Chapter 1 Resource Masters

Transcription

1 hapter Resource Masters

2 onsumable Workbooks Many of the worksheets contained in the hapter Resource Masters booklets are available as consumable workbooks. Study Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook NSWERS FOR WORKOOKS The answers for hapter of these workbooks can be found in the back of this hapter Resource Masters booklet. Glencoe/McGraw-Hill opyright by The McGraw-Hill ompanies, Inc. ll rights reserved. Printed in the United States of merica. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe s lgebra. ny other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill ompanies 88 Orion Place olumbus, OH 0-0 ISN: lgebra hapter Resource Masters

3 ontents Vocabulary uilder vii Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment hapter ssessment hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form D hapter Test, Form hapter Open-Ended ssessment hapter Vocabulary Test/Review hapter Quizzes & hapter Quizzes & hapter Mid-hapter Test hapter umulative Review hapter Standardized Test Practice.... Standardized Test Practice Student Recording Sheet NSWERS Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Glencoe/McGraw-Hill iii Glencoe lgebra

4 Teacher s Guide to Using the hapter Resource Masters The Fast File hapter Resource system allows you to conveniently file the resources you use most often. The hapter Resource Masters includes the core materials needed for hapter. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. ll of the materials found in this booklet are included for viewing and printing in the lgebra TeacherWorks D-ROM. Vocabulary uilder Pages vii viii include a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. WHEN TO USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their lgebra Study Notebook. Remind them to add definitions and examples as they complete each lesson. Study Guide and Intervention Each lesson in lgebra addresses two objectives. There is one Study Guide and Intervention master for each objective. WHEN TO USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN TO USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice and pply section of the Student Edition exercises. These exercises are of average difficulty. WHEN TO USE These provide additional practice options or may be used as homework for second day teaching of the lesson. Reading to Learn Mathematics One master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. dditional questions ask students to interpret the context of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques. WHEN TO USE This master can be used as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one extension master for each lesson. These activities may extend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students. WHEN TO USE These may be used as extra credit, short-term projects, or as activities for days when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe lgebra

5 ssessment Options The assessment masters in the hapter Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. hapter ssessment HPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms and contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with axes are provided for questions assessing graphing skills. Form is an advanced level test with free-response questions. Grids without axes are provided for questions assessing graphing skills. ll of the above tests include a freeresponse onus question. The Open-Ended ssessment includes performance assessment tasks that are suitable for all students. scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Vocabulary Test, suitable for all students, includes a list of the vocabulary words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate ssessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Mid-hapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. ontinuing ssessment The umulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of lgebra. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which may appear on the standardized tests that they may encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. ubble-in and grid-in answer sections are provided on the master. nswers Page is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages. This improves students familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe lgebra

6 Reading to Learn Mathematics Vocabulary uilder This is an alphabetical list of the key vocabulary terms you will learn in hapter. s you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. dd these pages to your lgebra Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term absolute value Found on Page Definition/Description/Example Vocabulary uilder algebraic expression ssociative Property uh SOH shee uh tihv ommutative Property kuh MYOO tuh tihv compound inequality Distributive Property dih STRIH byuh tihv empty set Identity Property intersection Inverse Property (continued on the next page) Glencoe/McGraw-Hill vii Glencoe lgebra

7 Reading to Learn Mathematics Vocabulary uilder (continued) Vocabulary Term irrational numbers Found on Page Definition/Description/Example open sentence rational numbers Reflexive Property set-builder notation Substitution Property Symmetric Property suh MEH trihk Transitive Property Trichotomy Property try KH tuh mee union Glencoe/McGraw-Hill viii Glencoe lgebra

8 - Study Guide and Intervention Expressions and Formulas Order of Operations Order of Operations. Simplify the expressions inside grouping symbols.. Evaluate all powers.. Do all multiplications and divisions from left to right.. Do all additions and subtractions from left to right. Example Example Evaluate [8 ( )]. [8 ( )] [8 0] 8 Evaluate x x(y ) if x and y 0.. Replace each variable with the given value. x x(y ) () (0. ) (9) (.).. Lesson - Exercises Find the value of each expression.. ( ). ( ). ( ). 9( ). ( ) ( ) (). (8 0 ). () 8. 8( 8 ) 0.. Evaluate each expression if a 8., b, c, and d. ab d. 9.. (c 8b 0d) ac bd. 0. (b c) a 8.8. b c. c d b d 9 8 c b d. b. cd. d(a c). a. a b c.. b c d. d b c 8. a d Glencoe/McGraw-Hill Glencoe lgebra

9 - Study Guide and Intervention (continued) Expressions and Formulas Formulas formula is a mathematical sentence that uses variables to express the relationship between certain quantities. If you know the value of every variable except one in a formula, you can use substitution and the order of operations to find the value of the unknown variable. Example To calculate the number of reams of paper needed to print n copies np of a booklet that is p pages long, you can use the formula r, where r is the 00 number of reams needed. How many reams of paper must you buy to print copies of a -page booklet? np Substitute n and p into the formula r. 00 ()() r 00, You cannot buy 8. reams of paper. You will need to buy 9 reams to print copies. Exercises For Exercises, use the following information. For a science experiment, Sarah counts the number of breaths needed for her to blow up a beach ball. She will then find the volume of the beach ball in cubic centimeters and divide by the number of breaths to find the average volume of air per breath.. Her beach ball has a radius of 9 inches. First she converts the radius to centimeters using the formula.i, where is a length in centimeters and I is the same length in inches. How many centimeters are there in 9 inches?.8 cm. The volume of a sphere is given by the formula V r, where V is the volume of the sphere and r is its radius. What is the volume of the beach ball in cubic centimeters? (Use. for.) 0,0 cm. Sarah takes 0 breaths to blow up the beach ball. What is the average volume of air per breath? about 0 cm. person s basal metabolic rate (or MR) is the number of calories needed to support his or her bodily functions for one day. The MR of an 80-year-old man is given by the formula MR w (0.0)()w, where w is the man s weight in pounds. What is the MR of an 80-year-old man who weighs 0 pounds? 9 calories Glencoe/McGraw-Hill Glencoe lgebra

10 - Skills Practice Expressions and Formulas Find the value of each expression ( 8) () 9. ( ) ( ). [ 9 0()].. (8 ) 8. [() 8 ] 8 Lesson - Evaluate each expression if r, s, t, v 0, and w. 9. r s 0 0. st rs 8. w(s r). s r v. (s). s r wt v t s t. (r w). rv. w[t (t r)] 8. 0 s w r 9. 9r (s )t 0 0. s v. TEMPERTURE The formula K gives the temperature in kelvins (K) for a given temperature in degrees elsius. What is the temperature in kelvins when the temperature is degrees elsius? 8 K. TEMPERTURE The formula (F ) gives the temperature in degrees elsius 9 for a given temperature in degrees Fahrenheit. What is the temperature in degrees elsius when the temperature is 8 degrees Fahrenheit? 0 Glencoe/McGraw-Hill Glencoe lgebra

11 - Find the value of each expression.. ( ) 0. ( ). (). [0 ( )] ( ) () 8. ( ) ()(8) ()(0) 8. 8 { [ ( )]} 8. [( ) ( 8)] 9. [ ] 0. [ ( )] 8( ) Practice (verage) Expressions and Formulas.. ( ) ( 9) ( 8) 9 Evaluate each expression if a, b 8, c, d, and e.. ab d. (c d)b 8 ab c. d.. (b de)e 8. ac b de 0 9. b[a (c d) ] 0 0. e d(b c) ac. 9bc. ab (d c) 9. TEMPERTURE The formula F gives the temperature in degrees Fahrenheit for a given temperature in degrees elsius. What is the temperature in degrees Fahrenheit when the temperature is 0 degrees elsius? 0 F ac d c e. PHYSIS The formula h 0t t gives the height h in feet of an object t seconds after it is shot upward from Earth s surface with an initial velocity of 0 feet per second. What will the height of the object be after seconds? ft. GRIULTURE Faith owns an organic apple orchard. From her experience the last few seasons, she has developed the formula P 0x 0.0x 0 to predict her profit P in dollars this season if her trees produce x bushels of apples. What is Faith s predicted profit this season if her orchard produces 00 bushels of apples? $80 Glencoe/McGraw-Hill Glencoe lgebra

