Example Ordering Real Numbers Fill in the blank with the appropriate inequality symbol: or a. b. c...8 7 Solution: a. 7 6 0 6 b. To compare 7 and, write the fractions as equivalent fractions with a common denominator. 7 0 and 7 7 Because 0 6, then 7 6 6 7 0 6. c.. 6.8 6 0.8 6 Skill Practice Fill in the blanks with the appropriate symbol, or. 6. 7. 8. 7..6 9. Interval Notation The set {x 0 x } represents all real numbers greater than or equal to. This set can be illustrated graphically on the number line. By convention, a closed circle or a square bracket [ is used to indicate that an endpoint (x ) is included in the set. This interval is a closed interval because its endpoint is included. The set {x 0 x } represents all real numbers strictly greater than. This set can be illustrated graphically on the number line. 0 0 0 0 ( By convention, an open circle or a parenthesis ( is used to indicate that an endpoint (x ) is not included in the set.this interval is an open interval because its endpoint is not included. Skill Practice Answers 6. 7. 8. Notice that the sets {x 0 x } and {x 0 x } consist of an infinite number of elements that cannot all be listed. Another method to represent the elements of such sets is by using interval notation. To understand interval notation, first consider the real number line, which extends infinitely far to the left and right. The symbol is used to represent infinity. The symbol is used to represent negative infinity. 0

Section. Sets of Numbers and Interval Notation 7 To express a set of real numbers in interval notation, sketch the graph first, using the symbols ( ) or [ ]. Then use these symbols at the endpoints to define the interval. Example Expressing Sets by Using Interval Notation 0 0 0 0 Graph the sets on the number line, and express the set in interval notation. a. {x x } b. {x x } c. {x x 6 Solution: a. Set-Builder Notation Graph Interval Notation {x x }, 0 [, ) The graph of the set {x 0 x } begins at and extends infinitely far to the right. The corresponding interval notation begins at and extends to. Notice that a square bracket [ is used at for both the graph and the interval notation to include x. A parenthesis is always used at (and at ) because there is no endpoint. b. Set-Builder Notation Graph Interval Notation {x 0 x }, 0 ( (, ) 0 c. Set-Builder Notation Graph Interval Notation {x x 6, 0 The graph of the set {x 0 x 6 extends infinitely far to the left. Interval notation is always written from left to right. Therefore, is written first, followed by a comma, and then followed by the right-hand endpoint. Skill Practice (, ] Graph and express the set, using interval notation. 9. w 0 w 76 0. x 0 x 6 06. y 0 y 7.6 Skill Practice Answers 9. 0.. 7 7, 0, 0 (., (.

8 Chapter Review of Basic Algebraic Concepts Using Interval Notation The endpoints used in interval notation are always written from left to right. That is, the smaller number is written first, followed by a comma, followed by the larger number. Parentheses ) or ( indicate that an endpoint is excluded from the set. Square brackets ] or [ indicate that an endpoint is included in the set. Parentheses are always used with and. Table - summarizes the solution sets for four general inequalities. Table - 0 Set-Builder Interval Notation Graph Notation ( x x 7 a6 a a, x 0 x a6 a a, x 0 x 6 a6 a (, a x 0 x a6 a, a. Union and Intersection of Sets Two or more sets can be combined by the operations of union and intersection. A Union B and A Intersection B The union of sets A and B, denoted A B, is the set of elements that belong to set A or to set B or to both sets A and B. The intersection of two sets A and B, denoted A B, is the set of elements common to both A and B. The concepts of the union and intersection of two sets are illustrated in Figures - and -: A B A B A B A union B The elements in A or B or both A B A intersection B The elements in A and B Figure - Figure - Example Finding the Union and Intersection of Sets Given the sets: A {a, b, c, d, e, f} B {a, c, e, g, i, k} C {g, h, i, j, k} Find: a. A B b. A B c. A C

Section. Sets of Numbers and Interval Notation 9 Solution: a. A B {a, b, c, d, e, f, g, i, k} The union of A and B includes all the elements of A along with all the elements of B. Notice that the elements a, c, and e are not listed twice. b. A B {a, c, e} The intersection of A and B includes only those elements that are common to both sets. c. A C { } (the empty set) Because A and C share no common elements, the intersection of A and C is the empty set or null set. TIP: The empty set may be denoted by the symbol { } or by the symbol. Skill Practice Given: A {r, s, t, u, v, w} B {s, v, w, y, z} C {x, y, z} Find:. B C. A B. A C Example 6 Finding the Union and Intersection of Sets Given the sets: A {x 0 x } B {x 0 x } C {x 0 x } Graph the following sets. Then express each set in interval notation. a. A B b. A C c. A B d. A C Solution: It is helpful to visualize the graphs of individual sets on the number line before taking the union or intersection. a. Graph of A {x 0 x } Graph of B {x 0 x } Graph of A B (the overlap ) Interval notation: [, ) 6 6 6 Note that the set A B represents the real numbers greater than or equal to and less than. This relationship can be written more concisely as a compound inequality: x. We can interpret this inequality as x is between and, including x. 0 0 0 ( ( 6 6 6 b. Graph of A {x 0 x } 6 0 ( 6 Graph of C {x 0 x } 6 0 6 Graph of A C 6 0 ( 6 Interval notation: (, ) [, ) A C includes all elements from set A along with the elements from set C. Skill Practice Answers. {s, v, w, x, y, z}. {s, v, w}. { }

0 Chapter Review of Basic Algebraic Concepts c. Graph of A {x 0 x } Graph of B {x 0 x } Graph of A B Interval notation: (, ) 6 6 6 0 0 0 A B includes all elements from set A along with the elements of set B. This encompasses all real numbers. ( 6 6 6 d. Graph of A {x 0 x } Graph of C {x 0 x } 6 6 0 0 ( 6 6 Graph of A C (the sets do not overlap ) A C is the empty set { }. 6 0 6 Skill Practice Find the intersection or union by first graphing the sets on the real number line. Write the answers in interval notation.. x 0 x 6 x 0 x 6 06 6. x 0 x 6 x 0 x 7 6 7. x 0 x 7 6 x 0 x 6 6 8. x 0 x 6 6 x 0 x 7 6. Translations Involving Inequalities In Table -, we learned that phrases such as at least, at most, no more than, no less than, and between can be translated into mathematical terms by using inequality signs. Example 7 Translating Inequalities The intensity of a hurricane is often defined according to its maximum sustained winds for which wind speed is measured to the nearest mile per hour. Translate the italicized phrases into mathematical inequalities. a. A tropical storm is updated to hurricane status if the sustained wind speed, w, is at least 7 mph. b. Hurricanes are categorized according to intensity by the Saffir-Simpson scale. On a scale of to, a category hurricane is the most destructive. A category hurricane has sustained winds, w, exceeding mph. c. A category hurricane has sustained winds, w, of at least mph but no more than mph. Solution: a. w 7 mph b. w mph c. mph w mph Skill Practice Answers. [, 0) 6. (, ] (, ) 7. (, ) 8. { } 9. m 0 0. m 7. 0 m 0 Skill Practice Translate the italicized phrase to a mathematical inequality. 9. The gas mileage, m, for a Honda Civic is at least 0 mpg. 0. The gas mileage, m, for a Harley Davidson motorcycle is more than mpg.. The gas mileage, m, for a Ford Explorer is at least 0 mpg, but not more than 0 mpg.

Section. Sets of Numbers and Interval Notation Section. Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos. In this text we will provide skills for you to enhance your learning experience. In the first four chapters, each set of Practice Exercises will begin with an activity that focuses on one of the following areas: learning about your course, using your text, taking notes, doing homework, and taking an exam. In subsequent chapters we will insert skills pertaining to the specific material in the chapter. In Chapter 7 we will give tips on studying for the final exam. To begin, write down the following information. a. Instructor s name b. Days of the week that the class meets c. The room number in which the d. Is there a lab requirement for this course? If so, what is the class meets requirement and what is the location of the lab? e. Instructor s office number f. Instructor s telephone number g. Instructor s email address h. Instructor s office hours. Define the key terms. a. Set b. Set-builder notation c. Real numbers d. Real number line e. Subset f. Natural numbers g. Whole numbers h. Integers i. Rational numbers j. Irrational numbers k. Inequality l. Interval notation m. Union n. Intersection o. Empty set Concept : Set of Real Numbers. Plot the numbers on the number line..7, p,,.6 6 0 6. Plot the numbers on the number line. e, 0,,, f 0 6 7 For Exercises 9, show that each number is a rational number by finding a ratio of two integers equal to the given number.. 0 6. 7. 8. 0. 9. 0

Chapter Review of Basic Algebraic Concepts 0. Check the sets to which each number belongs. 9.7 7 0 0. Real Irrational Rational Whole Natural Numbers Numbers Numbers Integers Numbers Numbers. Check the sets to which each number belongs. 6 8 Real Irrational Rational Whole Natural Numbers Numbers Numbers Integers Numbers Numbers p 0 0.8 8. Concept : Inequalities For Exercises 7, fill in the blanks with the appropriate symbol: 6 or 7.. 9. 0 6. 0. 0.0.. 0.6 6. 7. 0 7 7 Concept : Interval Notation For Exercises 8, express the set in interval notation. 8. ( 9. 0. ( 6 0... 9 0 ) (...8

Section. Sets of Numbers and Interval Notation For Exercises 6, graph the sets and express each set in interval notation. 6. x 0 x 7 6 7. x 0 x 6 6 8. y 0 y 6 9. z 0 z 6 0. w 0 w 6 9 6. p 0 p 7 6. x 0. 6 x.6. x 0 6 x 6 06. All real numbers less than.. All real numbers greater 6. All real numbers greater than. 7. All real numbers less than 7. than.. 8. All real numbers not 9. All real numbers no more 0. All real numbers between less than. than. and.. All real numbers between 7. All real numbers between. All real numbers between and. and 0, inclusive. and 6, inclusive. For Exercises, write an expression in words that describes the set of numbers given by each interval. (Answers may vary.). (, ). [, ) 6. (, 7] 7. (.9, 0) 8. [ 80, 90] 9. (, ) 0. (., ). (, ] Concept : Union and Intersection of Sets. Given: M {,,,, } and. Given: P a, b, c, d, e, f, g, h, i6 and N {,,,, 0}. Q a, e, i, o, u6. List the elements of the following sets: List the elements of the following sets. a. M N b. M N a. P Q b. P Q Let A x 0 x 7 6, B x 0 x 06, C x 0 x 6 6, and D x 0 6 x 6 6. For Exercises 6, graph the sets described here. Then express the answer in set-builder notation and in interval notation.. A B. A B 6. B C 7. B C 8. C D 9. C D 60. B D 6. A D

Chapter Review of Basic Algebraic Concepts Let X x 0 x 06, Y x 0 x 6 6, Z x 0 x 7 6, and W x 0 x 6. For Exercises 6 67, find the intersection or union of the sets X, Y, Z, and W. Write the answers in interval notation. 6. X Y 6. X Y 6. Y Z 6. Y Z 66. Z W 67. Z W Concept : Translations Involving Inequalities The following chart defines the ranges for normal blood pressure, high normal blood pressure, and high blood pressure (hypertension). All values are measured in millimeters of mercury, mm Hg. (Source: American Heart Association.) Normal Systolic less than 0 Diastolic less than 8 High normal Systolic 0 9 Diastolic 8 89 Hypertension Systolic 0 or greater Diastolic 90 or greater For Exercises 68 7, write an inequality using the variable p that represents each condition. 68. Normal systolic blood pressure 69. Diastolic pressure in hypertension 70. High normal range for systolic pressure 7. Systolic pressure in hypertension 7. Normal diastolic blood pressure A ph scale determines whether a solution is acidic or alkaline. The ph scale runs from 0 to, with 0 being the most acidic and being the most alkaline. A ph of 7 is neutral (distilled water has a ph of 7). For Exercises 7 77, write the ph ranges as inequalities and label the substances as acidic or alkaline. 7. Lemon juice:. through., inclusive 7. Eggs: 7.6 through 8.0, inclusive 7. Carbonated soft drinks:.0 through., inclusive 76. Milk: 6.6 through 6.9, inclusive 77. Milk of magnesia: 0.0 through.0, inclusive Expanding Your Skills For Exercises 78 89, find the intersection or union of the following sets. Write the answers in interval notation. 78.,, 7 79., 0, 80.,, 8. 6, 0, 9 8., 7, 8., 0, 8.,, 8. 6,, 86.,, 87.,, 88., 8 0, 89., 8 0,

0 Section. Operations on Real Numbers Operations on Real Numbers. Opposite and Absolute Value Several key definitions are associated with the set of real numbers and constitute the foundation of algebra. Two important definitions are the opposite of a real number and the absolute value of a real number. Definition of the Opposite of a Real Number Two numbers that are the same distance from 0 but on opposite sides of 0 on the number line are called opposites of each other. Symbolically, we denote the opposite of a real number a as a. Section. Concepts. Opposite and Absolute Value. Addition and Subtraction of Real Numbers. Multiplication and Division of Real Numbers. Exponential Expressions. Square Roots 6. Order of Operations 7. Evaluating Expressions The numbers and are opposites of each other. Similarly, the numbers and are opposites. Opposites 6 0 6 Opposites The absolute value of a real number a, denoted 0a 0, is the distance between a and 0 on the number line. Note: The absolute value of any real number is nonnegative. For example: 0 0 and 0 0 units units 6 0 6 Example Solution: a. 0. 0. b. 0 0 c. 0 0 Evaluating Absolute Value Expressions Simplify the expressions: a. 0. 0 b. c. 0 0 Skill Practice Simplify.. 0 9 0. 0 7.6 0. 0 0 0 6 units units. units 0 6 Calculator Connections Some calculators have an absolute value function. For example, The absolute value of a number a is its distance from zero on the number line. The definition of 0a 0 may also be given algebraically depending on whether a is negative or nonnegative. Skill Practice Answers. 9. 7.6.

6 Chapter Review of Basic Algebraic Concepts Definition of the Absolute Value of a Real Number Let a be a real number. Then. If a is nonnegative (that is, a 0, then 0a 0 a.. If a is negative (that is, a 6 0, then 0a 0 a. This definition states that if a is a nonnegative number, then 0a 0 equals a itself. If a is a negative number, then 0a 0 equals the opposite of a. For example, 09 0 9 Because 9 is positive, 09 0 equals the number 9 itself. 0 7 0 7 Because 7 is negative, 0 7 0 equals the opposite of 7, which is 7.. Addition and Subtraction of Real Numbers Addition of Real Numbers. To add two numbers with the same sign, add their absolute values and apply the common sign to the sum.. To add two numbers with different signs, subtract the smaller absolute value from the larger absolute value. Then apply the sign of the number having the larger absolute value. Example Adding Real Numbers Perform the indicated operations: a. ( 6) b. 0..8 c. Solution: a. 6 First find the absolute value of the addends. 0 0 and 0 6 0 6 6 6 Add their absolute values and apply the common sign (in this case, the common sign is negative). Common sign is negative. 8 b. 0..8 First find the absolute value of the addends. 0 0. 0 0. and 0.8 0.8 The absolute value of.8 is greater than the absolute value of 0.. Therefore, the sum is positive..8 0. Subtract the smaller absolute value from the larger absolute value.. Apply the sign of the number with the larger absolute value.