12 - Reading to Learn Mathematics Expressions and Formulas Pre-ctivity How are formulas used by nurses? Read the introduction to Lesson - at the top of page in your textbook. V d Nurses use the formula F to control the flow rate for IVs. Name t the quantity that each of the variables in this formula represents and the units in which each is measured. F represents the per minute. V represents the milliliters d represents the per milliliter. flow rate volume. drop factor and is measured in drops of solution and is measured in and is measured in drops Lesson - Reading the Lesson t represents time and is measured in minutes. Write the expression that a nurse would use to calculate the flow rate of an IV if a doctor orders 0 milliliters of IV saline to be given over 8 hours, with a drop factor of 0 drops per milliliter. Do not find the value of this expression There is a customary order for grouping symbols. rackets are used outside of parentheses. races are used outside of brackets. Identify the innermost expression(s) in each of the following expressions. a. [( ) 8] ( ) b. 9 [(8 ) (0 )] (8 ) and (0 ) c. { [8 ( ) ]} ( 00) ( ). Read the following instructions. Then use grouping symbols to show how the instructions can be put in the form of a mathematical expression. Multiply the difference of and by the sum of 9 and. dd the result to 0. Then divide what you get by. [( )(9 ) 0]. Why is it important for everyone to use the same order of operations for evaluating expressions? Sample answer: If everyone did not use the same order of operations, different people might get different answers. Helping You Remember. Think of a phrase or sentence to help you remember the order of operations. Sample answer: Please excuse my dear unt Sally. (parentheses; exponents; multiplication and division; addition and subtraction) Glencoe/McGraw-Hill Glencoe lgebra

13 - Enrichment Significant Digits ll measurements are approximations. The significant digits of an approximate number are those which indicate the results of a measurement. For example, the mass of an object, measured to the nearest gram, is 0 grams. The measurement 0 g has significant digits. The mass of the same object, measured to the nearest 00 g, is 00 g. The measurement 00 g has one significant digit.. Nonzero digits and zeros between significant digits are significant. For example, the measurement 9.0 m has significant digits, 9, 0,, and.. Zeros at the end of a decimal fraction are significant. The measurement 0.00 mm has significant digits, and 0.. Underlined zeros in whole numbers are significant. The measurement 0,00 0 km has significant digits,, 0,, 0, and 0. In general, a computation involving multiplication or division of measurements cannot be more accurate than the least accurate measurement in the computation. Thus, the result of computation involving multiplication or division of measurements should be rounded to the number of significant digits in the least accurate measurement. Example The mass of quarters is 0 g. Find the mass of one quarter. mass of quarter 0 g.8 g 0 has significant digits. does not represent a measurement. Round the result to significant digits. Why? Write the number of significant digits for each measurement m. 0.0 cm. 0.0 mm mg. 0,000 km. 0,00 0 km g g Solve. Round each result to the correct number of significant digits. 9. m. m 0.,00 0 ft 0 ft.. cm m. cm..0 mm. 908 yd m.0 m. mm 80 yd 0 m Glencoe/McGraw-Hill Glencoe lgebra

14 - Study Guide and Intervention Properties of Real Numbers Real Numbers ll real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and integers. R real numbers {all rationals and irrationals} Q rational numbers m {all numbers that can be represented in the form n n is not equal to 0} I irrational numbers {all nonterminating, nonrepeating decimals} N natural numbers {,,,,,,, 8, 9, } W whole numbers {0,,,,,,,, 8, } Z integers {,,,, 0,,,, }, where m and n are integers and Example Name the sets of numbers to which each number belongs. a. rationals (Q), reals (R) b. naturals (N), wholes (W), integers (Z), rationals (Q), reals (R) Lesson - Exercises Name the sets of numbers to which each number belongs.. Q, R. 8 Z, Q, R. 0 W, Z, Q, R Q, R 9. N, W, Z, Q, R. Q, R. Q, R 8.. Q, R 9. I, R 0. N, W, Z, Q, R.. Q, R. N, W, Z, Q, R. Z, Q, R. I, R 8.. Q, R. Q, R. I, R 8.. Q, R 9. 89,000 N, W, Z, Q, R Q, R Glencoe/McGraw-Hill Glencoe lgebra

15 - Study Guide and Intervention (continued) Properties of Real Numbers Properties of Real Numbers Real Number Properties For any real numbers a, b, and c Property ddition Multiplication ommutative a b b a a b b a ssociative (a b) c a (b c) (a b) c a (b c) Identity a 0 a 0 a a a a Inverse a ( a) 0 ( a) a If a is not zero, then a a. a a Distributive a(b c) ab ac and (b c)a ba ca Example Simplify 9x y y 0.9x. 9x y y 0.9x 9x ( 0.9x) y y (9 ( 0.9))x ( )y 8.x y ommutative Property ( ) Distributive Property Simplify. Exercises Simplify each expression.. 8(a b) (b a). 0s 8t t s. (j k j k) 0a s t k j a. 0(g h) (g h).. 8(.r.s) (.r.s) 80g h a b 0.r 9.s. (0 p) ( p) 8..j 8.9k.k 0.9j 9..(x ) (0.x) p.k.j.x 0. 9(e f) 0.(e f )..m( 8.). p r r p.e 9f 8.m p r. (0g 80h) 0(0h g). ( c) ( 8c) 0g 0h 0 0c. (.x) (.x ). (8 n n) 0.x 9 0 n. ( j ) j( ) 8. 0(a b) 0(b a) j 90a 0b b Glencoe/McGraw-Hill 8 Glencoe lgebra

16 - Skills Practice Properties of Real Numbers Name the sets of numbers to which each number belongs.. N, W, Z, Q, R. Z, Q, R. 0.8 Q, R. N, W, Z, Q, R. 9 Z, Q, R. 0 I, R Name the property illustrated by each equation.. x x 8. a 0 a omm. ( ) dd. Iden. 9. (r w) r w 0. r (r r) (r r) r Distributive ssoc. ( ) y. y. x() x Mult. Inv. Mult. Iden. Lesson -. 0.[(0.)] [0.()]0.. (0b b) b (b 0b) b ssoc. ( ) omm. ( ) Name the additive inverse and multiplicative inverse for each number..,..., 0.8., 8., Simplify each expression. 9. x x x 0. x y z y x z 0. (g h) g 0h g h. a a a a a a. (m z) (m z) m 8z. x y (x y z) x z. ( v) (v ) 8 v. (d ) (8 0d) 0d Glencoe/McGraw-Hill 9 Glencoe lgebra

17 - Name the sets of numbers to which each number belongs N, W, Z, Q, R I, R I, R W, Z, Q, R Practice (verage) Properties of Real Numbers. Q, R. Z, Q, R. Z, Q, R 8..8 Q, R Name the property illustrated by each equation. 9. x (y x) x (x y) 0. x (9x 8) (x 9x) 8 omm. ( ) ssoc. ( ). (x y) (x y). n n ( )n Mult. Iden. Distributive. (x)y ( )(xy). x y x y. ( )y 0y ssoc. ( ) omm. ( ) dd. Inv.. y y. (x y) x y 8. n 0 n Mult. Inv. Distributive dd. Iden. Name the additive inverse and multiplicative inverse for each number ,. 0..., 0..,., Simplify each expression.. x y x y x. a b a b a b. 8x y ( y) 8x y. c c (c c) c. (r 0s) (s r) r 8s 8. (0a ) (8 a) a 9. ( x y) ( x y) 0. x y (x y) 8x y. TRVEL Olivia drives her car at 0 miles per hour for t hours. Ian drives his car at 0 miles per hour for (t ) hours. Write a simplified expression for the sum of the distances traveled by the two cars. (0t 00) mi. NUMER THEORY Use the properties of real numbers to tell whether the following statement is true or false: If a b, it follows that a b. Explain your reasoning. a b false; counterexample: y Glencoe/McGraw-Hill 0 Glencoe lgebra