Section. Operations on Real Numbers 7 c. 6 a b 6 a b Write as an improper fraction. 6 a b 0 a b The LCD is. Write each fraction with the LCD. Find the absolute value of the addends. 0 ` ` 0 and ` ` The absolute value of is greater than the 0 absolute value of. Therefore, the sum is negative. a Subtract the smaller absolute value from the 0 b larger absolute value. Apply the sign of the number with the larger absolute value. Skill Practice Add....6.8 6. a 7 b Subtraction of real numbers is defined in terms of the addition process. To subtract two real numbers, add the opposite of the second number to the first number. Subtraction of Real Numbers If a and b are real numbers, then a b a ( b) Example Subtracting Real Numbers Perform the indicated operations. a. b..7 (.8) c. Solution: a. 8 Add the opposite of the second number to the first number. Add. Skill Practice Answers.. 0.8 6. 0 7

8 Chapter Review of Basic Algebraic Concepts b..7.8.7.8 6. Add the opposite of the second number to the first number. Add. c. a b a b 6 a 8 6 b Add the opposite of the second number to the first number. Write the mixed number as a fraction. Get a common denominator and add. 6 or 6 Skill Practice Subtract. 7.. 8. 9. 6. Multiplication and Division of Real Numbers The sign of the product of two real numbers is determined by the signs of the factors. Multiplication of Real Numbers. The product of two real numbers with the same sign is positive.. The product of two real numbers with different signs is negative.. The product of any real number and zero is zero. Example Multiplying Real Numbers Multiply the real numbers. a. ()(.) b. c. 9 8 a b a 0 b Solution: a. ()(.) 0. Different signs. The product is negative. Skill Practice Answers 7.. 8. 9. 7

Section. Operations on Real Numbers 9 b. 9 8 8 Different signs. The product is negative. Simplify to lowest terms. c. a b a 0 b a 0 b a 0 b 0 0 Write the mixed number as a fraction. Same signs. The product is positive. Simplify to lowest terms. Skill Practice Multiply. 0.. a. a 8 b b Notice from Example (c) that 0 0. If the product of two numbers is, then the numbers are said to be reciprocals. That is, the reciprocal of a real number a is a. Furthermore, a a. TIP: A number and its reciprocal have the same sign. For example: a 0 b a 0 b and Recall that subtraction of real numbers was defined in terms of addition. In a similar way, division of real numbers can be defined in terms of multiplication. To divide two real numbers, multiply the first number by the reciprocal of the second number. For example: Multiply 0 or equivalently 0 Reciprocal Because division of real numbers can be expressed in terms of multiplication, the sign rules that apply to multiplication also apply to division. 0 0 0 0 a b Dividing two numbers of the same sign produces a positive quotient. Skill Practice Answers 0.. 8.

0 Chapter Review of Basic Algebraic Concepts 0 0 a b 0 0 Dividing two numbers of opposite signs produces a negative quotient. Division of Real Numbers Assume that a and b are real numbers such that b 0.. If a and b have the same signs, then the quotient is positive.. If a and b have different signs, then the quotient is negative. 0. b 0. b. is undefined. 0 a b a b The relationship between multiplication and division can be used to investigate properties and in the preceding box. For example, 0 6 6 0 0 Because 6 0 0 is undefined Because there is no number that when multiplied by 0 will equal 6 Note: The quotient of 0 and 0 cannot be determined. Evaluating an expression of the form 0? is equivalent to asking, What number times zero will 0 equal 0? That is, (0)(?) 0. Any real number will satisfy this requirement; 0 however, expressions involving 0 are usually discussed in advanced mathematics courses. Example Dividing Real Numbers Divide the real numbers. Write the answer as a fraction or whole number. 96 a. b. c. d. 7 7 0 a b Solution: a. 6 Different signs. The quotient is negative. 7 TIP: If the numerator and denominator are both negative, then the fraction is positive: 7 7 TIP: Recall that multiplication may be used to check a division problem. For example: 7 6 7 6 96 b. Same signs. The quotient is positive. Simplify. c. Same signs. The quotient is positive. 7 7

Section. Operations on Real Numbers d. 0 a b 0 a b a 0 b Write the mixed number as an improper fraction, and multiply by the reciprocal of the second number. Different signs. The quotient is negative. TIP: If the numerator and denominator of a fraction have opposite signs, then the quotient will be negative. Therefore, a fraction has the same value whether the negative sign is written in the numerator, in the denominator, or in front of a fraction. Skill Practice Divide. 8... 6.. Exponential Expressions To simplify the process of repeated multiplication, exponential notation is often used. For example, the quantity can be written as ( to the fifth power). Definition of b n Let b represent any real number and n represent a positive integer. Then b n b b b b b n factors of b b n is read as b to the nth power. b is called the base and n is called the exponent, or power. b is read as b squared, and b is read as b cubed. Example 6 Evaluating Exponential Expressions Simplify the expression. a. b. ( ) c. d. a b Skill Practice Answers. 7.. 6. 6

Chapter Review of Basic Algebraic Concepts Solution: a. The base is, and the exponent is. b. The base is, and the exponent is. 6 The exponent applies to the entire contents of the parentheses. TIP: The quantity can also be interpreted as. 6 c. The base is, and the exponent is. 6 Because no parentheses enclose the negative sign, the exponent applies to only. d. a The base is, and the exponent is. b a b a b 9 Calculator Connections x On many calculators, the key is used to square a number. The ^ key is used to raise a base to any power. Skill Practice Simplify. 7. 8. 9. 0. a 0 0 b. Square Roots The inverse operation to squaring a number is to find its square roots. For example, finding a square root of 9 is equivalent to asking, What number when squared equals 9? One obvious answer is, because 9. However, is also a square root of 9 because 9. For now, we will focus on the principal square root which is always taken to be nonnegative. The symbol, called a radical sign, is used to denote the principal square root of a number. Therefore, the principal square root of 9 can be written as 9. The expression 6 represents the principal square root of 6. Example 7 Evaluating Square Roots Skill Practice Answers 7. 8 8. 00 7 9. 00 0. 6 Evaluate the expressions, if possible. a. 8 b. c. 6 A 6

Section. Operations on Real Numbers Solution: a. 8 9 because (9) 8 b. because A 6 8 a 8 b 6 c. 6 is not a real number because no real number when squared will be negative. Calculator Connections The key is used to find the square root of a nonnegative real number. Skill Practice Evaluate, if possible. 9... B 00 Example 7(c) illustrates that the square root of a negative number is not a real number because no real number when squared will be negative. The Square Root of a Negative Number Let a be a negative real number. Then a is not a real number. 6. Order of Operations When algebraic expressions contain numerous operations, it is important to evaluate the operations in the proper order. Parentheses ( ), brackets [ ], and braces { } are used for grouping numbers and algebraic expressions. It is important to recognize that operations must be done first within parentheses and other grouping symbols. Other grouping symbols include absolute value bars, radical signs, and fraction bars. Order of Operations. First, simplify expressions within parentheses and other grouping symbols. These include absolute value bars, fraction bars, and radicals. If embedded parentheses are present, start with the innermost parentheses.. Evaluate expressions involving exponents, radicals, and absolute values.. Perform multiplication or division in the order in which they occur from left to right.. Perform addition or subtraction in the order in which they occur from left to right. Example 8 Applying the Order of Operations Simplify the following expressions. a. 0 6 6 7 b. 0 0 Skill Practice Answers 7.. 0. Not a real number

Chapter Review of Basic Algebraic Concepts Solution: a. 0 6 6 7 TIP: Don t try to do too many steps at once. Taking a shortcut may result in a careless error. For each step rewrite the entire expression, changing only the operation being evaluated. 0 6 9 0 9 6 0 0 Simplify inside the parentheses and radical. Simplify exponents and radicals. Do multiplication and division from left to right. Do addition and subtraction from left to right. 0 0 b. Simplify numerator and denominator separately. 0 0 0 0 0 0 7 0 0 0 0 0 0 or Numerator: Simplify inner parentheses. Denominator: Do multiplication and division (left to right). Numerator: Simplify inner parentheses. Denominator: Multiply. Simplify exponents. Add within the absolute value. Evaluate the absolute value. Calculator Connections To evaluate the expression 0 0 on a graphing calculator, use parentheses to enclose the absolute value expression. Likewise, it is necessary to use parentheses to enclose the entire denominator. Skill Practice Simplify the expressions.. 6 8 6. 0 7 0 Skill Practice Answers. 7.

Section. Operations on Real Numbers 7. Evaluating Expressions The order of operations is followed when evaluating an algebraic expression or when evaluating a geometric formula. For a list of common geometry formulas, see the inside front cover of the text. It is important to note that some geometric formulas use Greek letters (such as p) and some formulas use variables with subscripts. A subscript is a number or letter written to the right of and slightly below a variable. Subscripts are used on variables to represent different quantities. For example, the area of a trapezoid is given by A b b h. The values of b and b (read as b sub and b sub ) represent the two different bases of the trapezoid (Figure -). This is illustrated in Example 9. b h Subscripts b Figure - Example 9 Evaluating an Algebraic Expression A homeowner in North Carolina wants to buy protective film for a trapezoidshaped window. The film will adhere to shattered glass in the event that the glass breaks during a bad storm. Find the area of the window whose dimensions are given in Figure -6. b. ft b.0 ft h.0 ft Figure -6 Solution: A b b h.0 ft. ft.0 ft 6. ft.0 ft 6. ft The area of the window is 6. ft. Substitute b.0 ft, b. ft, and h.0 ft. Simplify inside parentheses. Multiply from left to right. TIP: Subscripts should not be confused with superscripts, which are written above a variable. Superscripts are used to denote powers. b b Skill Practice 6. Use the formula given in Example 9 to find the area of the trapezoid. b in. h 0 in. b in. Skill Practice Answers 6. The area is 8 in.

6 Chapter Review of Basic Algebraic Concepts Section. Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos. Sometimes you may run into a problem with homework or you may find that you are having trouble keeping up with the pace of the class. A tutor can be a good resource. Answer the following questions. a. Does your college offer tutoring? b. Is it free? c. Where should you go to sign up for a tutor? d. Is there tutoring available online?. Define the key terms: a. Opposite b. Absolute value c. Reciprocal d. Base e. Exponent f. Power g. Principal square root h. Radical sign i. Order of operations j. Subscript Review Exercises For Exercises 6, describe the set.. Rational numbers Integers. Rational numbers Irrational numbers. Natural numbers {0} 6. Integers Whole numbers Concept : Opposite and Absolute Value 7. If the absolute value of a number can be thought of as its distance from zero, explain why an absolute value can never be negative. 8. If a number is negative, then its opposite will be a. Positive b. Negative. 9. If a number is negative, then its reciprocal will be a. Positive b. Negative. 0. If a number is negative, then its absolute value will be a. Positive b. Negative.. Complete the table. Number Opposite Reciprocal Absolute Value 6 8 0 0 0.

Section. Operations on Real Numbers 7. Complete the table. Number Opposite Reciprocal Absolute Value 9 0 9 0 0 0 0 0 For Exercises 0, fill in the blank with the appropriate symbol 6, 7,.. 06 0 6. 0. 6. 0 7. 0 8. 0 0 0 0 0 7 0 9. 0 0 0 0 0 0 0. 0 0 0 0 0 0 Concept : Addition and Subtraction of Real Numbers For Exercises 6, add or subtract as indicated.. 8. 7. 7.. 7 0 6. 7. 9 8. 9. 6 0... 9.6.... 6. 9 a b 7 a 7 b 8.8 0 6 9 Concept : Multiplication and Division of Real Numbers For Exercises 7 0, perform the indicated operation. 7. 8 8. 9. 0. 9 7... 7 0. a 7 b 8. 0 6. 0 7... 8. a 9 b a 7 b 6 0.6. 9..8 0.9 0. 6.9 7. Concept : Exponential Expressions For Exercises 8, evaluate the expressions.... 7.. 6. 7. 8. a 0 a 7 b 9 b

8 Chapter Review of Basic Algebraic Concepts Concept : Square Roots For Exercises 9 66, evaluate the expression, if possible. 9. 8 60. 6. 6. 9 6. 6. 6. 9 66. B B 6 00 Concept 6: Order of Operations For Exercises 67 9, simplify by using the order of operations. 67. 68. a a 69. 70. b b 0 7. 7. 7. 7. 7. 6 0 76. 8 77. 78. 8 79. 80. 8. a 9 9 8. b 9 7 0 a 9 b 6 0 8 8..7 0.. 8.. 0. 0.09 8. 86. 8 6 87. 0 0 07 0 88. 0 8 0 8 89. 90. 7 6 7 6 6 8 0 9. a 9. b a 6 b a 0 b a b a 8 b For Exercises 9 9, find the average of the set of data values by adding the values and dividing by the number of values. 9. Find the average low temperature for a week in January in St. John s, Newfoundland. (Round to the nearest tenth of a degree.) Day Mon. Tues. Wed. Thur. Fri. Sat. Sun. Low temperature 8 C 6 C 0 C C C C C 9. Find the average high temperature for a week in January in St. John s, Newfoundland. (Round to the nearest tenth of a degree.) Day Mon. Tues. Wed. Thur. Fri. Sat. Sun. High temperature C 6 C 7 C 0 C C 8 C 0 C Concept 7: Evaluating Expressions 9. The formula C 9 F converts temperatures in the Fahrenheit scale to the Celsius scale. Find the equivalent Celsius temperature for each Fahrenheit temperature. a. 77 F b. F c. F d. 0 F 96. The formula F 9 C converts Celsius temperatures to Fahrenheit temperatures. Find the equivalent Fahrenheit temperature for each Celsius temperature. a. C b. 0 C c. 7 C d. 0 C

Section. Operations on Real Numbers 9 Use the geometry formulas found in the inside front cover of the book to answer Exercises 97 06. For Exercises 97 00, find the area. 97. Trapezoid 98. Parallelogram 99. Triangle 00. Rectangle in. 8. m in. 6 m. cm yd in.. cm 76 yd For Exercises 0 06, find the volume. (Use the key on your calculator, and round the final answer to decimal place.) 0. Sphere 0. Right circular cone 0. Right circular cone r. ft h cm h. ft r cm r. ft 0. Sphere 0. Right circular cylinder 06. Right circular cylinder r yd h in. r in. h 9. m r m Graphing Calculator Exercises 07. Which expression when entered into a graphing calculator will yield the correct value of 6 or 6 08. Which expression when entered into a graphing calculator will yield the correct value of 6 or 6 09. Verify your solution to Exercise 8 by entering the expression into a graphing calculator: 0 8 0. Verify your solution to Exercise 86 by entering the expression into a graphing calculator: 6 7 6 6? 6?

0 Chapter Review of Basic Algebraic Concepts Section. Concepts. Recognizing Terms, Factors, and Coefficients. Properties of Real Numbers. Simplifying Expressions Simplifying Expressions. Recognizing Terms, Factors, and Coefficients An algebraic expression is a single term or a sum of two or more terms. A term is a constant or the product of a constant and one or more variables. For example, the expression 6x xyz or 6x xyz consists of the terms 6x, xyz, and. The terms 6x and xyz are variable terms, and the term is called a constant term. It is important to distinguish between a term and the factors within a term. For example, the quantity xyz is one term, but the values, x, y, and z are factors within the term. The constant factor in a term is called the numerical coefficient or simply coefficient of the term. In the terms 6x, xyz, and, the coefficients are 6,, and, respectively. A term containing only variables such as xy has a coefficient of. Terms are called like terms if they each have the same variables and the corresponding variables are raised to the same powers. For example: Like Terms Unlike Terms 6t and t 6t and s (different variables).8ab and ab.8xy and x (different variables) c d and c d c d and c d (different powers) and 6 p and 6 (different variables) Example Identifying Terms, Factors, Coefficients, and Like Terms a. List the terms of the expression. b. Identify the coefficient of the term. c. Identify the pair of like terms. x 7x 6b, b or c, 6 c Solution: a. The terms of the expression are x x 7x, 7x, and. b. The term yz can be written as yz ; therefore, the coefficient is. c. c, 6 c are like terms because they have the same variable raised to the same power. yz Skill Practice Given: x x y a. List the terms of the expression. b. Which term is the constant term? c. Identify the coefficient of the term y. Skill Practice Answers a. x, x,, y b. c.. Properties of Real Numbers Simplifying algebraic expressions requires several important properties of real numbers that are stated in Table -. Assume that a, b, and c represent real numbers or real-valued algebraic expressions.