18 - Reading to Learn Mathematics Properties of Real Numbers Pre-ctivity How is the Distributive Property useful in calculating store savings? Read the introduction to Lesson - at the top of page in your textbook. Why are all of the amounts listed on the register slip at the top of page followed by negative signs? Sample answer: The amount of each coupon is subtracted from the total amount of purchases so that you save money by using coupons. Describe two ways of calculating the amount of money you saved by using coupons if your register slip is the one shown on page. Sample answer: dd all the individual coupon amounts or add the amounts for the scanned coupons and multiply the sum by. Reading the Lesson. Refer to the Key oncepts box on page. The numbers. and both involve decimals that go on forever. Explain why one of these numbers is rational and the other is irrational. Sample answer:.. is a repeating decimal because there is a block of digits,, that repeats forever, so this number is rational. The number is a non-repeating decimal because, although the digits follow a pattern, there is no block of digits that repeats. So this number is an irrational number.. Write the ssociative Property of ddition in symbols. Then illustrate this property by finding the sum 8 in two different ways. (a b) c a (b c); Sample answer: ( 8) 0 ; (8 ). onsider the equations (a b) c a (b c) and (a b) c c (a b). One of the equations uses the ssociative Property of Multiplication and one uses the ommutative Property of Multiplication. How can you tell which property is being used in each equation? The first equation uses the ssociative Property of Multiplication. The quantities a, b, and c are used in the same order, but they are grouped differently on the two sides of the equation. The second equation uses the quantities in different orders on the two sides of the equation. So the second equation uses the ommutative Property of Multiplication. Lesson - Helping You Remember. How can the meanings of the words commuter and association help you to remember the difference between the commutative and associative properties? Sample answer: commuter is someone who travels back and forth to work or another place, and the commutative property says you can switch the order when two numbers that are being added or multiplied. n association is a group of people who are connected or united, and the associative property says that you can switch the grouping when three numbers are added or multiplied. Glencoe/McGraw-Hill Glencoe lgebra

19 - Enrichment Properties of a Group set of numbers forms a group with respect to an operation if for that operation the set has () the losure Property, () the ssociative Property, () a member which is an identity, and () an inverse for each member of the set. Example Does the set {0,,,, } form a group with respect to addition? losure Property: For all numbers in the set, is a b in the set? 0, and is in the set; 0, and is in the set; and so on. The set has closure for addition. ssociative Property: For all numbers in the set, does a (b c) (a b) c? 0 ( ) (0 ) ; ( ) ( ) ; and so on. The set is associative for addition. Identity: Inverse: Is there some number, i, in the set such that i a a a i for all a? 0 0; 0 0; and so on. The identity for addition is 0. Does each number, a, have an inverse, a, such that a a a a i? The integer inverse of is since 0, and 0 is the identity for addition. ut the set does not contain. Therefore, there is no inverse for. The set is not a group with respect to addition because only three of the four properties hold. Example Is the set {, } a group with respect to multiplication? losure Property: ( )( ) ; ( )() ; ()( ) ; ()() The set has closure for multiplication. ssociative Property: ( )[( )( )] ( )() ; and so on The set is associative for multiplication. Identity: ( ) ; () The identity for multiplication is. Inverse: is the inverse of since ( )( ), and is the identity. is the inverse of since ()(), and is the identity. Each member has an inverse. The set {, } is a group with respect to multiplication because all four properties hold. Tell whether the set forms a group with respect to the given operation.. {integers}, addition yes. {integers}, multiplication no.,,,, addition no. {multiples of }, multiplication no. {x, x, x, x, } addition no. {,,, }, multiplication no. {irrational numbers}, addition no 8. {rational numbers}, addition yes Glencoe/McGraw-Hill Glencoe lgebra

20 - Study Guide and Intervention Solving Equations Verbal Expressions to lgebraic Expressions The chart suggests some ways to help you translate word expressions into algebraic expressions. ny letter can be used to represent a number that is not known. Word Expression and, plus, sum, increased by, more than Operation addition minus, difference, decreased by, less than subtraction times, product, of (as in of a number) multiplication divided by, quotient division Example Example Write an algebraic expression to represent 8 less than the quotient of a number and. n 8 Write a verbal sentence to represent (n ). Six times the difference of a number and two is equal to. Exercises Write an algebraic expression to represent each verbal expression.. the sum of six times a number and n. four times the sum of a number and (n ). less than fifteen times a number n 9n. the difference of nine times a number and the quotient of and the same number. the sum of 00 and four times a number 00 n n Lesson -. the product of and the sum of and a number ( n). four times the square of a number increased by five times the same number n n 8. more than the product of and a number n Write a verbal sentence to represent each equation. Sample answers are given. 9. n 9 The difference of three times a number and is equal to (n n ) n Twice the sum of the cube of a number and three times the square of the number is equal to four times the number. n. n n 8 The quotient of five times a number and the sum of the number and is equal to the difference of the number and 8. Glencoe/McGraw-Hill Glencoe lgebra

21 - Study Guide and Intervention (continued) Solving Equations Properties of Equality You can solve equations by using addition, subtraction, multiplication, or division. ddition and Subtraction For any real numbers a, b, and c, if a b, Properties of Equality then a c b c and a c b c. Multiplication and Division For any real numbers a, b, and c, if a b, a b Properties of Equality then a c b c and, if c is not zero,. c c Example Example Solve 00 8x x x x 0 x x y 00 x y x 00 x y 00 x y (00 x) y 0 x Solve x y 00 for y. Exercises Solve each equation. heck your solution.. s. 9 a 8. t t. m. x 8. 8 (z ). 0.b x x 9. ( k) 0k 0. 0 y n 98 n 8.. p 8... n 0 n r. x 0 x 9 Solve each equation or formula for the specified variable. a c s. a b c, for b b. 0, for t t t h pq 8. h g, for g g 9., for p r p d f 0. xy x, for x x., for f f d y k. (j k) 08, for j j 8..s t, for s s t m 0n. m 0, for m m. x y 0, for y y x 0 n n s 0 r q Glencoe/McGraw-Hill Glencoe lgebra

22 - Skills Practice Solving Equations Write an algebraic expression to represent each verbal expression.. times a number, increased by. 8 less than times a number n n 8. times the sum of a number and. the product of and a number, divided by 9 (n ) n 9. times the difference of and a number ( n). the product of and the square of a number n Write a verbal expression to represent each equation. 0. Sample answers are given.. n x The difference of a number The sum of 8 and times a and 8 is. number is. 9. b b y 0. y Three added to the square of number divided by is the a number is the number. difference of and twice the number. Name the property illustrated by each statement.. If a 0.b, and 0.b 0, then a 0.. If d f, then d f. Transitive ( ) Subtraction ( ). If x, then x.. If (8 )r 0, then r 0. Symmetric ( ) Substitution ( ) Lesson - Solve each equation. heck your solution.. m 8. x x. t t 8. b b 9. x x 0. v 0 a. a..n 0.8n n Solve each equation or formula for the specified variable. I. I prt, for p p. y x, for x x y 8 rt x y., for y y x. r r rh, for h h r Glencoe/McGraw-Hill Glencoe lgebra

23 - Practice (verage) Solving Equations Write an algebraic expression to represent each verbal expression.. more than the quotient of a number and. the sum of two consecutive integers y n (n ). times the sum of a number and. less than twice the square of a number (m ) y Write a verbal expression to represent each equation. 8. Sample answers are given.. x. y y The difference of and twice a Three times a number is times number is. the cube of the number.. c (c ) 8. (m ) The quotient Three times a number is twice the of a number and is times the difference of the number and. sum of twice the number and. Name the property illustrated by each statement. 9. If t, then t. 0. If 8(q ), then (q ). Symmetric ( ) Division ( ). If h, then h 0.. If m, then m. Subtraction ( ) Multiplication ( ) m Solve each equation. heck your solution.. 8 r. 9 n 9. n. s 8..r x 9x 9. ( v) v 0. y (y ) Solve each equation or formula for the specified variable.. E mc E d, for m m. c, for d d c. h vt gt h gt, for v v. E Iw U, for I I t Define a variable, write an equation, and solve the problem. c (E U ) w. GEOMETRY The length of a rectangle is twice the width. Find the width if the perimeter is 0 centimeters. w width; (w) w 0; 0 cm. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $ each. The total cost of the rented clubs and the green fees for each person was $. What was the cost of the green fees for each person? g green fees per person; () g ; $ Glencoe/McGraw-Hill Glencoe lgebra

24 - Reading to Learn Mathematics Solving Equations Pre-ctivity How can you find the most effective level of intensity for your workout? Reading the Lesson Read the introduction to Lesson - at the top of page 0 in your textbook. To find your target heart rate, what two pieces of information must you supply? age () and desired intensity level (I ) Write an equation that shows how to calculate your target heart rate. (0 ) I P or P (0 ) I. a. How are algebraic expressions and equations alike? Sample answer: oth contain variables, constants, and operation signs. b. How are algebraic expressions and equations different? Sample answer: Equations contain equal signs; expressions do not. c. How are algebraic expressions and equations related? Sample answer: n equation is a statement that says that two algebraic expressions are equal. Read the following problem and then write an equation that you could use to solve it. Do not actually solve the equation. In your equation, let m be the number of miles driven.. When Louisa rented a moving truck, she agreed to pay $8 per day plus $0. per mile. If she kept the truck for days and the rental charges (without tax) were $., how many miles did Louisa drive the truck? (8) 0.m. Lesson - Helping You Remember. How can the words reflection and symmetry help you remember and distinguish between the reflexive and symmetric properties of equality? Think about how these words are used in everyday life or in geometry. Sample answer: When you look at your reflection, you are looking at yourself. The reflexive property says that every number is equal to itself. In geometry, symmetry with respect to a line means that the parts of a figure on the two sides of a line are identical. The symmetric property of equality allows you to interchange the two sides of an equation. The equal sign is like the line of symmetry. Glencoe/McGraw-Hill Glencoe lgebra

25 - Enrichment Venn Diagrams Relationships among sets can be shown using Venn diagrams. Study the diagrams below. The circles represent sets and, which are subsets of set S. S S S The union of and consists of all elements in either or. The intersection of and consists of all elements in both and. The complement of consists of all elements not in. You can combine the operations of union, intersection, and finding the complement. Example Shade the region ( ). ( ) means the complement of the intersection of and. First find the intersection of and. Then find its complement. S Draw a Venn diagram and shade the region indicated. See students diagrams ( ). Draw a Venn diagram and three overlapping circles. Then shade the region indicated. See students diagrams.. ( ) 8. ( ) 9. ( ) 0. ( ). Is the union operation associative? yes. Is the intersection operation associative? yes Glencoe/McGraw-Hill 8 Glencoe lgebra

26 - Study Guide and Intervention Solving bsolute Value Equations bsolute Value Expressions The absolute value of a number is the number of units it is from 0 on a number line. The symbol x is used to represent the absolute value of a number x. Words For any real number a, if a is positive or zero, the absolute value of a is a. bsolute Value If a is negative, the absolute value of a is the opposite of a. Symbols For any real number a, a a, if a 0, and a a, if a 0. Example Example if x. Evaluate x x 8 Evaluate x y if x and y. x y ( ) () 8 9 Exercises Evaluate each expression if w, x, y, and z.. x 8. z. w z. x w. x y z. x x. w x 8. wz xy 9. z yz 9 0. w z y. z z y 0. 0 xw. y z yz. wx x 8y. yz 0 9. w xy. x y y 8. xyz wxz Lesson - 9. z z x x 0. 0x 0y. z 8w. yz w w. wz 8y 0. xz xz Glencoe/McGraw-Hill 9 Glencoe lgebra

27 - Study Guide and Intervention (continued) Solving bsolute Value Equations bsolute Value Equations Use the definition of absolute value to solve equations containing absolute value expressions. For any real numbers a and b, where b 0, if a b then a b or a b. lways check your answers by substituting them into the original equation. Sometimes computed solutions are not actual solutions. Example ase Solve x. heck your solutions. a b x x x 0 x 0 HEK x (0) 0 There are two solutions, 0 and. Exercises Solve each equation. heck your solutions. ase a b x x x x HEK ( ). x {, }. t 0 {, 9}. x { 0, 0}. m m {}. b 9. k {, 0}. n 8 n { } 8. 8 a a, 9. p p, 0. x x {, }. x. 0 x x 0 {, 0}. f f 0 {}. b b {, 9}. x x. x x {} Glencoe/McGraw-Hill 0 Glencoe lgebra

28 - Skills Practice Solving bsolute Value Equations Evaluate each expression if w 0., x, y, and z 0.. w. 9y. 9y z. z 0. 0z. 8x y y x. z 8. x y 9. w. 0. w.. x y.. w Solve each equation. heck your solutions.. y {, }. a 0 {, } 8. k,. g 0 { }. 0 c { 9, } 8. x x 9 {, } 9. p 0. w {, }. x x 8 {, }. y {, 0} Lesson -. n,. 8d d {, }. a,. k 0 9 Glencoe/McGraw-Hill Glencoe lgebra

29 - Practice (verage) Solving bsolute Value Equations Evaluate each expression if a, b 8, c, and d... a. b. 0d a. c b. 0a. b 8b. a c 8. c a 9. 0.c 0.b. 0. d a.. a b b a. d b 9. Solve each equation. heck your solutions.. n { 9, }. x {, }. y 9 {, }. x { 9, }. u {, } 8. x 9. x 9 0. y 9,. 8 p p {}. w w { }. y 9 {, }. y,. s s { 8}. w {, }. r 0 {, } 8. d 9. WETHER thermometer comes with a guarantee that the stated temperature differs from the actual temperature by no more than. degrees Fahrenheit. Write and solve an equation to find the minimum and maximum actual temperatures when the thermometer states that the temperature is 8. degrees Fahrenheit. t 8..; minimum: 8.9 F, maximum: 88.9 F 0. OPINION POLLS Public opinion polls reported in newspapers are usually given with a margin of error. For example, a poll with a margin of error of % is considered accurate to within plus or minus % of the actual value. poll with a stated margin of error of % predicts that candidate Tonwe will receive % of an upcoming vote. Write and solve an equation describing the minimum and maximum percent of the vote that candidate Tonwe is expected to receive. x ; minimum: 8%, maximum: % Glencoe/McGraw-Hill Glencoe lgebra

30 - Reading to Learn Mathematics Solving bsolute Value Equations Pre-ctivity How can an absolute value equation describe the magnitude of an earthquake? Read the introduction to Lesson - at the top of page 8 in your textbook. What is a seismologist and what does magnitude of an earthquake mean? a scientist who studies earthquakes; a number from to 0 that tells how strong an earthquake is Why is an absolute value equation rather than an equation without absolute value used to find the extremes in the actual magnitude of an earthquake in relation to its measured value on the Richter scale? Sample answer: The actual magnitude can vary from the measured magnitude by up to 0. unit in either direction, so an absolute value equation is needed. If the magnitude of an earthquake is estimated to be.9 on the Richter scale, it might actually have a magnitude as low as. or as high as.. Reading the Lesson. Explain how a could represent a positive number. Give an example. Sample answer: If a is negative, then a is positive. Example: If a, then a ( ).. Explain why the absolute value of a number can never be negative. Sample answer: The absolute value is the number of units it is from 0 on the number line. The number of units is never negative.. What does the sentence b 0 mean? Sample answer: The number b is 0 or greater than 0.. What does the symbol mean as a solution set? Sample answer: If a solution set is, then there are no solutions. Lesson - Helping You Remember. How can the number line model for absolute value that is shown on page 8 of your textbook help you remember that many absolute value equations have two solutions? Sample answer: The number line shows that for every positive number, there are two numbers that have that number as their absolute value. Glencoe/McGraw-Hill Glencoe lgebra

31 - NME DTE PERIOD Enrichment onsidering ll ases in bsolute Value Equations You have learned that absolute value equations with one set of absolute value symbols have two cases that must be considered. For example, x must be broken into x or (x ). For an equation with two sets of absolute value symbols, four cases must be considered. onsider the problem x x. First we must write the equations for the case where x 0 and where x 0. Here are the equations for these two cases: x x x (x ) Each of these equations also has two cases. y writing the equations for both cases of each equation above, you end up with the following four equations: x x x (x ) (x ) x x (x ) Solve each of these equations and check your solutions in the original equation, x x. The only solution to this equation is. Solve each absolute value equation. heck your solution.. x x x.. x 9 x x,. x x 0 x. x x x.. How many cases would there be for an absolute value equation containing three sets of absolute value symbols? 8. List each case and solve x x x. heck your solution. x x x (x ) x x x x (x ) (x ) x (x ) (x ) ( x ) x x ( x ) x (x ) ( x ) (x ) x ( x ) (x ) No solution Glencoe/McGraw-Hill Glencoe lgebra

32 - Study Guide and Intervention Solving Inequalities Solve Inequalities The following properties can be used to solve inequalities. ddition and Subtraction Properties for Inequalities Multiplication and Division Properties for Inequalities For any real numbers a, b, and c: For any real numbers a, b, and c, with c 0: a b. If a b, then a c b c and a c b c.. If c is positive and a b, then ac bc and. c c. If a b, then a c b c and a c b c. a b. If c is positive and a b, then ac bc and. c c These properties are also true for and. Example Example Solve x. Then graph the solution set on a number line. x x x The solution set is {x x }. a b. If c is negative and a b, then ac bc and. c c a b. If c is negative and a b, then ac bc and. c c Solve w. Then graph the solution set on a number line. w w w 8 w The solution set is (, ] Exercises Solve each inequality. Describe the solution set using set-builder or interval notation. Then graph the solution set on a number line.. (a 9) 8. (9z ) z. ( n) {a a } or (, ] {z z } or (, ) {n n } or (, ) k (k ). (b ). (m ) (m ) {k k } or (, ) {b b } or (, ) {m m } or (, ] x (x ) 8. (y ) y 9..d x x or, {y y 9} or (, 9) {d d } or (, ] Lesson - Glencoe/McGraw-Hill Glencoe lgebra