Section. Simplifying Expressions Table - Algebraic Property Name Representation Example Description/Notes Commutative property a b b a The order in which of addition two real numbers are added or multiplied Commutative property a b b a does not affect the of multiplication result. Associative property a b c 7 The manner in which of addition a b c 7 two real numbers are grouped under Associative property of multiplication a b c a b c 7 7 addition or multiplication does not affect the result. Distributive property ab c A factor outside the of multiplication ab ac parentheses is over addition multiplied by each term inside the parentheses. Identity property 0 is the identity 0 0 Any number added of addition element for to the identity addition because a 0 0 a a element 0 will remain unchanged. Identity property is the identity Any number multiplied of multiplication element for by the identity multiplication element will because remain unchanged. a a a Inverse property a and a are 0 The sum of a number of addition additive inverses and its additive because a a 0 inverse (opposite) is the identity and element 0. a a 0 Inverse property of a and a are The product of a multiplication multiplicative number and its inverses because multiplicative inverse (reciprocal) a is the identity a and element. a a (provided a 0) The properties of real numbers are used to multiply algebraic expressions. To multiply a term by an algebraic expression containing more than one term, we apply the distributive property of multiplication over addition. Example Applying the Distributive Property Apply the distributive property. a. x b..q.7r c. a b c d. a 9x 8 y b

Chapter Review of Basic Algebraic Concepts Solution: a. x x 8x 0 Apply the distributive property. Simplify, using the associative property of multiplication. b..q.7r.q.7r The negative sign preceding the parentheses can be interpreted as a factor of..q.7r Apply the distributive property..q.7r TIP: When applying the distributive property, a negative factor preceding the parentheses will change the signs of the terms within the parentheses. c. d. a b c a b c a 6b c a 9x 8 y b Apply the distributive property. Simplify. a b c a 6b c 9x a b a 8 yb a b 8 x 6 y 0 6x y 0 Apply the distributive property. Simplify. Reduce to lowest terms. Skill Practice. x y 6. Apply the distributive property.. 00y 0. 7t.6s 9. a 7 Notice that the parentheses are removed after the distributive property is applied. Sometimes this is referred to as clearing parentheses. Two terms can be added or subtracted only if they are like terms. To add or subtract like terms, we use the distributive property, as shown in Example. Skill Practice Answers. 00y 00. 7t.6s 9.. 8x 6y Example Using the Distributive Property to Add and Subtract Like Terms Add and subtract as indicated. a. 8x x b..7y 9.y y. a 7

Section. Simplifying Expressions Solution: a. b. 8x x x 8 Apply the distributive property. x Simplify. x.7y 9.y y.7y 9.y y Notice that is interpreted as y. y.7 9. y. Apply the distributive property. Simplify..y Skill Practice Combine like terms. 6. y 7y 7. a 6a a y Although the distributive property is used to add and subtract like terms, it is tedious to write each step. Observe that adding or subtracting like terms is a matter of combining the coefficients and leaving the variable factors unchanged. This can be shown in one step. This shortcut will be used throughout the text. For example: w 7w w 8ab 0ab ab ab. Simplifying Expressions Clearing parentheses and combining like terms are important tools to simplifying algebraic expressions. This is demonstrated in Example. Example Clearing Parentheses and Combining Like Terms Simplify by clearing parentheses and combining like terms. a. x 8 b. s t t 8s 0s c..x.7x.x x d. w 6 a w b Solution: a. x 8 6x 6x 7 6x 6x 7 Apply the distributive property. Group like terms. Combine like terms. TIP: The expression 7 6x is equal to 6x 7. However, it is customary to write the variable term first. Skill Practice Answers 6. y 7. a

Chapter Review of Basic Algebraic Concepts b. c. s t t 8s 0s s t 0t 0s 0s s 0s 0s t 0t s t.x.7x.x x.x.7x.x x.x.x x.7x.9x.7x.88x 9.x 9.x.88x Apply the distributive property. Group like terms. Combine like terms. Apply the distributive property to inner parentheses. Group like terms. Combine like terms. Apply the distributive property. TIP: By using the commutative property of addition, the expression.88x 9.x can also be written as 9.x.88x or simply 9.x.88x. Although the expressions are all equal, it is customary to write the terms in descending order of the powers of the variable. d. w 6 a w b w 6 w Apply the distributive property. w w Reduce fractions to lowest terms. w w w Group like terms and find a common denominator. Combine like terms. Skill Practice Simplify by clearing parentheses and combining like terms. 8. 7 x 9. 6z 0y z y 8y 0..a. p.a 6a.a p Skill Practice Answers 8. 6x 0 9. y 8z 0..7a 7.a. 9 p

Section. Simplifying Expressions Section. Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor. It is very important to attend class every day. Math is cumulative in nature and you must master the material learned in the previous class to understand a new day s lesson. Because this is so important, many instructors tie attendance to the final grade. Write down the attendance policy for your class.. Define the key terms. a. Term b. Variable term c. Constant term d. Factor e. Coefficient f. Like terms e-professors Videos Review Exercises For Exercises, a. Classify the number as a whole number, natural number, rational number, irrational number, or real number. (Choose all that apply.) b. Write the reciprocal of the number. c. Write the opposite of the number. d. Write the absolute value of the number... 0 For Exercises 8, write the set in interval notation.. x 0 x 7 0 06 6. e x 0 x ` `f 7. e w 0 8. 6 w 9 f e z 0 z 6 f Concept : Recognizing Terms, Factors, and Coefficients For Exercises 9 : a. Determine the number of terms in the expression. b. Identify the constant term. c. List the coefficients of each term. Separate by commas. 9. x xy 6 0. a ab b 8. pq 7 q q p. 7x xy

6 Chapter Review of Basic Algebraic Concepts Concept : Properties of Real Numbers For Exercises, match each expression with the appropriate property... 7. 7. 7. a. Commutative property of addition. 6 8 6 8 6. 9 7 9 7 7. 9 9 8. a b 0 0 9. 6 6 0. 6x 6x 8. y 0 0 y. 7 7 b. Associative property of multiplication c. Distributive property of multiplication over addition d. Commutative property of multiplication e. Associative property of addition Concept : Simplifying Expressions For Exercises 60, clear parentheses and combine like terms.. 8y x y y. 9a a b a. 6. 6q 9 q q 0 7. p 7p p 6p 8. 9. m n n 9 0. x y x 8y.. 6m n mn m n. xy y xy.. 8x 6. b 7. 8. z z 9. 0w 9 w 0.. 9 z. w.. t t 7. w 8w w 6. 7. 8x x x 6 8. 6y y 9. 0. d 6 d..x.x. p p p 6 p a a 7a 6a ab ab 8a 9uv u uv u c c y 7 y s 7 s t 7 t 9 c c. y.9y. c a. a b a a d c ab b b 6b d. y y y y 6. 7.. 86x.x 6 7.x6 8. x 6 x x. 6.y 9 y. 7y6 9. 60. 8 n 6m m 8n a 0b 7 a b 7

Section. Linear Equations in One Variable 7 Expanding Your Skills 6. What is the identity element for addition? Use it in an example. 6. What is the identity element for multiplication? Use it in an example. 6. What is another name for a multiplicative inverse? 6. What is another name for an additive inverse? 6. Is the operation of subtraction commutative? If not, give an example. 66. Is the operation of division commutative? If not, give an example. 67. Given the rectangular regions: x A x B C y z y z a. Write an expression for the area of region A. (Do not simplify.) b. Write an expression for the area of region B. c. Write an expression for the area of region C. d. Add the expressions for the area of regions B and C. e. Show that the area of region A is equal to the sum of the areas of regions B and C. What property of real numbers does this illustrate? Linear Equations in One Variable. Definition of a Linear Equation in One Variable An equation is a statement that indicates that two quantities are equal. The following are equations. x p z 0 All equations have an equal sign. Furthermore, notice that the equal sign separates the equation into two parts, the left-hand side and the right-hand side. A solution to an equation is a value of the variable that makes the equation a true statement. Substituting a solution to an equation for the variable makes the righthand side equal to the left-hand side. Section. Concepts. Definition of a Linear Equation in One Variable. Solving Linear Equations. Clearing Fractions and Decimals. Conditional Equations, Contradictions, and Identities

8 Chapter Review of Basic Algebraic Concepts Equation Solution Check x p z 0 8 0 x p 8 z 0 0 0 Substitute for x. Right-hand side equals lefthand side. Substitute 8 for p. Right-hand side equals lefthand side. Substitute 0 for z. Right-hand side equals lefthand side. Throughout this text we will learn to recognize and solve several different types of equations, but in this chapter we will focus on the specific type of equation called a linear equation in one variable. Definition of a Linear Equation in One Variable Let a and b be real numbers such that a 0. A linear equation in one variable is an equation that can be written in the form ax b 0 Notice that a linear equation in one variable has only one variable. Furthermore, because the variable has an implied exponent of, a linear equation is sometimes called a first-degree equation. Linear equation in one variable x 0 p 0 0 Not a linear equation in one variable x 8 0 p 0 q 0 (exponent for x is not ) (more than one variable). Solving Linear Equations To solve a linear equation, the goal is to simplify the equation to isolate the variable. Each step used in simplifying an equation results in an equivalent equation. Equivalent equations have the same solution set. For example, the equations x 7 and x are equivalent because x is the solution to both equations. To solve an equation, we may use the addition, subtraction, multiplication, and division properties of equality. These properties state that adding, subtracting, multiplying, or dividing the same quantity on each side of an equation results in an equivalent equation. Addition and Subtraction Properties of Equality Let a, b, and c represent real numbers. Addition property of equality: If a b, then a c b c. Subtraction property of equality: If a b, then a c b c.

Section. Linear Equations in One Variable 9 Multiplication and Division Properties of Equality Let a, b, and c represent real numbers. Multiplication property of equality: If a b, then a c b c. Division property of equality: If a b, a then c b provided c 0. c Example Solving Linear Equations Solve each equation. a. x 0 b. c. w d. p. x 6 Solution: a. x 0 x 0 x 8 Check: x 0 8 0 0 0 To isolate x, subtract from both sides. Simplify. Check the solution in the original equation. True statement b. p a pb p 0 To isolate p, multiply both sides by. Simplify. Check: p Check the solution in the original equation. 0 c. d. w. 8.8 w Check: w. 8.8. x 6 x 6 x 6. a w. b. True statement To isolate w, multiply both sides by.. Simplify. Check the solution in the original equation. True statement To isolate x, multiply both sides by. Simplify.

0 Chapter Review of Basic Algebraic Concepts Check: x 6 6 6 6 6 Check the solution in the original equation. True statement Skill Practice Solve the equations.. x. 6. t. a y 6 For more complicated linear equations, several steps are required to isolate the variable. Steps to Solve a Linear Equation in One Variable. Simplify both sides of the equation. Clear parentheses. Consider clearing fractions or decimals (if any are present) by multiplying both sides of the equation by a common denominator of all terms. Combine like terms.. Use the addition or subtraction property of equality to collect the variable terms on one side of the equation.. Use the addition or subtraction property of equality to collect the constant terms on the other side of the equation.. Use the multiplication or division property of equality to make the coefficient of the variable term equal to.. Check your answer. Example Solving Linear Equations Solve the linear equations and check the answers. a. z z b. x 7 x c. y y 6 y Skill Practice Answers. x 6. y. t 80. a Solution: a. z z z z 0 z z z z 0 6z 0 6z 0 6z 6z 6 6 z Clear parentheses. Subtract z from both sides. Combine like terms. Subtract from both sides. To isolate z, divide both sides of the equation by 6. Simplify.

Section. Linear Equations in One Variable Check: z z 0 0 Check the solution in the original equation. True statement b. c. x 7 x x 7 x Check: x 7 x Check: x x 6 x x x x 6 x 6 x 6 x 8 x 8 x 7 0 7 0 8y 60 8y 7 8y 8 7 8 y y y 6 y y y 0y y 0y 8y 60 0y 8y 0y 60 0y 0y 8y 60 60 60 y y 6 y 6 6 0 7 8 8 8 Clear parentheses. Combine like terms. Add x to both sides of the equation. Combine like terms. Subtract from both sides. To isolate x, divide both sides by. Simplify. Check the solution in the original equation. True statement Clear parentheses. Combine like terms. Clear parentheses. Add 0y to both sides of the equation. Combine like terms. Add 60 to both sides of the equation. To isolate y, divide both sides by 8. Simplify. True statement

Chapter Review of Basic Algebraic Concepts Skill Practice Solve the equations.. 7 y 6y 6. 7. p p p t 6t 6 t. Clearing Fractions and Decimals When an equation contains fractions or decimals, it is sometimes helpful to clear the fractions and decimals. This is accomplished by multiplying both sides of the equation by the least common denominator (LCD) of all terms within the equation. This is demonstrated in Example. Example Solving Linear Equations by Clearing Fractions Solve the equation. Solution: w w w w w w w w w Apply the distributive property to clear parentheses. Check: a w w b a w b w w w w w 6w 7w 6w 7w 6w 6w 6w w w w w w w Multiply both sides of the equation by the LCD of all terms. In this case, the LCD is. Apply the distributive property. Subtract 6w Add to both sides. Skill Practice Answers 6. y 6. 7. p 0 t 8 8 8 True statement

Section. Linear Equations in One Variable TIP: The fractions in this equation can be eliminated by multiplying both sides of the equation by any common multiple of the denominators. For example, multiplying both sides of the equation by produces the same solution. a w w b w 6w 8w w w w 8 w w Skill Practice 8. a a Solve the equation by first clearing the fractions. Example Solving a Linear Equation with Fractions Solve. x x x 0 Solution: x 0 a x x x 0 x b 0 a x 0 b The LCD of all terms in the equation is 0. Multiply both sides by 0. 0 ax b 0 ax b 0 a b 0 ax 0 b x x 0 x x x 0 0 x x 6 x x x 6 x x x 6 x 6 6 6 x 8 Clear fractions. Apply the distributive property. Apply the distributive property. Simplify both sides of the equation. Subtract x from both sides. Subtract 6 from both sides. Skill Practice Answers 8. a

Chapter Review of Basic Algebraic Concepts x 8 x Divide both sides by. The check is left to the reader. Skill Practice 9. Solve. 8 x x The same procedure used to clear fractions in an equation can be used to clear decimals. Example Solve the equation. Solving Linear Equations by Clearing Decimals 0.x 0.6.0x Solution: Recall that any terminating decimal can be written as a fraction. Therefore, the equation 0.x 0.6.0x is equivalent to 00 x 6 0 0 00 x A convenient common denominator for all terms in this equation is 00. Multiplying both sides of the equation by 00 will have the effect of moving the decimal point places to the right. 000.x 0.6 00.0x x 60 0x x x 60 0x x 60 0x 60 0 0x 0 60 0 x Multiply both sides by 00 to clear decimals. Subtract x from both sides. To isolate x, divide both sides by 0. Check: x 0. 0.x 0.6.0x 0. 0. 0.6.0 0. 0. 0.6 0.8 0.8 0.8 True statement Skill Practice Answers 9. x 0. x 0. Skill Practice 0..x 0..6x 0. Solve the equation by first clearing the decimals.

Section. Linear Equations in One Variable. Conditional Equations, Contradictions, and Identities The solution to a linear equation is the value of x that makes the equation a true statement. A linear equation has one unique solution. Some equations, however, have no solution, while others have infinitely many solutions. I. Conditional Equations An equation that is true for some values of the variable but false for other values is called a conditional equation. The equation x 6 is a conditional equation because it is true on the condition that x. For other values of x, the statement x 6 is false. II. Contradictions Some equations have no solution, such as x x. There is no value of x that when increased by will equal the same value increased by. If we tried to solve the equation by subtracting x from both sides, we get the contradiction. This indicates that the equation has no solution. An equation that has no solution is called a contradiction. III. Identities An equation that has all real numbers as its solution set is called an identity. For example, consider the equation x x. Because the left- and righthand sides are identical, any real number substituted for x will result in equal quantities on both sides. If we solve the equation, we get the identity. In such a case, the solution is the set of all real numbers. Identifying Conditional Equations, Contradictions, and Identities Solve the equations. Identify each equation as a conditional equation, a contradiction, or an identity. a. x x b. c c c 7 c. Example 6 x x x x x 7 x x Solution: a. x x x x x x x x contradiction No solution identity The solution is all real numbers. x x Clear parentheses. Combine like terms. Contradiction This equation is a contradiction. There is no solution.