33 - Real-World Problems with Inequalities Many real-world problems involve inequalities. The chart below shows some common phrases that indicate inequalities. is less than is greater than is at most is at least is fewer than is more than is no more than is no less than is less than or equal to is greater than or equal to Example SPORTS The Vikings play games this year. t midseason, they have won games. How many of the remaining games must they win in order to win at least 80% of all their games this season? Let x be the number of remaining games that the Vikings must win. The total number of games they will have won by the end of the season is x. They want to win at least 80% of their games. Write an inequality with. x 0.8() x 0.8() x.8 Since they cannot win a fractional part of a game, the Vikings must win at least of the games remaining. Exercises Study Guide and Intervention (continued) Solving Inequalities. PRKING FEES The city parking lot charges $.0 for the first hour and $0. for each additional hour. If the most you want to pay for parking is $.0, solve the inequality.0 0.(x ).0 to determine for how many hours you can park your car. t most hours PLNNING For Exercises and, use the following information. Ethan is reading a 8-page book for a book report due on Monday. He has already read 80 pages. He wants to figure out how many pages per hour he needs to read in order to finish the book in less than hours Write an inequality to describe this situation. n or n Solve the inequality and interpret the solution. Ethan must read at least pages per hour in order to finish the book in less than hours. OWLING For Exercises and, use the following information. Four friends plan to spend Friday evening at the bowling alley. Three of the friends need to rent shoes for $.0 per person. string (game) of bowling costs $.0 per person. If the friends pool their $0, how many strings can they afford to bowl?. Write an equation to describe this situation. (.0) (.0)n 0. Solve the inequality and interpret the solution. The friends can bowl at most strings. Glencoe/McGraw-Hill Glencoe lgebra

34 - Skills Practice Solving Inequalities Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line. z. {z z 8} or (, 8]. a {a a } or (, ] q {q q } or (, ). 0 s s {s s } or (, ) 0 0. x 9 {x x } or [, ). b 9 {b b } or (, ] 0 0. z 9 z {z z } or (, ) 8. f 9 f {f f } or (, ) 0 9. s 8 s {s s } or [, ) 0. t (t ) t t or, 0. 0.m 0.m m {m m }. (x ) x x or 0 or (, ] 0 0 0,..y 0.8 {y y.}. x 9 x {x x } or (, ) or (., ) 0 0 Define a variable and write an inequality for each problem. Then solve.. Nineteen more than a number is less than. n 9 ; n. The difference of three times a number and is at least 8. n 8; n 8. One half of a number is more than less than the same number. n n ; n 8. Five less than the product of and a number is no more than twice that same number. n n; n Lesson - Glencoe/McGraw-Hill Glencoe lgebra

35 - Practice (verage) Solving Inequalities Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line.. 8x 0 {x x } or [, ). u {u u } or (, ) r 0 {r r } or (, ]. s 9s {s s } or (, ) 0. 9x x 9 x x or,. (w ) 8 w w 0 or,. 8u u 0 {u u } or [, ) x {x x 0.} or (, 0.) (r ) r {r r } 0. (x 8) (x ) {x x } or (, ) or (, ] x.. {x x } or [, ). q ( q) 0 q q or, (w ) (w ). n (n ) (n ) {w w } or (, ) {n n } or (, ) Define a variable and write an inequality for each problem. Then solve.. Twenty less than a number is more than twice the same number. n 0 n; n 0. Four times the sum of twice a number and is less than. times that same number. [n ( )].n; n.8. HOTELS The Lincoln s hotel room costs $90 a night. n additional 0% tax is added. Hotel parking is $ per day. The Lincoln s expect to spend $0 in tips during their stay. Solve the inequality 90x 90(0.)x x 0 00 to find how many nights the Lincoln s can stay at the hotel without exceeding total hotel costs of $00. nights 8. NKING Jan s account balance is $800. Of this, $0 is for rent. Jan wants to keep a balance of at least $00. Write and solve an inequality describing how much she can withdraw and still meet these conditions w 00; w $0 Glencoe/McGraw-Hill 8 Glencoe lgebra

36 - Reading to Learn Mathematics Solving Inequalities Pre-ctivity How can inequalities be used to compare phone plans? Reading the Lesson Read the introduction to Lesson - at the top of page in your textbook. Write an inequality comparing the number of minutes per month included in the two phone plans or 00 0 Suppose that in one month you use 0 minutes of airtime on your wireless phone. Find your monthly cost with each plan. Plan : $ Plan : $ Which plan should you choose? Plan. There are several different ways to write or show inequalities. Write each of the following in interval notation. a. {x x } (, ) b. {x x } [, ) c. (, ] 0 d. (, ) 0. Show how you can write an inequality symbol followed by a number to describe each of the following situations. a. There are fewer than 00 students in the senior class. 00 b. student may enroll in no more than six courses each semester. c. To participate in a concert, you must be willing to attend at least ten rehearsals. 0 d. There is space for at most students in the high school band. Helping You Remember. One way to remember something is to explain it to another person. common student error in solving inequalities is forgetting to reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. Suppose that your classmate is having trouble remembering this rule. How could you explain this rule to your classmate? Sample answer: Draw a number line. Plot two positive numbers, for example, and 8. Then plot their additive inverses, and 8. Write an inequality that compares the positive numbers and one that compares the negative numbers. Notice that 8, but 8. The order changes when you multiply by. Lesson - Glencoe/McGraw-Hill 9 Glencoe lgebra

37 - Enrichment Equivalence Relations relation R on a set is an equivalence relation if it has the following properties. Reflexive Property For any element a of set, a R a. Symmetric Property For all elements a and b of set, if a R b, then b R a. Transitive Property For all elements a, b, and c of set, if a R b and b R c, then a R c. Equality on the set of all real numbers is reflexive, symmetric, and transitive. Therefore, it is an equivalence relation. In each of the following, a relation and a set are given. Write yes if the relation is an equivalence relation on the given set. If it is not, tell which of the properties it fails to exhibit.., {all numbers} no; reflexive, symmetric., {all triangles in a plane} yes. is the sister of, {all women in Tennessee} no; reflexive., {all numbers} no; symmetric. is a factor of, {all nonzero integers} no; symmetric., {all polygons in a plane} yes. is the spouse of, {all people in Roanoke, Virginia} no; reflexive, transitive 8., {all lines in a plane} no; reflexive, transitive 9. is a multiple of, {all integers} no; symmetric 0. is the square of, {all numbers} no; reflexive, symmetric, transitive., {all lines in a plane} no; reflexive. has the same color eyes as, {all members of the leveland Symphony Orchestra} yes. is the greatest integer not greater than, {all numbers} no; reflexive, symmetric, transitive. is the greatest integer not greater than, {all integers} yes Glencoe/McGraw-Hill 0 Glencoe lgebra

38 - Study Guide and Intervention Solving ompound and bsolute Value Inequalities ompound Inequalities compound inequality consists of two inequalities joined by the word and or the word or.to solve a compound inequality, you must solve each part separately. nd ompound Inequalities Example: x and x 0 The graph is the intersection of solution sets of two inequalities. Lesson - Or ompound Inequalities Example: x or x 0 The graph is the union of solution sets of two inequalities. Example Example Solve x 9. Graph the solution set on a number line. x and x 9 8 x x x x x Solve y or y 9. Graph the solution set on a number line. y or y 9 y 9 or y 8 y or y Exercises Solve each inequality. Graph the solution set on a number line.. 0 x. a 8 or a {x x } {a a or a } x 0 0. k or 8k 9 {x x } {k k or k.} y. b 0 or b {y 9 y } {b b or b 8} w 8 8. d 9 or d {w w } {all real numbers} Glencoe/McGraw-Hill Glencoe lgebra