6 Chapter Review of Basic Algebraic Concepts b. c c c 7 c c 7 Clear parentheses and combine like terms. c 7 c 7 Identity 0 0 This equation is an identity. The solution is the set of all real numbers. Skill Practice Answers. The equation is a contradiction. There is no solution.. The equation is an identity. The solution is the set of all real numbers.. The equation is conditional. The solution is x. c. Add to both sides. To isolate x, divide both sides by. This equation is a conditional equation. The solution is x. Skill Practice x 7 x 7 x 0 x 0 x Solve the equations. Identify each equation as a conditional equation, an identity, or a contradiction.. x x x 6.. x x 6x x 6x 8 Section. Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos. Some instructors allow the use of calculators. Does your instructor allow the use of a calculator? If so, what kind? Will you be allowed to use a calculator on tests or just for occasional calculator problems in the text? Helpful Hint: If you are not permitted to use a calculator on tests, you should do your homework in the same way, without the calculator.. Define the key terms. a. Equation b. Solution to an equation c. Linear equation in one variable d. Conditional equation e. Contradiction f. Identity Review Exercises For Exercises 6, clear parentheses and combine like terms.. 8x y xy x xy. ab a a 7. z z 6. 6w w

Section. Linear Equations in One Variable 7 Concept : Definition of a Linear Equation in One Variable For Exercises 7, label the equation as linear or nonlinear. 7. x 8. 0 x 6 9. 0. x x. x. x 7 9. 7x 0. Use substitution to determine which value is the solution to x. a. b. c. 0 d.. Use substitution to determine which value is the solution to y. a. b. c. 0 d. Concept : Solving Linear Equations For Exercises, solve the equations and check your solutions.. x 7 9 6. y 8 7. 6x 8. t 8 9. 7 8 6 z 0. b. a....t..8 6. y. p.9.8 6. 7. 6q 6 8. w 9. y 7 0.. b. 6 y. x 6 x. 8 x 7.a.9 6z 8 y y. 6 t t 6. 7. 6a 0 a 8. 9. z z 0.. 6 y y y y.. x x x 6. p p 8b b 9b w 0 w 7w w w 6 w 8 p 6p 7 p Concept : Clearing Fractions and Decimals For Exercises 6, solve the equations.. 6. 7. y 9 0 y x 6 x 6 x 8. x 7 9. x 6x 6 q 7 9 q 0.. q 6 a 9 q 6 q. a a 0 6 0 0. p p 0 p y y y 6.w..8. 0.x.6 x. 0.7m 0.m 0. 6. 0.n 0 0.6n

8 Chapter Review of Basic Algebraic Concepts Concept : Conditional Equations, Contradictions, and Identities 7. What is a conditional equation? 8. Explain the difference between a contradiction and an identity. For Exercises 9 6, solve the following equations. Then label each as a conditional equation, a contradiction, or an identity. 9. x x 60. x 6 x 6. x x x 6. x 7 6. x 8x 7x 8 x 6. 7x 8 x x Mixed Exercises For Exercises 6 96, solve the equations. 6. b 9 7 66. x 8 66 67. 68. 9x 69. 0c c 70. 7. b b 8 6 b 6 7. z z z 7. c 7. x x 7. 76. c c 8 6 0 x w 6w 7 x x x 7 d d 0 d 0 7 0 77. 0.78x 6x 9 78. z z 79. 7p p p 80. 6z z 8 z 8. b b 6 b 8. 9 8. 8. 8. 0 w x 9 y 9 86. 87. 88. x x 6 0 y x 6 x 6 y y 8 x 6 x x 89. 0.8x 0.08x 0.60 x 90. 0.07w 0.060 w 90 9. 0.x 0. x 9. 0.b 9. 0.b. 0.b 9. 7 0.7a 0. 0.7a 9. 7 96. 8 y a yb x 8 x x Expanding Your Skills 97. a. Simplify the expression. y y 98. a. Simplify the expression. b. Solve the equation. y y 0 b. Solve the equation. c. Explain the difference between simplifying an expression and solving an equation. w 8 w w 8 w 0

Section. Applications of Linear Equations in One Variable 9 Applications of Linear Equations in One Variable. Introduction to Problem Solving One of the important uses of algebra is to develop mathematical models for understanding real-world phenomena. To solve an application problem, relevant information must be extracted from the wording of a problem and then translated into mathematical symbols. This is a skill that requires practice. The key is to stick with it and not to get discouraged. Step Step Problem-Solving Flowchart for Word Problems Read the problem carefully. Assign labels to unknown quantities. Familiarize yourself with the problem. Ask yourself, What am I being asked to find? If possible, estimate the answer. Identify the unknown quantity or quantities. Let x represent one of the unknowns. Draw a picture and write down relevant formulas. Section. Concepts. Introduction to Problem Solving. Applications Involving Consecutive Integers. Applications Involving Percents and Rates. Applications Involving Principal and Interest. Applications Involving Mixtures 6. Applications Involving Distance, Rate, and Time Step Develop a verbal model. Write an equation in words. Step Write a mathematical equation. Replace the verbal model with a mathematical equation using x or another variable. Step Step 6 Solve the equation. Interpret the results and write the final answer in words. Solve for the variable, using the steps for solving linear equations. Once you ve obtained a numerical value for the variable, recall what it represents in the context of the problem. Can this value be used to determine other unknowns in the problem? Write an answer to the word problem in words. Example Translating and Solving a Linear Equation The sum of two numbers is 9. One number is less than twice the other. What are the numbers? Solution: Step : Read the problem carefully. Step : Let x represent one number. Let x represent the other number. Step : (One number) (other number) 9 Step : Replace the verbal model with a mathematical equation. (One number) (other number) 9 x (x ) 9

0 Chapter Review of Basic Algebraic Concepts Step : Solve for x. x x 9 x 9 x x x Step 6: Interpret your results. Refer back to step. One number is x: The other number is x : Answer: The numbers are and. Skill Practice (). One number is more than times another number. The sum of the numbers is. Find the numbers.. Applications Involving Consecutive Integers The word consecutive means following one after the other in order. The numbers,, 0,, are examples of consecutive integers. Notice that any two consecutive integers differ by. If x represents an integer, then x represents the next consecutive integer. The numbers,, 6, 8 are consecutive even integers. The numbers, 7, 9, are consecutive odd integers. Both consecutive odd and consecutive even integers differ by. If x represents an even integer, then x represents the next consecutive even integer. If x represents an odd integer, then x represents the next consecutive odd integer. Example Solving a Linear Equation Involving Consecutive Integers The sum of two consecutive odd integers is 7. Find the integers. Solution: Step : Read the problem carefully. Step : Label unknowns: Let x represent the first odd integer. Let x represent the next odd integer. Skill Practice Answers. The numbers are 0 and.

Section. Applications of Linear Equations in One Variable Step : Step : Write an equation in words: (First integer) (second integer) 7 Write a mathematical equation based on the verbal model. (First integer) (second integer) 7 x (x ) 7 Step : Solve for x. x x 7 x 7 x 70 x 8 Step 6: Interpret your results. One integer is x: 8 The other integer is x : 8 87 Answer: The numbers are 8 and 87. Skill Practice The sum of three consecutive integers is 66. a. If the first integer is represented by x, write expressions for the next two integers. b. Write a mathematical equation that describes the verbal model. c. Solve the equation and find the three integers. TIP: After completing a word problem, it is always a good idea to check that the answer is reasonable. Notice that 8 and 87 are consecutive odd integers, and the sum is equal to 7 as desired.. Applications Involving Percents and Rates In many real-world applications, percents are used to represent rates. In 006, the sales tax rate for one county in Tennessee was 6%. An ice cream machine is discounted 0%. A real estate sales broker receives a % commission on sales. A savings account earns 7% simple interest. The following models are used to compute sales tax, commission, and simple interest. In each case the value is found by multiplying the base by the percentage. Sales tax (cost of merchandise)(tax rate) Commission (dollars in sales)(commission rate) Simple interest (principal)(annual interest rate)(time in years) I Prt Skill Practice Answers a. x and x b. x x x 66 c. The integers are,, and.

Chapter Review of Basic Algebraic Concepts Example Solving a Percent Application A realtor made a 6% commission on a house that sold for $7,000. How much was her commission? Solution: Let x represent the commission. (Commission) (dollars in sales)(commission rate) x ($7,000)(0.06) Label the variables. Verbal model Mathematical model x $0,0 Solve for x. The realtor s commission is $0,0. Skill Practice Interpret the results.. The sales tax rate in Atlanta, Georgia, is 7%. Find the amount of sales tax paid on an automobile priced at $,000. Example Solving a Percent Application A woman invests $000 in an account that earns % simple interest. If the money is invested for years, how much money is in the account at the end of the -year period? Solution: Let x represent the total money in the account. P $000 r 0.0 t (principal amount invested) (interest rate) (time in years) The total amount of money includes principal plus interest. (Total money) (principal) (interest) x P Prt Label variables. Verbal model Mathematical model x $000 ($000)(0.0)() Substitute for P, r, x $000 $787.0 and t. x $787.0 Solve for x. The total amount of money in the account is $787.0. Skill Practice Interpret the results.. A man earned $0 in year on an investment that paid a % dividend. Find the amount of money invested. Skill Practice Answers. $80. $800 As consumers, we often encounter situations in which merchandise has been marked up or marked down from its original cost. It is important to note that percent increase and percent decrease are based on the original cost. For example, suppose a microwave originally priced at $0 is marked down 0%.

Section. Applications of Linear Equations in One Variable The discount is determined by 0% of the original price: (0.0)($0) $6.00. The new price is $0.00 $6.00 $.00. Example Solving a Percent Increase Application A college bookstore uses a standard markup of % on all books purchased wholesale from the publisher. If the bookstore sells a calculus book for $0.70, what was the original wholesale cost? Solution: Let x original wholesale cost. Label the variables. The selling price of the book is based on the original cost of the book plus the bookstore s markup. (Selling price) (original price) (markup) Verbal model (Selling price) (original price) (original price markup rate) 0.70 x (x)(0.) Mathematical model 0.70 x 0.x 0.70.x Combine like terms. 0.70. x x $8.00 Simplify. The original wholesale cost of the textbook was $8.00. Interpret the results. Skill Practice. An online bookstore gives a 0% discount on paperback books. Find the original price of a book that has a selling price of $.8 after the discount.. Applications Involving Principal and Interest Example 6 Solving an Investment Growth Application Miguel had $0,000 to invest in two different mutual funds. One was a relatively safe bond fund that averaged 8% return on his investment at the end of year. The other fund was a riskier stock fund that averaged 7% return in year. If at the end of the year Miguel s portfolio grew to $,7 ($7 above his $0,000 investment), how much money did Miguel invest in each fund? Solution: This type of word problem is sometimes categorized as a mixture problem. Miguel is mixing his money between two different investments. We have to determine how the money was divided to earn $7. Skill Practice Answers. $6.60

Chapter Review of Basic Algebraic Concepts The information in this problem can be organized in a chart. (Note: There are two sources of money: the amount invested and the amount earned.) 8% Bond Fund 7% Stock Fund Total Amount invested ($) x (0,000 x) 0,000 Amount earned ($) 0.08x 0.7(0,000 x) 7 Because the amount of principal is unknown for both accounts, we can let x represent the amount invested in the bond fund. If Miguel spends x dollars in the bond fund, then he has (0,000 x) left over to spend in the stock fund. The return for each fund is found by multiplying the principal and the percent growth rate. To establish a mathematical model, we know that the total return ($7) must equal the growth from the bond fund plus the growth from the stock fund: (Growth from bond fund) (growth from stock fund) (total growth) 0.08x 0.7(0,000 x) 7 0.08x 0.7(0,000 x) 7 Mathematical model 8x 7(0,000 x) 7,00 Multiply by 00 to clear decimals. 8x 70,000 7x 7,00 9x 70,000 7,00 Combine like terms. 9x,00 Subtract 70,000 from both sides. 9x 9,00 9 x 00 Solve for x and interpret the results. The amount invested in the bond fund is $00. The amount invested in the stock fund is $0,000 x, or $700. Skill Practice 6. Winston borrowed $000 in two loans. One loan charged 7% interest, and the other charged.% interest. After year, Winston paid $ in interest. Find the amount borrowed in each loan account.. Applications Involving Mixtures Example 7 Solving a Mixture Application How many liters (L) of a 60% antifreeze solution must be added to 8 L of a 0% antifreeze solution to produce a 0% antifreeze solution? Skill Practice Answers 6. $000 was borrowed at 7% interest, and $000 was borrowed at.% interest.

Section. Applications of Linear Equations in One Variable Solution: The given information is illustrated in Figure -7. 60% Antifreeze solution 0% Antifreeze solution 0% Antifreeze solution x L of solution 8 L of solution (8 x) L of solution 0.60x L of pure antifreeze 0.0(8) L of pure antifreeze Figure -7 0.0(8 x) L of pure antifreeze The information can be organized in a chart. Final Solution: 60% Antifreeze 0% Antifreeze 0% Antifreeze Number of liters of solution x 8 (8 x) Number of liters of pure antifreeze 0.60x 0.0(8) 0.0(8 x) Notice that an algebraic equation is derived from the second row of the table which relates the number of liters of pure antifreeze in each container. The amount of pure antifreeze in the final solution equals the sum of the amounts of antifreeze in the first two solutions. Pure antifreeze antifreeze a b apure from solution from solution b a pure antifreeze in the final solution b 0.60x 0.0(8) 0.0(8 x) 0.60x 0.0(8) 0.0(8 x) Mathematical model 0.6x 0.8.6 0.x Apply the distributive property. 0.6x 0.x 0.8.6 0.x 0.x Subtract 0.x from both sides. 0.x 0.8.6 0.x 0.8 0.8.6 0.8 Subtract 0.8 from both sides. 0.x 0.8 x Divide both sides by 0.. Answer: L of 60% antifreeze solution is necessary to make a final solution of 0% antifreeze. Skill Practice 0.x 0. 0.8 0. 7. Find the number of ounces (oz) of 0% alcohol solution that must be mixed with 0 oz of a 70% solution to obtain a solution that is 0% alcohol. Skill Practice Answers 7. 0 oz of the 0% solution is needed.

6 Chapter Review of Basic Algebraic Concepts 6. Applications Involving Distance, Rate, and Time The fundamental relationship among the variables distance, rate, and time is given by Distance (rate)(time) or d rt For example, a motorist traveling 6 mph (miles per hour) for hr (hours) will travel a distance of d (6 mph)( hr) 9 mi Example 8 Solving a Distance, Rate, Time Application A hiker can hike mph down a trail to visit Archuletta Lake. For the return trip back to her campsite (uphill), she is only able to go mph. If the total time for the round trip is hr 8 min (.8 hr), find a. The time required to walk down to the lake b. The time required to return back to the campsite c. The total distance the hiker traveled Solution: The information given in the problem can be organized in a chart. Distance (mi) Rate (mph) Time (hr) Trip to the lake. t Return trip. (.8 t) Column : The rates of speed going to and from the lake are given in the statement of the problem. Column : There are two unknown times. If we let t be the time required to go to the lake, then the time for the return trip must equal the total time minus t, or (.8 t) Column : To express the distance in terms of the time t, we use the relationship d rt. That is, multiply the quantities in the second and third columns. Distance (mi) Rate (mph) Time (hr) Trip to the lake.t. t Return trip.(.8 t). (.8 t) To create a mathematical model, note that the distances to and from the lake are equal. Therefore, (Distance to lake) (return distance) Verbal model.t.(.8 t) Mathematical model.t 7..t Apply the distributive property..t.t 7..t.t Add.t to both sides..0t 7..0t.0 7..0 t.8 Solve for t and interpret the results.

Section. Applications of Linear Equations in One Variable 7 Answers: a. Because t represents the time required to go down to the lake,.8 hr is required for the trip to the lake. b. The time required for the return trip is (.8 t) or (.8 hr.8 hr) hr. Therefore, the time required to return to camp is hr. c. The total distance equals the distance to the lake and back. The distance to the lake is (. mph)(.8 hr). mi. The distance back is (. mph) (.0 hr). mi. Therefore, the total distance the hiker walked is 9.0 mi. Skill Practice 8. Jody drove a distance of 0 mi to visit a friend. She drives part of the time at 0 mph and part at 60 mph. The trip took 6 hr. Find the amount of time she spent driving at each speed. Skill Practice Answers 8. Jody drove hr at 0 mph and hr at 60 mph. Section. Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor. Look through the text and write down a page number that contains: a. Avoiding Mistakes b. TIP box c. A key term (shown in bold). Define the key terms. a. Sales tax b. Commission c. Simple interest e-professors Videos Review Exercises For Exercises, solve the equations.. 7a. z 6. (x ) 7 9 6. (y ) 7. (b ) (b 8) b 8. c c 9 ( 7c) c 9. 0. 8 p p x. 0.08()d 0.07()d 0 For the remaining exercises, follow the steps outlined in the Problem-Solving Flowchart found on page 9. Concept : Introduction to Problem Solving. The larger of two numbers is more than twice the smaller. The difference of the larger number and the smaller number is 8. Find the numbers.. One number is less than another. Their sum is. Find the numbers.

8 Chapter Review of Basic Algebraic Concepts. The sum of times a number and is the same as the difference of the number and. Find the number.. Twice the sum of a number and is the same as subtracted from the number. Find the number. 6. The sum of two integers is 0. Ten times one integer is times the other integer. Find the integers. (Hint: If one number is x, then the other number is 0 x.) 7. The sum of two integers is 0. Three times one integer is less than 8 times the other integer. Find the integers. (Hint: If one number is x, then the other number is 0 x.) Concept : Applications Involving Consecutive Integers 8. The sum of two consecutive page numbers in a book is. Find the page numbers. 9. The sum of the numbers on two consecutive raffle tickets is 808,. Find the numbers on the tickets. 0. The sum of two consecutive odd integers is 8. Find the two integers.. Three times the smaller of two consecutive even integers is the same as 6 minus times the larger integer. Find the integers.. The sum of three consecutive integers is 7. Find the integers.. Five times the smallest of three consecutive even integers is 0 more than twice the largest. Find the integers. Concept : Applications Involving Percents and Rates. Leo works at a used car dealership and earns an 8% commission on sales. If he sold $9,000 in used cars, what was his commission?. Alysha works for a pharmaceutical company and makes 0.6% commission on all sales within her territory. If the yearly sales in her territory came to $8,00,000, what was her commission? 6. An account executive earns $600 per month plus a % commission on sales. The executive s goal is to earn $00 this month. How much must she sell to achieve this goal? 7. If a salesperson in a department store sells merchandise worth over $00 in one day, she receives a % commission on the sales over $00. If the sales total $ on one particular day, how much commission did she earn? 8. Molly had the choice of taking out a -year car loan at 8.% simple interest or a -year car loan at 7.7% simple interest. If she borrows $,000, which option will demand less interest? 9. Robert can take out a -year loan at 8% simple interest or a -year loan at 8% simple interest. If he borrows $7000, which option will demand less interest? 0. If Ivory Soap is 99 00% pure, then what quantity of impurities will be found in a bar of Ivory Soap that weighs. oz (ounces)?

Section. Applications of Linear Equations in One Variable 9. In the 996 presidential election, a third party candidate received a significant number of votes. The figure illustrates the number of votes received for Bill Clinton, Bob Dole, and Ross Perot in that election. Compute the percent of votes received by each candidate. (Round to the nearest tenth of a percent.) Number of votes by candidate 996 Presidential election Ross Perot 8 million. The total bill (including a 6% sales tax) to have a radio installed in a car came to $6. What was the cost before tax?. Wayne County has a sales tax rate of 7%. How much does Mike s Honda Civic cost before tax if the total cost of the car plus tax is $,888.60? Bob Dole 9 million Bill Clinton 7 million. The price of a swimsuit after a 0% markup is $.08. What was the price before the markup?. The price of a used textbook after a % markdown is $9.. What was the original price? 6. In 006, 9.6 million people lived below the poverty level in the United States. This represents an 80% increase from the number in 00. How many people lived below the poverty level in 00? 7. In 006, Americans spent approximately $69 billion on weddings. This represents a 0% increase from the amount spent in 00. What amount did Americans spend on weddings in 00? Concept : Applications Involving Principal and Interest 8. Darrell has a total of $,00 in two accounts. One account pays 8% simple interest per year, and the other pays % simple interest. If he earned $60 in the first year, how much did he invest in each account? 9. Lillian had $,000 invested in two accounts, one paying 9% simple interest and one paying 0% simple interest. How much was invested in each account if the interest after year is $? 0. Ms. Riley deposited some money in an account paying % simple interest and twice that amount in an account paying 6% simple interest. If the total interest from the two accounts is $76 for year, how much was deposited into each account?. Sienna put some money in a certificate of deposit earning.% simple interest. She deposited twice that amount in a money market account paying % simple interest. After year her total interest was $88. How much did Sienna deposit in her money market account?. A total of $0,000 is invested between two accounts: one paying % simple interest and the other paying % simple interest. After year the total interest was $70. How much was invested at each rate?. Mr. Hall had some money in his bank earning.% simple interest. He had $000 more deposited in a credit union earning 6% simple interest. If his total interest for year was $0, how much did he deposit in each account?

60 Chapter Review of Basic Algebraic Concepts Concept : Applications Involving Mixtures. For a car to survive a winter in Toronto, the radiator must contain at least 7% antifreeze solution. Jacques truck has 6 L of 0% antifreeze mixture, some of which must be drained and replaced with pure antifreeze to bring the concentration to the 7% level. How much 0% solution should be drained and replaced by pure antifreeze to have 6 L of 7% antifreeze?. How many ounces of water must be added to 0 oz of an 8% salt solution to make a % salt solution? 6. Ronald has a % solution of the fertilizer Super Grow. How much pure Super Grow should he add to the mixture to get oz of a 7.% concentration? 7. How many liters of an 8% alcohol solution must be added to a 0% alcohol solution to get 0 L of a % alcohol solution? 8. For a performance of the play Company, 7 tickets were sold. The price of the orchestra level seats was $, and the balcony seats sold for $. If the total revenue was $887.00, how many of each type of ticket were sold? 9. Two different teas are mixed to make a blend that will be sold at a fair. Black tea sells for $.0 per pound and orange pekoe tea sells for $.00 per pound. How much of each should be used to obtain lb of a blend selling for $.0? 0. A nut mixture consists of almonds and cashews. Almonds are $.98 per pound, and cashews are $6.98 per pound. How many pounds of each type of nut should be mixed to produce 6 lb selling for $.7 per pound?. Two raffles are being held at a potluck dinner fund-raiser. One raffle ticket costs $.00 per ticket for a weekend vacation. The other costs $.00 per ticket for free passes to a movie theater. If 08 tickets were sold and a total of $0 was received, how many of each type of ticket were sold? Concept 6: Applications Involving Distance, Rate, and Time. Two cars are 9 miles apart and travel toward each other on the same road. They meet in hr. One car travels mph faster than the other. What is the average speed of each car?. Two cars are 90 miles apart and travel toward each other along the same road. They meet in hr. One car travels mph slower than the other car. What is the average speed of each car?. A Piper Cub airplane has an average air speed that is 0 mph faster than a Cessna 0 airplane. If the combined distance traveled by these two small planes is 690 miles after hr, what is the average speed of each plane?. A woman can hike mph faster down a trail to Archuletta Lake than she can on the return trip uphill. It takes her hr to get to the lake and 6 hr to return. What is her speed hiking down to the lake? 6. Two boats traveling the same direction leave a harbor at noon. After hr they are 60 miles apart. If one boat travels twice as fast as the other, find the average rate of each boat. 7. Two canoes travel down a river, starting at 9:00. One canoe travels twice as fast as the other. After. hr, the canoes are. miles apart. Find the average rate of each canoe.

Section.6 Literal Equations and Applications to Geometry 6 Literal Equations and Applications to Geometry. Applications Involving Geometry Some word problems involve the use of geometric formulas such as those listed in the inside front cover of this text. Example Solving an Application Involving Perimeter Section.6 Concepts. Applications Involving Geometry. Literal Equations The length of a rectangular corral is ft more than times the width. The corral is situated such that one of its shorter sides is adjacent to a barn and does not require fencing. If the total amount of fencing is 77 ft, then find the dimensions of the corral. Solution: Read the problem and draw a sketch (Figure -8). x x x x Figure -8 Let x represent the width. Let x represent the length. Label variables. To create a verbal model, we might consider using the formula for the perimeter of a rectangle. However, the formula P L W incorporates all four sides of the rectangle. The formula must be modified to include only one factor of the width. Distance around a b a times times b a three sides the length the width b Verbal model 77 (x ) x Mathematical model 77 x x 77 6x x 77 7x 770 7x 0 x Solve for x. Apply the distributive property. Combine like terms. Subtract from both sides. Divide by 7 on both sides. x 0 Because x represents the width, the width of the corral is 0 ft. The length is given by x or 0 Interpret the results. The width of the corral is 0 ft, and the length is ft. (To check the answer, verify that the three sides add to 77 ft.)

6 Chapter Review of Basic Algebraic Concepts Skill Practice. The length of Karen s living room is ft longer than the width. The perimeter is 80 ft. Find the length and width. The applications involving angles utilize some of the formulas found in the front cover of this text. Example Solving an Application Involving Angles Two angles are complementary. One angle measures 0 less than times the other angle. Find the measure of each angle (Figure -9). Solution: Let x represent one angle. Let represent the other angle. Recall that two angles are complementary if the sum of their measures is 90. Therefore, a verbal model is Verbal model Mathematical equation Solve for x. If x 0, then x 0 0 0 70. The two angles are 0 and 70. Skill Practice x 0 One angle the complement of the angle 90 x x 0 90 x 0 90 x 00 x 0 (x 0). Two angles are supplementary, and the measure of one is 6 less than times the other. Find their measures. x Figure -9 Skill Practice Answers. The length is ft, and the width is 9 ft.. 9 and. Literal Equations Literal equations (or formulas) are equations that contain several variables. For example, the formula for the perimeter of a rectangle P L W is an example of a literal equation. In this equation, P is expressed in terms of L and W. However, in science and other branches of applied mathematics, formulas may be more useful in alternative forms. For example, the formula P L W can be manipulated to solve for either L or W: Solve for L P L W P W L P W L L P W Solve for W P L W Subtract W. P L W Subtract L. P L Divide by. W Divide by. W P L

Section.6 Literal Equations and Applications to Geometry 6 To solve a literal equation for a specified variable, use the addition, subtraction, multiplication, and division properties of equality. Example Solving a Literal Equation The formula for the volume of a rectangular box is V LWH. a. Solve the formula V LWH for W. b. Find the value of W if V 00 in., L 0 in., and H in. (Figure -0). V 00 in. H in. W? L 0 in. Figure -0 Solution: a. V LWH The goal is to isolate the variable W. V LH LWH LH V LH W W V LH Divide both sides by LH. Simplify. 00 in. b. W Substitute V 00 in., L 0 in., and H in. 0 in. in. W in. Skill Practice The width is in. The formula for the area of a triangle is A bh. a. Solve the formula for h. b. Find the value of h when A 0 in. and b 6 in. Example Solving a Literal Equation The formula to find the area of a trapezoid is given by A b b h, where b and b are the lengths of the parallel sides and h is the b height. Solve this formula for b. h Solution: A b b h A b b h The goal is to isolate b. Multiply by to clear fractions. b A b h b h A b h h A b h h A b b h A b h b h b h h b Apply the distributive property. Subtract b h from both sides. Divide by h. Skill Practice Answers a. h A b. h in. b

6 Chapter Review of Basic Algebraic Concepts Skill Practice. The formula for the volume of a right circular cylinder is V pr h. Solve for h. r h TIP: When solving a literal equation for a specified variable, there is sometimes more than one way to express your final answer. This flexibility often presents difficulty for students. Students may leave their answer in one form, but the answer given in the text looks different. Yet both forms may be correct. To know if your answer is equivalent to the form given in the text you must try to manipulate it to look like the answer in the book, a process called form fitting. The literal equation from Example may be written in several different forms. The quantity A b h h can be split into two fractions. b A b h h A h b h h A h b Example Solving a Linear Equation in Two Variables Given x y, solve for y. Solution: Skill Practice. x y x y y x y x y x Solve for y. or y x Add x to both sides. Divide by on both sides. Example 6 Applying a Literal Equation Skill Practice Answers. h V pr x. y or y x Buckingham Fountain is one of Chicago s most familiar landmarks. With jets spraying a total of,000 gal (gallons) of water per minute, Buckingham Fountain is one of the world s largest fountains. The circumference of the fountain is approximately 880 ft. a. The circumference of a circle is given by C pr. Solve the equation for r. b. Use the equation from part (a) to find the radius and diameter of the fountain. (Use the p key on the calculator, and round the answers to decimal place.)

Section.6 Literal Equations and Applications to Geometry 6 Solution: a. C pr C p pr p C p r r C p 880 ft b. r Substitute C 880 ft and use the p key on the calculator. p r 0. ft The radius is approximately 0. ft. The diameter is twice the radius (d r); therefore the diameter is approximately 80. ft. Skill Practice The formula to compute the surface area S of a sphere is given by S pr. 6a. Solve the equation for p. b. A sphere has a surface area of in. and a radius of in. Use the formula found in part (a) to approximate p. Round to decimal places. Skill Practice Answers 6a. p S b.. r Section.6 Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos. Several topics are given here about taking notes. Which would you do first to help make the most of notetaking? Put them in order of importance to you by labeling them with the numbers 6. Read your notes after class to complete any abbreviations or incomplete sentences. Highlight important terms and definitions. Review your notes from the previous class. Bring pencils (more than one) and paper to class. Sit in class where you can clearly read the board and hear your instructor. Keep your focus on the instructor looking for phrases such as, The most important point is... and Here is where the problem usually occurs. Review Exercises For Exercises, solve the equations.. 7 x x 6 6x. y y. z z z 7. a 8a 7a 8 a

66 Chapter Review of Basic Algebraic Concepts Concept : Applications Involving Geometry For Exercises 6, use the geometry formulas listed in the inside front cover of the text. 6. A volleyball court is twice as long as it is wide. If the perimeter is 77 ft, find the dimensions of the court. 7. Two sides of a triangle are equal in length, and the third side is. times the length of one of the other sides. If the perimeter is m (meters), find the lengths of the sides. 8. The lengths of the sides of a triangle are given by three consecutive even integers. The perimeter is m. What is the length of each side? 9. A triangular garden has sides that can be represented by three consecutive integers. If the perimeter of the garden is ft, what are the lengths of the sides? 0. Raoul would like to build a rectangular dog run in the rear of his backyard, away from the house. The width of the yard is yd, and Raoul wants an area of 9 yd (square yards) for his dog. a. Find the dimensions of the dog run. b. How much fencing would Raoul need to enclose the dog run? x House yd Dog run. George built a rectangular pen for his rabbit such that the length is 7 ft less than twice the width. If the perimeter is 0 ft, what are the dimensions of the pen?. Antoine wants to put edging in the form of a square around a tree in his front yard. He has enough money to buy 8 ft of edging. Find the dimensions of the square that will use all the edging.. Joanne wants to plant a flower garden in her backyard in the shape of a trapezoid, adjacent to her house (see the figure). She also wants a front yard garden in the same shape, but with sides one-half as long. What should the dimensions be for each garden if Joanne has only a total of 60 ft of fencing? x x Back x. The measures of two angles in a triangle are equal. The third angle measures times the sum of the equal angles. Find the measures of the three angles.. The smallest angle in a triangle is one-half the size of the largest. The middle angle measures less than the largest. Find the measures of the three angles. House Front 6. Two angles are complementary. One angle is times as large as the other angle. Find the measure of each angle. 7. Two angles are supplementary. One angle measures less than times the other. Find the measure of each angle.