39 - Study Guide and Intervention (continued) Solving ompound and bsolute Value Inequalities bsolute Value Inequalities Use the definition of absolute value to rewrite an absolute value inequality as a compound inequality. For all real numbers a and b, b 0, the following statements are true.. If a b, then b a b.. If a b, then a b or a b. These statements are also true for and. Example Example Solve x. Graph the solution set on a number line. y statement above, if x, then x or x. Subtracting from both sides of each inequality gives x or x. Solve x. Graph the solution set on a number line. y statement above, if x, then x. dding to all three parts of the inequality gives x. Dividing by gives x Exercises Solve each inequality. Graph the solution set on a number line.. x 8 x x. s {s s. or s.} c. {c c }. a 9 0 {a a 9 or a } f 9 {f f or f 0}. w 8 {w w.} k {k k } 8. 0 {x x or x } x b b b m 0 m m or m Glencoe/McGraw-Hill Glencoe lgebra

40 - Skills Practice Solving ompound and bsolute Value Inequalities Write an absolute value inequality for each of the following. Then graph the solution set on a number line.. all numbers greater than or equal to. all numbers less than and greater or less than or equal to n than n Lesson -. all numbers less than or greater. all numbers between and n than n Write an absolute value inequality for each graph.. n. n 0. n 8. n Solve each inequality. Graph the solution set on a number line. 9. c or c 0 {c c 0. y {y y } or c 0} 0. 0 x {x x }. a 8 or a {a a or a } 0. 8 x {x x }. w 0 or w all real numbers t {t t or t }. x {x x } 0 0. r {r r or r } 8. p n {n n } 0. h {h h or h } Glencoe/McGraw-Hill Glencoe lgebra

41 - Practice (verage) Solving ompound and bsolute Value Inequalities Write an absolute value inequality for each of the following. Then graph the solution set on a number line.. all numbers greater than or less than n. all numbers between. and., including. and. n. Write an absolute value inequality for each graph.. n 0. n Solve each inequality. Graph the solution set on a number line.. 8 y 0 {y y }. (x ) or x x {x x or x 8}. x or x {x x } 8. x 0 and x {x x } 0 9. w w w or w 0. y {x x } x 8 {x x or x }. z z z. x {x x 0}. x x all real numbers. b. n n n. RINFLL In 90% of the last 0 years, the rainfall at Shell each has varied no more than. inches from its mean value of inches. Write and solve an absolute value inequality to describe the rainfall in the other 0% of the last 0 years. r.; {r r. or r 0.} 8. MNUFTURING company s guidelines call for each can of soup produced not to vary from its stated volume of. fluid ounces by more than 0.08 ounces. Write and solve an absolute value inequality to describe acceptable can volumes. v. 0.08; {v. v.8} Glencoe/McGraw-Hill Glencoe lgebra

42 - Reading to Learn Mathematics Solving ompound and bsolute Value Inequalities Pre-ctivity How are compound inequalities used in medicine? Read the introduction to Lesson - at the top of page 0 in your textbook. Five patients arrive at a medical laboratory at :0.M. for a glucose tolerance test. Each of them is asked when they last had something to eat or drink. Some of the patients are given the test and others are told that they must come back another day. Each of the patients is listed below with the times when they started to fast. (The P.M. times refer to the night before.) Which of the patients were accepted for the test? Ora :00.M. Juanita :0 P.M. Jason and Juanita Jason :0.M. Samir :00 P.M. Lesson - Reading the Lesson. a. Write a compound inequality that says, x is greater than and x is less than or equal to. x b. Graph the inequality that you wrote in part a on a number line. 0. Use a compound inequality and set-builder notation to describe the following graph. {x x or x } 0. Write a statement equivalent to x that does not use the absolute value symbol. x or x. Write a statement equivalent to x 8 that does not use the absolute value symbol. 8 x 8 Helping You Remember. Many students have trouble knowing whether an absolute value inequality should be translated into an and or an or compound inequality. Describe a way to remember which of these applies to an absolute value inequality. lso describe how to recognize the difference from a number line graph. Sample answer: If the absolute value quantity is followed by a or symbol, the expression inside the absolute value bars must be between two numbers, so this becomes an and inequality. The number line graph will show a single interval between two numbers. If the absolute value quantity is followed by a or symbol, it becomes an or inequality, and the graph will show two disconnected intervals with arrows going in opposite directions. Glencoe/McGraw-Hill Glencoe lgebra

43 - Enrichment onjunctions and Disjunctions n absolute value inequality may be solved as a compound sentence. Example Solve x 0. x 0 means x 0 and x 0. Solve each inequality. x and x. Every solution for x 0 is a replacement for x that makes both x and x true. compound sentence that combines two statements by the word and is a conjunction. Example Solve x. x means x or x. Solve each inequality. x 8 or x x or x Every solution for the inequality is a replacement for x that makes either x or x true. compound sentence that combines two statements by the word or is a disjunction. Solve each inequality. Then write whether the solution is a conjunction or disjunction.. x. x 8 x or x ; disjunction. x. x x and x ; conjunction. x x. x x and x ; conjunction x or x 0; disjunction x and x ; conjunction x and x ; conjunction. x 8. x x or x ; disjunction 9. 8 x 0. x x or x 0; disjunction x and x 8; conjunction x and x ; conjunction Glencoe/McGraw-Hill Glencoe lgebra

44 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the value of [ (8 )] D... Evaluate (a y) y if a and y D. 0.. Evaluate b if b D... The formula S n(n ) can be used to find the sum of the first n natural numbers. Find the sum of the first 0 natural numbers D. 90. ssessment. Name the sets of numbers to which belongs.. rationals. naturals, reals. rationals, reals D. integers, rationals, reals.. Simplify (x ) (x ).. x. x. x D. 9x.. Select the algebraic expression that represents the verbal expression: the product of nine and a number. n 9. 9n. 9 n D. 9 n. For Questions 8, solve each equation. 8. y 8... D (x 9) x... 0 D x. {9}. {}. {9, } D. 0.. x 0. {}. { 8}. {, 8} D.. Glencoe/McGraw-Hill Glencoe lgebra

45 NME DTE PERIOD hapter Test, Form (continued). Which equation could be used to solve the following problem? The sum of times a number and is. Find the number.. (n ). n. n D. n.. mar is five years older than his sister. The sum of their ages is 9. Find mar s age.... D. 9. For Questions 8, solve each inequality.. 8w. {w w }. {w w }. {w w } D. {w w }.. x or x. {x x }. {x x or x }. {x x } D... y 9. y y. all real numbers. y y 9 D. {y y }.. m 8. {m m }. {m m or m }. {m m or m } D.. 8. x 9. {x x }. {x x }. {x x or x } D. all real numbers Identify the graph of the solution set of 9 x D parking garage charges $ for the first hour and $ for each additional hour. Fran has $.0 to spend for parking. What is the greatest number of hours Fran can park?... D. 0. onus Solve x. : Glencoe/McGraw-Hill 8 Glencoe lgebra

46 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the value of.... D... Evaluate b(a c ) if a, b, and c D. 0.. Evaluate c d if c and d D. 8.. The formula for the surface area of a sphere is r, where r is the length of the radius. Find the surface area of a sphere with a radius of feet. Use for. ssessment. 8 ft. ft. ft D. 0 ft.. Name the sets of numbers to which belongs.. naturals, rationals. rational, reals. integers, rationals D. integers, rationals, reals.. Simplify (x 9) (x ).. 0x. x. x D. (0x ).. Name the property illustrated by (x y) (y x).. ommutative Property of Multiplication. Distributive Property. ommutative Property of ddition D. ssociative Property of ddition. For Questions 8, solve each equation. 8. m... D x 0. {, }. {, }. {, } D. {} (x ) x D. 0.. x 0. { }. {, }. {} D.. Glencoe/McGraw-Hill 9 Glencoe lgebra

47 NME DTE PERIOD hapter Test, Form (continued). Jamie is years younger than her brother. Five years from now, the sum of their ages will be. Find Jamie s present age D... One side of a triangle is four centimeters longer than the shortest side. The third side of the triangle is twice as long as the shortest side. Find the length of the longest side of the triangle if its perimeter is 0 centimeters.. 9 cm. cm. cm D. 8 cm. For Questions 8, solve each inequality x. {x x.}. {x x 98.}. {x x } D. {x x.9}.. 9 x. {x x 8}.. {x x or x 8} D. {x x }.. x or 9 x. {x x 9}. {x x or x 9}. all real numbers D. {x x 9}.. x. {x x }. {x x }. {x x } D. all real numbers. 8. m 8. {m m }. all real numbers. {m m or m } D. {m m or m } Identify the graph of the solution set of. 0.9y D One number is four times a second number. If you take one-half of the second number and increase it by the first number, the result is at least. Find the least possible value for the second number D. 0. onus arlos expects the grade on his next lgebra test to be between and 8. Using g to represent arlos test grade, write an absolute value inequality to describe this situation. : Glencoe/McGraw-Hill 0 Glencoe lgebra