Section.6 Literal Equations and Applications to Geometry 67 In Exercises 8, solve for x, and then find the measure of each angle. 8. 9. (7x ) (x ) (0x 6) [(x )] 0.. (x ) (x ) [(x 7)] (x.).. (x) (x ) [(x )] (x ) (x ).. [(x 6)] (x ) (0x) (x ) (0x ) Concept : Literal Equations 6. Which expression(s) is (are) equivalent to x? a. b. c. x x x 7. Which expression(s) is (are) equivalent to z? z z a. b. z c. 8. Which expression(s) is (are) equivalent to x 7 y? x 7 x 7 a. x 7 b. c. y y y 9. Which expression(s) is (are) equivalent to w x y? w a. w b. c. w x y x y x y

68 Chapter Review of Basic Algebraic Concepts For Exercises 0 7, solve for the indicated variable. 0. A lw for l. C R for R. I Prt for P. a b c P for b. W K K for K. y mx b for x 6. F 9 7. C C for C 9 F for F 8. 9. I Prt for r 0. v v 0 at for a. K mv for v a b c for b. w pv v for v. A lw for w. ax by c for y. P L W for L 6. V 7. Bh for B V pr h for h In Chapter it will be necessary to change equations from the form Ax By C to y mx b. For Exercises 8 9, express each equation in the form y mx b by solving for y. 8. x y 6 9. x y 0. x y 0. x y. 6x y. x 7y. x y 6. x y 8 6. 9x y 7. x 8. x 9. y y 0 x y 0 For Exercises 60 6, use the relationship between distance, rate, and time given by d rt. 60. a. Solve d rt for rate r. b. In 006, Sam Hornish won the Indianapolis 00 in hr, 0 min, 9 sec.8 hr. Find his average rate of speed if the total distance is 00 mi. Round to the nearest tenth of a mile per hour. 6. a. Solve d rt for time t. b. In 006, Jimmie Johnson won the Daytona 00 with an average speed of.7 mph. Find the total time it took for him to complete the race if the total distance is 00 miles. Round to the nearest hundredth of an hour. For Exercises 6 6, use the fact that the force imparted by an object is equal to its mass times acceleration, or F ma. 6. a. Solve F ma for mass m. b. The force on an object is. N (newtons), and the acceleration due to gravity is 9.8 m/sec. Find the mass of the object. (The answer will be in kilograms.) 6. a. Solve F ma for acceleration a. b. Approximate the acceleration of a 000-kg mass influenced by a force of,000 N. (The answer will be in meters per second squared, m/sec.)

Section.7 Linear Inequalities in One Variable 69 In statistics the z-score formula z x m is used in studying probability. Use this formula for Exercises 6 6. s 6. a. Solve z x m for x. s b. Find x when z., m 00, and s. 6. a. Solve z x m for s. s b. Find s when x 0, z., and m 0. Expanding Your Skills For Exercises 66 7, solve for the indicated variable. 66. 6t rt for t 67. a ca for a 68. 69. cx dx 9 for x 70. A P Prt for P 7. ax 6x A P Prt for r for x 7. T mg mf for m 7. T mg mf for f 7. 7. Lt h mt g for t ax by cx z for x Linear Inequalities in One Variable. Solving Linear Inequalities In Sections..6, we learned how to solve linear equations and their applications. In this section, we will learn the process of solving linear inequalities. A linear inequality in one variable, x, is defined as any relationship of the form: ax b 6 0, ax b 0, ax b 7 0, or ax b 0, where a 0. The solution to the equation x can be graphed as a single point on the number line. Section.7 Concepts. Solving Linear Inequalities. Inequalities of the Form a x b. Applications of Inequalities 0 Now consider the inequality x. The solution set to an inequality is the set of real numbers that makes the inequality a true statement. In this case, the solution set is all real numbers less than or equal to. Because the solution set has an infinite number of values, the values cannot be listed. Instead, we can graph the solution set or represent the set in interval notation or in set-builder notation. Graph Interval Notation Set-Builder Notation, x 0 x 6 0 The addition and subtraction properties of equality indicate that a value added to or subtracted from both sides of an equation results in an equivalent equation. The same is true for inequalities.

70 Chapter Review of Basic Algebraic Concepts Addition and Subtraction Properties of Inequality Let a, b, and c represent real numbers. *Addition property of inequality: *Subtraction property of inequality: If then If then *These properties may also be stated for a b, a 7 b, and a b. a 6 b a c 6 b c a 6 b a c 6 b c Example Solving a Linear Inequality Solve the inequality. Graph the solution and write the solution set in interval notation. x 7 7 x Solution: x 7 7 x x 7 7 x 8 x 7 7 x 9 x x 7 7 x x 9 x 7 7 9 x 7 7 7 9 7 Graph ( x 7 0 Apply the distributive property. Subtract x from both sides. Add 7 to both sides. Interval Notation, Skill Practice. Solve the inequality. Graph the solution and write the solution in interval notation. x 7 7x ( Skill Practice Answers., Multiplying both sides of an equation by the same quantity results in an equivalent equation. However, the same is not always true for an inequality. If you multiply or divide an inequality by a negative quantity, the direction of the inequality symbol must be reversed. For example, consider multiplying or dividing the inequality 6 by. Multiply/divide by : 6 0 The number lies to the left of on the number line. However, lies to the right of. Changing the signs of two numbers changes their relative position on the number line. This is stated formally in the multiplication and division properties of inequality. 6

Section.7 Linear Inequalities in One Variable 7 Multiplication and Division Properties of Inequality Let a, b, and c represent real numbers. *If c is positive and a 6 b, then *If c is negative and a 6 b, then ac 6 bc and ac 7 bc and The second statement indicates that if both sides of an inequality are multiplied or divided by a negative quantity, the inequality sign must be reversed. *These properties may also be stated for a b, a 7 b, and a b. a c 6 b c a c 7 b c Example Solving Linear Inequalities Solve the inequalities. Graph the solution and write the solution set in interval notation. a. x 6 b. 6x x 8 Solution: a. x 6 x 6 x 6 7 x 7 7 x 7 7 Add to both sides. Divide by (reverse the inequality sign). or x 7. Graph Interval Notation 7 ) 0 a 7, b TIP: The inequality x 6 could have been solved by isolating x on the right-hand side of the inequality. This creates a positive coefficient on the x term and eliminates the need to divide by a negative number. x 6 6 x Add x to both sides. 7 6 x Subtract from both sides. 7 6 x Divide by (because is positive, do not reverse the inequality sign). 7 6 x (Note that the inequality 7 is equivalent to x 7 7 6 x.

7 Chapter Review of Basic Algebraic Concepts b. Graph 6x x 8 6x 8 x 6 6x 8 8 x 6x x 8 8 x x x 8 8 x 8 8 8 8 x 0 x x 0 0 0 Apply the distributive property. Combine like terms. Add x to both sides. Subtract 8 from both sides. Divide by (reverse the inequality sign). Interval Notation, 0 Skill Practice Solve the inequalities. Graph the solution and write the solution in interval notation.. x 0. x 6 x Example Solving a Linear Inequality x Solve the inequality 7 x. solution set in interval notation. Graph the solution and write the Skill Practice Answers. 8, 8. (, Solution: a x b 6 x x x 6 x x 6 x 6 6 Graph x x 6 x 6 x 6 6 x 6 8 x 7 8 0 x 7 7 x ( 6 Multiply by to clear fractions (reverse the inequality sign). Add x to both sides. Subtract from both sides. Divide by (the inequality sign is reversed again). Simplify. Interval Notation,

Section.7 Linear Inequalities in One Variable 7 Skill Practice. x x Solve the inequality. Graph the solution and write the solution in interval notation. In Example, the inequality sign was reversed twice: once for multiplying the inequality by and once for dividing by. If you are in doubt about whether you have the inequality sign in the correct direction, you can check your final answer by using the test point method. That is, pick a point in the proposed solution set, and verify that it makes the original inequality true. Furthermore, any test point picked outside the solution set should make the original inequality false. 0 ( 6 Pick x 0 as a test point x 0 7? 0 7? 7 x False Pick x as a test point x 7 x 7? 7? 7 7 7? 7 True Because a test point to the right of x makes the inequality true, we have shaded the correct part of the number line.. Inequalities of the Form a < x < b An inequality of the form a 6 x 6 b is a type of compound inequality, one that defines two simultaneous conditions on the quantity x. a x b a 6 x and x 6 b The solution to the compound inequality a 6 x 6 b is the intersection of the inequalities a 6 x and x 6 b. To solve a compound inequality of this form, we can actually work with the inequality as a three-part inequality and isolate the variable x. Skill Practice Answers.,

7 Chapter Review of Basic Algebraic Concepts Example Solving a Compound Inequality of the Form a < x < b Solve the inequality x 6. solution set in interval notation. Graph the solution and express the Solution: To solve the compound inequality x 6, isolate the variable x in the middle. The operations performed on the middle portion of the inequality must also be performed on the left-hand side and right-hand side. x 6 x 6 x 6 x 6 Subtract from all three parts of the inequality. Simplify. Divide by in all three parts of the inequality. x 6 Simplify. Graph Interval Notation 0 ( c, b Skill Practice. Solve the compound inequality. Graph the solution and express the solution set in interval notation. 8 6 x. Applications of Inequalities Example Solving a Compound Inequality Application Beth received grades of 87%, 8%, 96%, and 79% on her last four algebra tests. To graduate with honors, she needs at least a B in the course. a. What grade does she need to make on the fifth test to get a B in the course? Assume that the tests are weighted equally and that to earn a B the average of the test grades must be at least 80% but less than 90%. b. Is it possible for Beth to earn an A in the course if an A requires an average of 90% or more? Skill Practice Answers. (, Solution: a. Let x represent the score on the fifth test. The average of the five tests is given by 87 8 96 79 x

Section.7 Linear Inequalities in One Variable 7 To earn a B, Beth requires 80 average of test scores 6 90 80 Verbal model Mathematical model Multiply by to clear fractions. 00 x 6 0 Simplify. 00 x 6 0 Subtract from all three parts. 6 x 6 06 Simplify. To earn a B in the course, Beth must score at least 6% but less than 06% on the fifth exam. Realistically, she may score between 6% and 00% because a grade over 00% is not possible. b. To earn an A, Beth s average would have to be greater than or equal to 90%. Average of test scores 90 Verbal model 87 8 96 79 x 90 Mathematical equation 87 8 96 79 x a b 90 Clear fractions. x 0 Simplify. x 06 Solve for x. It would be impossible for Beth to earn an A in the course because she would have to earn at least a score of 06% on the fifth test. It is impossible to earn over 00%. Skill Practice 87 8 96 79 x 80 a 6 90 87 8 96 79 x b 6 90 6. Jamie is a salesman who works on commission, so his salary varies from month to month. To qualify for an automobile loan, his salary must average at least $00 for 6 months. His salaries for the past months have been $800, $00, $00, $00, and $800. What amount does he need to earn in the last month to qualify for the loan? Example 6 Solving a Linear Inequality Application The number of registered passenger cars N (in millions) in the United States has risen between 960 and 00 according to the equation N.t 6., where t represents the number of years after 960 (t 0 corresponds to 960, t corresponds to 96, and so on) (Figure -). Skill Practice Answers 6. Jamie s salary must be at least $000.

d 76 Chapter Review of Basic Algebraic Concepts Number of cars (millions) N 00 0 00 0 Number of Registered Passenger Cars, United States, 960 00 N.t 6. 0 t 0 0 0 0 0 Year (t 0 corresponds to 960) Figure - (Source: U.S. Department of Transportation) a. For what years after 960 was the number of registered passenger cars less than 89. million? b. For what years was the number of registered passenger cars between 9. million and 0.9 million? c. Predict the years for which the number of passenger cars will exceed. million. Solution: a. We require N 6 89. million. N 6 89..t 6. 6 89..t 6. 6. 6 89. 6..t 6.t. 6. t 6 0 Substitute the expression for N. Subtract 6. from both sides. Divide both sides by...t 6. t 0 corresponds to the year 970. Before 970, the number of registered passenger cars was less than 89. million. b. We require 9. 6 N 6 0.9. Hence 9. 6.t 6. 6 0.9 9. 6. 6.t 6. 6. 6 0.9 6. 0.0 6.t 6 7. 0.0. 6.t. 6 7.. 6 t 6 Substitute the expression.t 6. for N. Subtract 6. from all three parts of the inequality. Divide by.. t corresponds to 97 and t corresponds to 97. Between the years 97 and 97, the number of registered passenger cars was between 9. million and 0.9 million.

Section.7 Linear Inequalities in One Variable 77 c. We require N 7...t 6. 7. Substitute the expression.t 6. for N..t 7 90 Subtract 6. from both sides..t Divide by... 7 90. t 7 6 t 6 corresponds to the year 996. After the year 996, the number of registered passenger cars exceeded. million. Skill Practice 7. The population of Alaska has steadily increased since 90 according to the equation P 0t 7, where t represents the number of years after 90 and P represents the population in thousands. For what years after 90 was the population less than 7 thousand people? Skill Practice Answers 7. The population was less than 7 thousand for t 6 0. This corresponds to the years before 980. Section.7 Boost your GRADE at mathzone.com! Study Skills Exercises Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos. After doing a section of homework, check the answers to the odd-numbered exercises in the back of the text. Choose a method to identify the exercises that you got wrong or had trouble with (e.g., circle the number or put a star by the number). List some reasons why it is important to label these problems.. Define the key terms. a. Linear inequality b. Test point method c. Compound inequality Review Exercises. Solve for v.. Solve for x. d vt 6t x x. Five more than times a number is 6 less than twice the number. Find the number. 6. Solve for y. x y 6 0 7. a. The area of a triangle is given by A bh. Solve for h. b. If the area of a certain triangle is 0 cm and the base is cm, find the height. 8. Solve for t. t 0 t 0 t

78 Chapter Review of Basic Algebraic Concepts Concept : Solving Linear Inequalities For Exercises 9, solve the inequalities. Graph the solution and write the solution set in interval notation. Check each answer by using the test point method. 9. 6 y 0. x. z 6. 6z 7 6. 8w. t 6 8. 6. 8y 9 6 7. p x 7 0 8. 0.8a 0. 0.a 9. 0.w 0.7 6 0. 0.9w 0. x 7 6. w 6 7 9.. 6 x k p. 7. 6. y 7 6 0.t 7.t 0 7. 0 8 8. y 6 y 9. x 6 b 0. 7.k..7. 6h.9 6.8 Concept : Inequalities of the form a < x < b. Write x 6 as two inequalities.. Write 6 x 7 as two inequalities. For Exercises, solve the compound inequalities. Graph the solution and write the solution set in interval notation.. 0 a 6 7. 8 6 k 7 6 6. 6 y 6 7. 7 m 6 0 8. x 9. a 6 9 0. 7 x 8. y 8. 6 b 7 6 6

Section.7 Linear Inequalities in One Variable 79. 7 w 7. 8 7 w 7. h 0 6. Explain why 6 x 6 has no solution. 7. Explain why 7 x 7 has no solution. Concept : Applications of Inequalities 8. Nolvia sells copy machines, and her salary is $,000 plus a % commission on sales. The equation S,000 0.0x represents her salary S in dollars in terms of her total sales x in dollars. a. How much money in sales does Nolvia need to earn a salary that exceeds $0,000? b. How much money in sales does Nolvia need to earn a salary that exceeds $80,000? c. Why is the money in sales required to earn a salary of $80,000 more than twice the money in sales required to earn a salary of $0,000? 9. The amount of money in a savings account A depends on the principal P, the interest rate r, and the time in years t that the money is invested. The equation A P Prt shows the relationship among the variables for an account earning simple interest. If an investor deposits $000 at 6 % simple interest, the account will grow according to the formula A 000 0000.06t. a. How many years will it take for the investment to exceed $0,000? (Round to the nearest tenth of a year.) b. How many years will it take for the investment to exceed $,000? (Round to the nearest tenth of a year.) 0. The revenue R for selling x fleece jackets is given by the equation R 9.9x. The cost to produce x jackets is C 00 8.0x. Find the number of jackets that the company needs to sell to produce a profit. (Hint: A profit occurs when revenue exceeds cost.). The revenue R for selling x mountain bikes is R 9.9x. The cost to produce x bikes is C 6,000 0x. Find the number of bikes that the company needs to sell to produce a profit.. The average high and low temperatures for Vancouver, British Columbia, in January are.6 C and 0 C, respectively. The formula relating Celsius temperatures to Fahrenheit temperatures is given by C 9 F. Convert the inequality 0.0 C.6 to an equivalent inequality using Fahrenheit temperatures.. For a day in July, the temperatures in Austin, Texas ranged from 0 C to 9 C. The formula relating Celsius temperatures to Fahrenheit temperatures is given by C 9 F. Convert the inequality 0 C 9 to an equivalent inequality using Fahrenheit temperatures.. The poverty threshold P for four-person families between the years 960 and 006 can be approximated by the equation P 87t, where P is measured in dollars; and t 0 corresponds to the year 960, t corresponds to 96, and so on. (Source: U.S. Bureau of the Census.) a. For what years after 960 was the poverty threshold under $700? b. For what years after 960 was the poverty threshold between $ and $0,6?