48 NME DTE PERIOD hapter Test, Form SORE Write the letter for the correct answer in the blank at the right of each question.. Find the value of D... Evaluate (a y) y if a and y D... Evaluate a b if a and b D... The formula 80(n ) relates the measure of an interior angle of n a regular polygon to the number of sides n. If an interior angle measures 0, find the number of sides D. 0. ssessment. Name the sets of numbers to which 8 belongs.. integers. naturals, integers, reals. integers, rationals D. integers, rationals, reals.. Simplify (x ) (x ).. 0x 9. 9x 9. 0x D. 0x.. Name the property illustrated by (9 ) (9 ).. Distributive Property. ommutative Property of Multiplication. ssociative Property of Multiplication D. ommutative Property of ddition. For Questions 8, solve each equation. 8. y. 8.. D x. {9}. {}. {, 9} D (x ) x... D. 0.. y 8. {}. { }. {, } D.. Glencoe/McGraw-Hill Glencoe lgebra

49 NME DTE PERIOD hapter Test, Form (continued). Yoshi is years older than his sister. Six years from now, the sum of their ages will be. Find Yoshi s present age D... Two sides of a triangle are equal in length. The length of the third side is three meters less than the sum of the lengths of the other two sides. Find the length of the longest side of the triangle if its perimeter is 9 meters.. 8 m. m. m D. 0 m. For Questions 8, solve each inequality.. (r ) 9. {r r }. {r r }. {r r } D. {r r }.. z 0. {z z }. {z z }. z z D. z z.. x 0 or x. x x or x. x x. all real numbers D... m. {m m }. {m m or m }. {m m or m } D. all real numbers. 8. w. w w. {w w }. {w w } D. all real numbers Identify the graph of the solution set of y D One number is two less than a second number. If you take one-half of the first number and increase it by the second number, the result is at least. Find the least possible value for the second number D. 0. onus Solve x x 0. : Glencoe/McGraw-Hill Glencoe lgebra

50 NME DTE PERIOD hapter Test, Form SORE. Find the value of 8... Evaluate a b c if a, b, and c.. For Questions and, evaluate each expression if a. and b 8.. b a.. b a.. Use I = prt, the formula for simple interest over t years,. to find I when p = $00, r = 8.%, and t = 0 months. ssessment Name the sets of numbers to which each number belongs For Questions 9 and 0, name each property illustrated by each equation ab 0 ab 0.. Simplify (v 8) (v )... Write an algebraic expression to represent the verbal expression. ten less than the cube of a number. Solve each equation.. x 8.. x x.. x.. x. Glencoe/McGraw-Hill Glencoe lgebra

51 NME DTE PERIOD hapter Test, Form (continued) Define a variable, write an equation, and solve the problem.. The sum of twice a number and is 8. What is the number?. 8. Lana ordered concert tickets that cost $.0 for children 8. and $.00 for adults. She ordered 8 more children s tickets than adults tickets. Her total bill was $8. How many of each type of ticket did she order? For Questions 9, solve each inequality. Describe the solution set using set builder or interval notation. Then, graph the solution set on a number line. 9. t (x ) n. 8. v or v x. 0. x. 0. Define a variable and write an inequality. Then solve the. resulting inequality. The raves play games in a season. So far, they have won and lost 0. To win at least 0% of all games, how many more games must they win? onus Find the value of k so that the equation below has the : solution set { }. (x ) x( k) Glencoe/McGraw-Hill Glencoe lgebra

52 NME DTE PERIOD hapter Test, Form D SORE. Find the value of 9... Evaluate a if a, b, and c.. c b For Questions and, evaluate each expression if a. and b 0.. b a.. a b.. Use I prt, the formula for simple interest over t years, to. find I when p = $000, r = %, and t = 8 months. ssessment Name the sets of numbers to which each number belongs For Questions 9 and 0, name the property illustrated by each equation. 9. ab ( ab) xyz xyz 0.. Simplify (0x ) (x )... Write an algebraic expression to represent the verbal expression. five times the sum of seven and a number. Solve each equation.. n.. x 0 x.. w 9.. x. Glencoe/McGraw-Hill Glencoe lgebra

53 NME DTE PERIOD hapter Test, Form D (continued) Define a variable, write an equation, and solve the problem.. The sum of times a number and is. Find the number.. 8. The length of a rectangular garden is feet longer than its 8. width. The perimeter of the garden is 8 feet. Find the width and length of the garden. For Questions 9, solve each inequality. Describe the solution set using set builder or interval notation. Then, graph the solution set on a number line. 9. 0t (x ) x x. 0. n or n. 0. x x Define a variable and write an inequality. Then solve the. resulting inequality. The coins in Danielle s piggy bank have a value of at least $.. The bank contains only nickels and dimes. What is the fewest number of dimes that could be in the bank? onus Find the value of k so that the equation below has the : solution set { }. (x ) x( k) Glencoe/McGraw-Hill Glencoe lgebra

54 NME DTE PERIOD hapter Test, Form SORE. Find the value of 8... Evaluate (n v) v if n and v... Determine whether the statement is sometimes, always, or. never true. Explain your reasoning. If a and b are real numbers, then a b is negative.. The formula for the volume of a cylinder is V r h, where r. is the radius of the base and h is the height of the cylinder. Find the volume of a cylinder with a radius of. inches and a height of inches. Use. for π.. Name the sets of numbers to which each number belongs. a. b. c. 0 d. e.. a. b. c. d. e. ssessment. Simplify 8 (x 8) (y )... Write a verbal expression to represent the algebraic expression. (n n). For Questions 8, solve each equation. 8. (n 8) ( n) n x h(a b), for a 0.. y 8.. Define a variable, write an equation, and solve the problem.. The width of a rectangle is meters more than one-fourth its length. The perimeter is 0 meters more than twice its length. Find the length and width. Glencoe/McGraw-Hill Glencoe lgebra

55 NME DTE PERIOD hapter Test, Form (continued). The formula for the area of a triangle is bh, where b represents the base length, and h represents the height. The perimeter of the triangle shown is 8 inches. Write an equation for the area of this triangle in terms of its base length b. b inches 0 inches. For Questions 9, solve each inequality. Describe the solution set using set builder or interval notation. Then, graph the solution set on a number line...8 x.. (y ).. x 8 or x.. w x w Define a variable and write an inequality. Then solve the resulting inequality. Mr. rooks plans to invest part of $000 in a stock that pays 8% interest annually. The rest will be invested in a savings account that pays % interest annually. Mr. rooks wants to make at least $0 on the investment for the first year. What is the least amount that should be invested in the stock? 0. onus jet is flying from Hawaii to San Francisco, a distance : of 00 miles. In still air, the jet flies at 00 mph, but there is now a 0-mph tailwind. In case of emergency, how many hours after takeoff will it be faster for the jet to go on to San Francisco rather than to return to Hawaii? Glencoe/McGraw-Hill 8 Glencoe lgebra

56 NME DTE PERIOD hapter Open-Ended ssessment SORE Demonstrate your knowledge by giving a clear, concise solution to each problem. e sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.. a. State the property of real numbers or the property of equality that justifies each step in the solution of the equation given. x 8x Given x ( x) 8x ( x) x [( x) ] 8x ( x) [x ( x)] 8x ( x) 0 8x ( x) 8x ( x) [8 ( )]x x Substitution ssessment (x) x x x x b. Write your own solution of the equation ( x) 9x as you would write it on a test. ompare your solution to the solution above. Did you use all of the same properties as you listed above to solve your equation? Explain.. Given the inequality x k, find a value of k, if possible, that satisfies each condition. In each case, explain your choice. a. Find a value of k for which the inequality has no solution. b. Find a value of k for which the inequality has exactly one solution. c. Find a value of k for which a solution exists but for which the solution set does not include.. a. Write a word problem for the inequality x 0. b. Solve your problem and explain the meaning of your answer. c. Graph the solution of the inequality x 0. Does the graph have meaning for your word problem? Why or why not? Glencoe/McGraw-Hill 9 Glencoe lgebra