80 Chapter Review of Basic Algebraic Concepts. Between the years 960 and 006, the average gas mileage (miles per gallon) for passenger cars has increased. The equation N.6 0.t approximates the average gas mileage corresponding to the year t, where t 0 represents 960, t represents 96, and so on. a. For what years after 960 was the average gas mileage less than. mpg? (Round to the nearest year.) b. For what years was the average gas mileage between 7. and 8.0 mpg? (Round to the nearest year.) Mixed Exercises For Exercises 6 7, solve the inequalities. Graph the solution, and write the solution set in interval notation. Check each answer by using the test point method. 6. 6p 7 7 7. y 8. x 8 9. 60..b 0. 0.b 6. a 7 0.t. 6 6. 6 t 6. h 6 6. c c 6. 66. y 6 y 6 67. q 7 q 6 6k k 68. 0 q 69. 0 6 7p 6 70. 6x 6 x 6x 7. p p p 7. 6a 9a a 7. 8q q 7 Expanding Your Skills For Exercises 7 77, assume a b. Determine which inequality sign ( or ) should be inserted to make a true statement. Assume a 0 and b 0. 7. a c b c, for c 7 0 7. a c b c, for c 6 0 76. ac bc, for c 7 0 77. ac bc, for c 6 0

Section.8 Properties of Integer Exponents and Scientific Notation 8 Properties of Integer Exponents and Scientific Notation. Properties of Exponents In Section., we learned that exponents are used to represent repeated multiplication. The following properties of exponents are often used to simplify algebraic expressions. Section.8 Concepts. Properties of Exponents. Simplifying Expressions with Exponents. Scientific Notation Properties of Exponents* Description Property Example Details/Notes Multiplication of like bases Division of like bases Power rule Power of a product Power of a quotient b m b n b m n b m b n bm n b m n b m n ab m a m b m a a b b m am b m b b b b 6 b b b b b b b 8 ab a b aa b b a b b b b bb b b b b 6 b b b b b b b b b b b b b b bb b b b b8 ab ababab a a ab b b a b a a b b a a b b a a b b a a b b a a a b b b a b *Assume that a and b are real numbers b 0 and that m and n represent positive integers. In addition to the properties of exponents, two definitions are used to simplify algebraic expressions. b 0 and b n Let n be an integer, and let b be a real number such that b 0.. b 0. b n a b b n b n The definition of b 0 is consistent with the properties of exponents. For example, if b is a nonzero real number and n is an integer, then b n b n b n b n bn n b 0 The expression b 0

8 Chapter Review of Basic Algebraic Concepts b n The definition of is also consistent with the properties of exponents. If b is a nonzero real number, then b b b b b b b b b b b b b b b The expression b b Example Using the Properties of Exponents Simplify the expressions a. b. c. d. 7x 0 e. 7x 0 Solution: a. b. c. 6 6 6 b0 6 d. 7x 0 because e. 7x 0 7 x 0 7 7 Skill Practice Simplify the expressions..... 8y 0. 6 0. Simplifying Expressions with Exponents Example Simplifying Expressions with Exponents Simplify the following expressions. Write the final answer with positive exponents only. Skill Practice Answers a. x 7 x b. a b 0. 9. 9... 9 c. d. a a7 b a y w 0 8a 9 b b 6a b 0 y w b

Section.8 Properties of Integer Exponents and Scientific Notation 8 Solution: a. x 7 x x 7 x x 8 Multiply like bases by adding exponents. Apply the power rule. Multiply exponents. b. a b 0 a b 0 Simplify negative exponents. Evaluate the exponents. 00 0 Write the expressions with a common denominator. Simplify. c. a y w 0 Work within the parentheses first. y w b y w 0 y w 6 y w 6 y w 6 Divide like bases by subtracting exponents. Simplify within parentheses. Apply the power rule. Multiply exponents. y a y w6b or w 6 Simplify negative exponents. d. a a7 b Subtract exponents within first 8a 9 b b 6a b 0 parentheses. Simplify to. a a7 9 b b 6a a a b b 6a c a b d 6 a a a6 b 6 b 6 a a 6 b 6 c 6 d 6a8 b 6 6 6a8 b 6 9 a In the second parentheses, replace b 0 by. Simplify inside parentheses. Apply the power rule. Multiply exponents. Simplify negative exponents. Multiply factors in the numerator and denominator. Simplify. 8

8 Chapter Review of Basic Algebraic Concepts Skill Practice Simplify the expressions. Write the final answers with positive exponents only. 6. 7. a b a 0 a b b 8. a b c 9. b c b x y x y. Scientific Notation Scientists in a variety of fields often work with very large or very small numbers. For instance, the distance between the Earth and the Sun is approximately 9,000,000 mi. The national debt in the United States in 00 was approximately $7,80,000,000,000. The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 9 kg. Scientific notation was devised as a shortcut method of expressing very large and very small numbers. The principle behind scientific notation is to use a power of 0 to express the magnitude of the number. Consider the following powers of 0: 0 0 0 00 0 0 0. 0 000 0 00 0.0 0 0000 0 000 0.00 0 0,000 0 0 0,000 0.000 Each power of 0 represents a place value in the base-0 numbering system. A number such as 0,000 may therefore be written as 0,000 or equivalently as.0 0. Similarly, the number 0.00 is equal to. 000 or, equivalently,. 0. Definition of a Number Written in Scientific Notation A number expressed in the form a 0 n, where 0a 0 6 0 and n is an integer, is said to be written in scientific notation. Consider the following numbers in scientific notation: The distance between the Sun and the Earth: 9,000,000 mi 9. 0 7 mi Skill Practice Answers 6. a 0 7. b 8. x 9. 8b c y The national debt of the United States in 00: $7,80,000,000,000 $7.8 0 The mass of an electron: 0.000 000 000 000 000 000 000 000 000 000 9 kg 9. 0 kg 7 places places places

Section.8 Properties of Integer Exponents and Scientific Notation 8 In each case, the power of 0 corresponds to the number of place positions that the decimal point is moved. The power of 0 is sometimes called the order of magnitude (or simply the magnitude) of the number. The order of magnitude of the national debt is 0 dollars (trillions). The order of magnitude of the distance between the Earth and Sun is 0 7 mi (tens of millions). The mass of an electron has an order of magnitude of 0 kg. Example Writing Numbers in Scientific Notation Fill in the table by writing the numbers in scientific notation or standard notation as indicated. Quantity Standard Notation Scientific Notation Number of NASCAR fans Width of an influenza virus Cost of hurricane Andrew Probability of winning the Florida state lottery Approximate width of a human red blood cell Profit of Citigroup Bank, 00 7,000,000 people 0.00000000 m 0.000007 m $.6 0 0.87878 0 8 $. 0 0 Solution: Quantity Standard Notation Scientific Notation Number of NASCAR fans 7,000,000 people Width of an influenza virus 0.00000000 m.0 0 9 m Cost of hurricane Andrew $6,00,000,000 Probability of winning the 0.000000087878 Florida state lottery Approximate width of a human red blood cell 0.000007 m 7. 0 7 people $.6 0 0.87878 0 8 7.0 0 6 m Profit of Citigroup Bank, 00 $,00,000,000 $. 0 0 Skill Practice Write the numbers in scientific notation. 0.,600,000. 0.00088 Write the numbers in standard notation...7 0 8..9 0 Calculator Connections Calculators use scientific notation to display very large or very small numbers. To enter scientific notation in a calculator, try using the EE key or the EXP key to express the power of 0. Skill Practice Answers 0..6 0 6. 8.8 0. 0.00000007. 90,000

86 Chapter Review of Basic Algebraic Concepts Example Applying Scientific Notation a. The U.S. national debt in 00 was approximately $7,80,000,000,000. Assuming there were approximately 90,000,000 people in the United States at that time, determine how much each individual would have to pay to pay off the debt. b. The mean distance between the Earth and the Andromeda Galaxy is approximately.8 0 6 light-years. Assuming light-year is 6.0 0 mi, what is the distance in miles to the Andromeda Galaxy? Solution: a. Divide the total U.S. national debt by the number of people: 7.8 0.9 0 8 a 7.8.9 b a0 0 8 b.7 0 Divide 7.8 by.9 and subtract the powers of 0. In standard notation, this amounts to approximately $7,000 per person. b. Multiply the number of light-years by the number of miles per light-year. Calculator Connections Use a calculator to check the solutions to Example..8 0 6 6.0 0.86.0 0 6 0 0.8 0 8.08 0 0 8.08 0 0 8.08 0 9 Multiply.8 and 6.0 and add the powers of 0. The number 0.8 0 8 is not in proper scientific notation because 0.8 is not between and 0. Rewrite 0.8 as.08 0. Apply the associative property of multiplication. The distance between the Earth and the Andromeda Galaxy is.08 0 9 mi. Skill Practice. The thickness of a penny is 6. 0 in. The height of the Empire State Building is 0 ft (. 0 in.). How many pennies would have to be stacked on top of each other to equal the height of the Empire State Building? Round to the nearest whole unit.. The distance from Earth to the nearby star, Barnard s Star, is 7.6 light years (where light year 6.0 0 mi). How many miles away is Barnard s Star? Skill Practice Answers. Approximately,90 pennies..6 0 mi

Section.8 Properties of Integer Exponents and Scientific Notation 87 Section.8 Boost your GRADE at mathzone.com! Practice Exercises Practice Problems Self-Tests NetTutor e-professors Videos Study Skills Exercises. Write down the page number(s) for the Chapter Summary for this chapter. Describe one way in which you can use the Summary found at the end of each chapter.. Define the key term scientific notation. Review Exercises For Exercises 6, solve the equation or inequality. Write the solutions to the inequalities in interval notation. a y 6. a. y. 6x x 7x 6. c c 7 6c 8 For Exercises 7 8, solve the equation for the indicated variable. 7. x 9y for x 8. x y 8 for y Concept : Properties of Exponents 9. Explain the difference between b b and b. 0. Explain the difference between ab and (ab). (Hint: Expand both expressions and compare.) For Exercises 6, write two examples of each property. Include examples with and without variables. (Answers may vary.). b n b m b n m. ab n a n b n. b n m b nm b n.. a a n 6. b b an n b 0 b m bn m b 0 b b 0 b 0 7. Simplify: a b 8. Simplify: a b For Exercises 9, simplify. 8 9. 0...... a 8 6. b 8 a 8 b 7. 8. 9. a a a 0. b 9 b b a b. 0ab 0. x 0. 0ab 0. x 0

88 Chapter Review of Basic Algebraic Concepts Concept : Simplifying Expressions with Exponents For Exercises 8, simplify and write the answer with positive exponents only. 8. y y 6. x x 8 7. 8. 6 9. y 0. z. x.. p. q. 7 0 7 6. w 7. 8. 9. a a 0. w r.... s r a b t t 8 s c. 6. 7. 6xyz 0 8. d 9. 60. 6. 6. z 6 z 7 y 9 7 b b 8 w 8 w 7ab 0 6. a 6. b a b a 0 b a b a b a 0 7 b p q m n 8ab 0 6. 66. 67. 68. p q m n a b 69. x y z 70. 6a b c 7. m n m 6 n 7. 7. 7. 7. a x p q pq mn m n 76. y b x y 8a b x y 77. 78. 79. a x6 y 80. 6a xy x y b b 7 8. 8. 8. xy a x y a a b c 0 a x y 0 8. x 6 y b a b c b 6x y b x y 0x y 7 6pq p q a a b b a b a 6a b a b b 7x y a x y 9x y b Concept : Scientific Notation 8. Write the numbers in scientific notation. a. Paper is 0.00 in. thick. b. One mole is 60,00,000,000,000,000,000,000 particles. c. The dissociation constant for nitrous acid is 0.0006. 86. Write the numbers in scientific notation. a. The estimated population of the United States in 007 is approximately 9,600,000. b. As of 00, the net worth of Bill Gates was $6,600,000,000. c. A trillion is defined as,000,000,000,000.

Section.8 Properties of Integer Exponents and Scientific Notation 89 87. Write the numbers in standard notation. a. The number of $0 bills in circulation in 00 was.8 0 9. b. The dissociation constant for acetic acid is.8 0. c. In 00, the population of the world was approximately 6.78 0 9. 88. Write the numbers in standard notation. a. The proposed budget for the 006 federal government allocated $.6 0 0 for the Department of Education. b. The mass of a neutron is.67 0 g. c. The number of $ bills in circulation in 00 was 6.8 0 8. For Exercises 89 9, determine which numbers are in proper scientific notation. If the number is not in proper scientific notation, correct it. 89. 0 90. 0.69 0 7 9. 7.0 0 0 9. 8. 0 9. 9 0 9. 6.9 0 0 For Exercises 9 0, perform the indicated operations and write the answer in scientific notation. 9. 6. 0. 0 8 96..6 0 6 8. 0 9 97. (0.00000)(6,700,000,000) 98. (,00,000,000)(70,000,000,000,000) 99. 8. 0. 0 00. 0 9. 0 0. 900,000,000 60,000 0. 0.000000000 8,000,000 0. If one H O molecule contains hydrogen atoms and oxygen atom, and 0 H O molecules contain 0 hydrogen atoms and 0 oxygen atoms, how many hydrogen atoms and oxygen atoms are contained in 6.0 0 H O molecules? 0. The star named Alpha Centauri is. light-years from the Earth. If there is approximately 6 0 9 mi in light-year, how many miles away is Alpha Centauri? 0. The county of Queens, New York, has a population of approximately,00,000. If the area is 0 mi, how many people are there per square mile? 06. The county of Catawba, North Carolina, has a population of approximately 0,000. If the area is how many people are there per square mile? 00 mi, Expanding Your Skills For Exercises 07, simplify the expression. Assume that a and b represent positive integers and x and y are nonzero real numbers. 07. x a x a 08. y a y a 7 09. y a y a x a x b y b 0... x a x b y b x a y a x a y a. At one count per second, how many days would it take to count to million? (Round to decimal place.). Do you know anyone who is more than.0 0 9 sec old? If so, who?. Do you know anyone who is more than. 0 hr old? If so, who?

90 Chapter Review of Basic Algebraic Concepts Chapter Section. SUMMARY Sets of Numbers and Interval Notation Key Concepts Natural numbers:,,,...6 Whole numbers: 0,,,,...6 Integers:...,,,, 0,,,,...6 Rational numbers: does not equal 0 f e p q 0 p and q are integers and q Irrational numbers: {x x is a real number that is not rational} Examples Example Some rational numbers are: 7, 0., 0.,, 7 Some irrational numbers are: 7,, p Real numbers: x 0 x is rational or x is irrational6 a b a is less than b a b a is greater than b a b a is less than or equal to b a b a is greater than or equal to b a x b x is between a and b A B is the union of A and B and is the set of elements that belong to set A or set B or both sets A and B. A B is the intersection of A and B and is the set of elements common to both A and B. Example Set-Builder Notation Interval Notation Graph x 0 x 7 6 x 0 x 6 x 0 x 6 6 x 0 x 6 Example Union A B,,,, Intersection A B ( ( A B A B

Summary 9 Section. Operations on Real Numbers Key Concepts The reciprocal of a number a 0 is a. The opposite of a number a is a. The absolute value of a, denoted 0a 0, is its distance from zero on the number line. Addition of Real Numbers Same Signs: Add the absolute values of the numbers, and apply the common sign to the sum. Unlike Signs: Subtract the smaller absolute value from the larger absolute value. Then apply the sign of the number having the larger absolute value. Subtraction of Real Numbers Add the opposite of the second number to the first number. Multiplication and Division of Real Numbers Same Signs: Product or quotient is positive. Opposite Signs: Product or quotient is negative. The product of any real number and 0 is 0. The quotient of 0 and a nonzero number is 0. The quotient of a nonzero number and 0 is undefined. Exponents and Radicals b b b b b (b is the base, is the exponent) b is the principal square root of b (b is the radicand, is the radical sign). Order of Operations. Simplify expressions within parentheses and other grouping symbols first.. Evaluate expressions involving exponents, radicals and absolute values.. Perform multiplication or division in order from left to right.. Perform addition or subtraction in order from left to right. Examples Example Given: The reciprocal is. The opposite is. The absolute value is. Example 7 Example 7 7 Example 0 70 0 0 is undefined Example 6 6 6 6 6 Example 6 0 6 0 6 0 0 0 0 6 7 00 0 6 0 9 0

9 Chapter Review of Basic Algebraic Concepts Section. Simplifying Expressions Key Concepts A term is a constant or the product of a constant and one or more variables. A variable term contains at least one variable. A constant term has no variable. The coefficient of a term is the numerical factor of the term. Like terms have the same variables, and the corresponding variables are raised to the same powers. Distributive Property of Multiplication over Addition ab c ab ac Two terms can be added or subtracted if they are like terms. Sometimes it is necessary to clear parentheses before adding or subtracting like terms. Examples Example x Variable term has coefficient. x y Variable term has coefficient. 6 Constant term has coefficient 6. Example ab and ab are like terms. Example x y x 8y a 6b c a 6b c Example d d d 9d Example w w w w 8 w 8 6w 6 6w

Summary 9 Section. Linear Equations in One Variable Key Concepts A linear equation in one variable can be written in the form ax b 0 a 0. Steps to Solve a Linear Equation in One Variable. Simplify both sides of the equation. Clear parentheses. Consider clearing fractions or decimals (if any are present) by multiplying both sides of the equation by a common denominator of all terms. Combine like terms.. Use the addition or subtraction property of equality to collect the variable terms on one side of the equation.. Use the addition or subtraction property of equality to collect the constant terms on the other side.. Use the multiplication or division property of equality to make the coefficient on the variable term equal to.. Check your answer. An equation that has no solution is called a contradiction. An equation that has all real numbers as its solutions is called an identity. Examples Example x x x x a x x b a b x 8 x 6 Example x 6 x x 6 x There is no solution. Example x x x 6 Contradiction x x x x x x Identity All real numbers are solutions.

9 Chapter Review of Basic Algebraic Concepts Section. Key Concepts Problem-Solving Steps for Word Problems. Read the problem carefully.. Assign labels to unknown quantities.. Develop a verbal model.. Write a mathematical equation.. Solve the equation. 6. Interpret the results and write the final answer in words. Sales tax: (Cost of merchandise)(tax rate) Commission: (Dollars in sales)(commission rate) Simple interest: I Prt Distance (rate)(time) d rt Applications of Linear Equations in One Variable Examples Example. Estella has $800 to invest between two accounts, one bearing 6% simple interest and the other bearing 0% simple interest. At the end of year, she has earned $70 in interest. Find the amount Estella has invested in each account.. Let x represent the amount invested at 6%. Then 800 x is the amount invested at 0%. 6% Account 0% Account Total Principal x 800 x 800 Interest 0.06x 0.0(800 x) 70.. Interest from from a b ainterest 6% account 0% account b a total interest b 0.06x 0.0800 x 70. 6x 0800 x 7,000 6x 8,000 0x 7,000 x 0,000 x 00 6. x 00 800 x 6000 $00 was invested at 6% and $6000 was invested at 0%.

Summary 9 Section.6 Literal Equations and Applications to Geometry Key Concepts Some useful formulas for word problems: Perimeter Rectangle: P l w Area Rectangle: Square: Triangle: Trapezoid: Angles A lw A x A bh A b b h Two angles whose measures total 90 are complementary angles. Two angles whose measures total 80 are supplementary angles. Examples Example A border of marigolds is to enclose a rectangular flower garden.if the length is twice the width and the perimeter is. ft, what are the dimensions of the garden? P l w. x x. x x. 6x. x x x The width is. ft, and the length is (.) ft or 8. ft. Vertical angles have equal measure. m a m c m b m d d a c b The sum of the angles of a triangle is 80. x x y z 80 y z Literal equations (or formulas) are equations with several variables. To solve for a specific variable, follow the steps to solve a linear equation. Example Solve for y. x y 0 y x 0 y x 0 y x 0 or y x

96 Chapter Review of Basic Algebraic Concepts Section.7 Linear Inequalities in One Variable Key Concepts A linear inequality in one variable can be written in the form ax b 6 0, ax b 7 0, ax b 0, or ax b 0 Properties of Inequalities. If a 6 b, then a c 6 b c.. If a 6 b, then a c 6 b c.. If c is positive and a 6 b, then ac 6 bc and a c 6 b c 0. c. If c is negative and a 6 b, then ac 7 bc and a c 7 b c 0. c Properties and indicate that if we multiply or divide an inequality by a negative value, the direction of the inequality sign must be reversed. The inequality a 6 x 6 b is represented by ( or, in interval notation, (a, b). a b ( Examples Example Solve. x Interval notation:, Example x 6 x 6 x 6 6 6 x a x b 7 x x 7 6x 7x 7 7x 7 6 7 x 6 x 6 6 x 6 ( (Reverse the inequality sign.) (Reverse the inequality sign.) (,

Review Exercises 97 Key Concepts Let a and b b 0 represent real numbers and m and n represent positive integers. a a b b m Section.8 b m b n b m n b m n b mn am b m b n a b b n b m b n bm n ab m a m b m b 0 A number expressed in the form a 0 n, where 0a 0 6 0 and n is an integer, is written in scientific notation. Properties of Integer Exponents and Scientific Notation Examples Example a x y z b x y 0 a x 6 y z bx x 0 y z x 0 y z or 8x 0 y z Example 0.000000,000.0 0 7. 0 7.0 0 or 0.007 Chapter Review Exercises Section.. Find a number that is a whole number but not a natural number. For Exercises, answers may vary.. List three rational numbers that are not integers. 0. Explain the difference between the union and intersection of two sets. You may use the sets C and D in the following diagram to provide an example.. List five integers, two of which are not whole numbers. C D For Exercises 9, write an expression in words that describes the set of numbers given by each interval. (Answers may vary.). (7, 6). 0,.6 6. 6, 7. 8, 8., 9., Let A x 0 x 6 6, B x 0 x 06, and C x 0 6 x 6 6. For Exercises 6, graph each set and write the set in interval notation.. A. B. C

98 Chapter Review of Basic Algebraic Concepts. A B Section.. 6. B C A B For Exercises 8, apply the distributive property and simplify.. x y 6. x 8y 7. x 0y z 8. a b c 7. True or false? x 6 is equivalent to 7 x 8. True or false? x 6 is equivalent to 7 x Section. For Exercises 9 0, find the opposite, reciprocal, and absolute value. 9. 8 0. For Exercises, simplify the exponents and the radicals..,., For Exercises, perform the indicated operations.. 6 8. 9 For Exercises 9, clear parentheses if necessary, and combine like terms. 9. 6q q 9 0. 8p 7p 8p.. 7 y y 8x 6x For Exercises, answers may vary.. Write an example of the commutative property of addition.. Write an example of the associative property of multiplication. Section.. Describe the solution set for a contradiction.. 8.7 6..7. 6. Describe the solution set for an identity. 7. 8. 8 a 0 b 7 9. 0... 0 0 0 9 a b a 6 b 8. Given h gt v 0 t h 0, find h if g ft/sec, v 0 6 ft/sec, h 0 6 ft, and t sec.. Find the area of a parallelogram with base in. and height 8 in. For Exercises 7 6, solve the equations and identify each as a conditional equation, a contradiction, or an identity. 7. x 7 8. y 7 8 9. 7. 0.6x 0.x 0. 0.y..y... m 9 m n 6 n x x. x x 8 in. in.

Review Exercises 99. 6. 0 8 m 8 7 8 m 8 m 68. a. Cory made $0,0 in taxable income in 007. If he pays 8% in federal income tax, determine the amount of tax he must pay. m m m m b. What is his net income (after taxes)? Section. 7. Explain how you would label three consecutive integers. Section.6 69. The length of a rectangle is ft more than the width. Find the dimensions if the perimeter is 0 ft. 8. Explain how you would label two consecutive odd integers. 9. Explain what the formula d rt means. 60. Explain what the formula I Prt means. 6. To do a rope trick, a magician needs to cut a piece of rope so that one piece is one-third the length of the other piece. If she begins with a -ft rope, what lengths will the two pieces of rope be? 6. Of three consecutive even integers, the sum of the smallest two integers is equal to 6 less than the largest. Find the integers. 6. Pat averages a rate of mph on his bike. One day he rode for min ( hr) and then got a flat tire and had to walk back home. He walked the same path that he rode and it took him hr. What was his average rate walking? 6. How much 0% acid solution should be mixed with a % acid solution to produce L of a solution that is % acid? 6. Sharyn invests $000 more in an account that earns 9% simple interest than she invests in an account that earns 6% simple interest. How much did she invest in each account if her total interest is $0 after year? 66. In 00, approximately 7. million men were in college in the United States. This represents an 8% increase over the number of men in college in 000. Approximately how many men were in college in 000? (Round to the nearest tenth of a million.) 67. In 00, there were 7,0 deaths due to alcoholrelated accidents in the United States. This was a % increase over the number of alcohol-related deaths in 999. How many such deaths were there in 999? For Exercises 70 7, solve for x, and then find the measure of each angle. 70. 7. (x ) (x ) (x ) For Exercises 7 7, solve for the indicated variable. 7. x y for y 7. 6x y for y 7. S pr pr h for h 7. A for b bh x a b 76. a. The circumference of a circle is given by C pr. Solve this equation for p. b. Tom measures the radius of a circle to be 6 cm and the circumference to be 7.7 cm. Use these values to approximate p. (Round to decimal places.)

00 Chapter Review of Basic Algebraic Concepts Section.7 For Exercises 77 8, solve the inequality. Graph the solution and write the solution set in interval notation. 77. 78. 79. 80. 8. 8. 8. 8. 8. 6x 7 6 0x x 9 7x 7 9x x 0 x x 8 x 7 q 6 z 0 86. One method to approximate your maximum heart rate is to subtract your age from 0. To maintain an aerobic workout, it is recommended that you sustain a heart rate of between 60% and 7% of your maximum heart rate. a. If the maximum heart rate h is given by the formula h 0 A, where A is a person s age, find your own maximum heart rate. (Answers will vary.) b. Find the interval for your own heart rate that will sustain an aerobic workout. (Answers will vary.) Section.8 9 8 For Exercises 87 9, simplify the expression and write the answer with positive exponents. 87. x x 88. 6x x 8 9. a b 9. a b 9. a x y 9. x y b 9. Write the numbers in scientific notation. a. The population of Asia was,686,600,000 in 000. b. A nanometer is 0.00000 of a millimeter. 96. Write the numbers in scientific notation. a. A millimeter is 0.00 of a meter. b. The population of Asia is predicted to be,,700,000 by 00. 97. Write the numbers in standard form. a. A micrometer is 0 of a millimeter. b. A nanometer is 0 9 of a meter. 98. Write the numbers in standard form. a. The total square footage of shopping centers in the United States is approximately. 0 9 ft. b. The total sales of those shopping centers is $.09 0. (Source: International Council of Shopping Centers.) For Exercises 99 0, perform the indicated operations. Write the answer in scientific notation. 99. 00. 0.,00,000 0.000 0.000,000.6 0 8 9.0 0 0. 7.0 0. 0 a x y x y b x y 89. 90. 8x y 8x y x y

Test 0 Chapter Test. a. List the integers between and, inclusive. b. List three rational numbers between and. (Answers may vary.) For Exercises 0, solve the equations. 0. x 7 0. Explain the difference between the intervals, and,.. Graph the sets and write each set in interval notation. a. All real numbers less than 6... 8 z z 8z 0.x 0.0860,000 x 0,00 x 6 x x b. All real numbers at least. Given sets A x 0 x 6 6 and B x 0 x 6, graph A B and write the set in interval notation.. Write the opposite, reciprocal, and absolute value for each of the numbers. a. b. c. 0 6. Simplify. 0 8 0 7. Given z x m, find z when n 6, x 8, s/ n s.8, and m 7.. (Round the answer to decimal place.) 8. True or false? a. x y x y is an example of the associative property of addition. b. is an example of the commutative property of multiplication. c. x x is an example of the distributive property. d. 0 y z 0 y z is an example of the associative property of addition. 9. Simplify the expressions. a. b 7b 6 b. x ax b. Label each equation as a conditional equation, an identity, or a contradiction. a. x 9 9 x b. a a c. w w. The difference between two numbers is 7. If the larger is times the smaller, find the two numbers. 6. Joëlle is determined to get some exercise and walks to the store at a brisk rate of. mph. She meets her friend Yun Ling at the store, and together they walk back at a slower rate of mph. Joëlle s total walking time was hr. a. How long did it take her to walk to the store? b. What is the distance to the store? 7. Shawnna banks at a credit union. Her money is distributed between two accounts: a certificate of deposit (CD) that earns % simple interest and a savings account that earns.% simple interest. Shawnna has $00 less in her savings account than in the CD. If after year her total interest is $8.0, how much did she invest in the CD? 8. A yield sign is in the shape of an equilateral triangle (all sides have equal length). Its perimeter is 8 in. Find the length of the sides. For Exercises 9 0, solve the equations for the indicated variable. 9. x y 6 for y 0. x m zs for z

0 Chapter Review of Basic Algebraic Concepts For Exercises, solve the inequalities. Graph the solution and write the solution set in interval notation.. x 8 7. x 6 x For Exercises 8, simplify the expression, and write the answer with positive exponents only. 0a 7. 6. a 6 x 6 x x. 6 x 7. a x6 8. y b 7 xy x y x 0 y. An elevator can accommodate a maximum weight of 000 lb. If four passengers on the elevator have an average weight of 80 lb each, how many additional passengers of the same average weight can the elevator carry before the maximum weight capacity is exceeded? 9. Multiply. 8.0 0 6 7. 0 0. Divide. (Write the answer in scientific notation.) 9,00,000 0.00