57 NME DTE PERIOD hapter Vocabulary Test/Review SORE absolute value ddition Property algebraic expression ssociative Property ommutative Property compound inequality counterexample Distributive Property Division Property empty set equation formula Identity Property intersection interval notation Inverse Property irrational numbers Multiplication Property open sentence order of operations rational numbers real numbers Reflexive Property set-builder notation solution Substitution Property Subtraction Property Symmetric Property Transitive Property Trichotomy Property union variable hoose from the terms above to complete each sentence.. The of addition says that adding 0 to any number does not change its value.. The are the numbers that can be written as ratios of two integers, with the integer in the denominator not being 0.. The property that allows you to switch the two sides of an equation is the.. x x is an example of the.. The graph of a compound inequality containing the word and is the of the graphs of the two separate inequalities.. {x x.} describes a set by using.. The of Multiplication says that you can reverse the order of two factors without changing the value of their product. 8. If y and y, then y y. This is an example of the. 9. Two inequalities combined by the word and or the word or form a 0. The of a number is the number of units between that number and 0 on a number line. In your own words Define each term.. irrational number. Trichotomy Property Glencoe/McGraw-Hill 0 Glencoe lgebra

58 NME DTE PERIOD hapter Quiz (Lessons and ) SORE. Find the value of 0... Evaluate n an if a and n... The formula for the perimeter P of a rectangle is. P ( w), where represents the length, and w represents the width of the rectangle. Find the perimeter of a rectangle with a length of 9. meters and a width of. meters.. Name the sets of numbers to which belongs... Simplify (v ) (8v ).. ssessment NME DTE PERIOD hapter Quiz (Lesson ) SORE Write the letter for the correct answer in the blank at the right of the question.. Standardized Test Practice If n, what is the value of n?.. 0. D.. For Questions and, solve each equation. heck your solution.. x x.. 8 w w 9.. Solve y mx b for x... Define a variable, write an equation, and solve the problem.. arla began a running program to prepare for track team try-outs. On her first day she ran miles, and on her second day she ran miles. Since then, arla has run miles each day. If her log book shows that arla has run a total of 99 miles, for how many days has arla been running miles? Glencoe/McGraw-Hill Glencoe lgebra

59 NME DTE PERIOD hapter Quiz (Lessons and ) SORE. Evaluate a 8b if a and b.. For Questions and, solve each equation.. x 0.. x x.. Solve x x, and graph its solution set on a. number line. 0. Define a variable and write an inequality. Then solve. The oston eltics play an 8-game schedule. If they have won of their first 0 games, how many more games must they win to win at least 0% of all 8 games?. NME DTE PERIOD hapter Quiz (Lesson ) SORE Solve each inequality. Describe the solution set using set builder or interval notation. Then, graph the solution set on a number line.. x or 9 x. 0. m x.. x. 0. x 9. 0 Glencoe/McGraw-Hill Glencoe lgebra

60 NME DTE PERIOD hapter Mid-hapter Test (Lessons through ) SORE Part I For Questions, write the letter for the correct answer in the blank at the right of each question.. Find the value of (9 )8.... D. 8.. Name the sets of numbers to which belongs.. integers, rationals. integers, rationals, reals. whole numbers, integers, reals D. integers, reals.. Name the property illustrated by ab ab 0.. dditive Inverse. dditive Identity. Multiplicative Inverse D. Multiplicative Identity. ssessment. Solve (x ) x D... Simplify (8y 0) (y ).. y 8. y. y 9 D. y. Part II. Write an algebraic expression to represent the verbal. expression the difference of three times a number x and.. Given the formula (F ), find the value of if F is Define a variable, write an equation, and solve the problem. 8. dults tickets to a play cost $ and students tickets cost $. If 9 tickets were sold and a total of $90 was collected, how many students tickets were sold? 9. Evaluate m np if m 0., n, and p Solve h b for b. 0. a. Find the value of [ ( )].. Glencoe/McGraw-Hill Glencoe lgebra

61 NME DTE PERIOD hapter umulative Review (hapter ). Simplify. Evaluate ( 0.).. (Prerequisite Skill) (Prerequisite Skill). For Questions and, find the value of each expression... 9 [( ) ] (Lesson ) (Lesson )... Use the formula F 9 to find the value of F if.. (Lesson ). Name the sets of numbers to which the number belongs.. (Lesson ). Simplify (x ) (9x ). (Lesson ). 8. Write an algebraic expression to represent the verbal 8. expression the square of a number increased by the cube of the same number. (Lesson ) Solve each equation. 9. x (x ) 0. y (Lesson ) (Lesson ). (m ) (0 m) m (Lesson ). Solve each inequality. Graph the solution set.. (t ) t (Lesson ) x 0 or x 0 (Lesson ). 0. x (Lesson ). Define a variable, write an equation, and solve the problem.. Forty-eight decreased by three times a number is thirty-six.. Find the number. (Lesson ) 0 Define a variable and write an inequality. Then solve.. The incinnati Reds play games in a season. So far. they have won games. How many more games must they win in order to win at least % of all games for the season? (Lesson ) Glencoe/McGraw-Hill Glencoe lgebra

62 NME DTE PERIOD Standardized Test Practice (hapter ) Part : Multiple hoice Instructions: Fill in the appropriate oval for the best answer.. If x is an even integer, what is the next consecutive even integer?. x. x. x D. x 8. D. 9 is 8% of what number? E. 00 F. 0 G.. H. 0%.. Which number is least?... 9 D The radius of a circle is tripled. What happens to the area of the circle? E. area is tripled F. area is multiplied by E F G H D ssessment G. area is multiplied by 9 H. area is multiplied by. E F G H. Which number is not a solution of x?... D.. D. Which represents a rational number? E. F. G. 0 H. 0. E F G H. In the figure shown, the length of X Y is of the perimeter of. What is the length of X Y?... 9 D.. X Y 0 Z D 8. Which number is not prime? E. F. 9 G. 9 H E F G H 9. If x 0, which of the following is negative?. x. x. x D. 9. x D 0. If a b is defined as b a, what is the value of? E. 9 F. 8 G. H. 0. E F G H. If more than the product of a number and is greater than 0, which of the following could be that number?... 0 D.. D Glencoe/McGraw-Hill Glencoe lgebra

63 NME DTE PERIOD Standardized Test Practice (continued) Part : Grid In Instructions: Enter your answer by writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entry.. The average of 8,, 9,, and x is x... What is the value of x? 0 0. D D D What is the length of the shortest path from to D? x. If, what is the value of x? Simplify Part : Quantitative omparison Instructions: ompare the quantities in columns and. Shade in if the quantity in column is greater; if the quantity in column is greater; if the quantities are equal; or D if the relationship cannot be determined from the information given. olumn olumn. (0.0) D. b b. D x D x (x + 0) x 0 x 9. g g 9. D Glencoe/McGraw-Hill Glencoe lgebra

64 Standardized Test Practice Student Record Sheet (Use with pages of the Student Edition.) Glencoe/McGraw-Hill Glencoe lgebra Select the best answer from the choices given and fill in the corresponding oval Solve the problem and write your answer in the blank. For Questions 8, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 8 Select the best answer from the choices given and fill in the corresponding oval. 9 0 D D D D D / / / / / / / / / / / / D D D D D D D D D D NME DTE PERIOD nswers Part Short Response/Grid In Part Short Response/Grid In Part Multiple hoice Part Multiple hoice Part Quantitative omparison Part Quantitative omparison

65 nswers (Lesson -) Lesson - - Study Guide and Intervention Expressions and Formulas Order of Operations. Simplify the expressions inside grouping symbols. Order of. Evaluate all powers. Operations. Do all multiplications and divisions from left to right.. Do all additions and subtractions from left to right. Example Example Evaluate [8 ( )]. [8 ( )] [8 0] 8 Evaluate x x(y ) if x and y 0.. Replace each variable with the given value. x x(y ) () (0. ) (9) (.).. Exercises Find the value of each expression.. ( ). ( ). ( ). 9( ). ( ) ( ) (). (8 0 ). () ( 8 ) 0.. Evaluate each expression if a 8., b, c, and d. ab c. 9.. (c 8b 0d) 8. d b d 9. ac bd. 0. (b c) a a 8.8. d b c. c. d b b. cd d. d(a c). a. a b c.. b c d. d b c 8. Glencoe/McGraw-Hill Glencoe lgebra - Study Guide and Intervention (continued) Expressions and Formulas Formulas formula is a mathematical sentence that uses variables to express the relationship between certain quantities. If you know the value of every variable except one in a formula, you can use substitution and the order of operations to find the value of the unknown variable. Example To calculate the number of reams of paper needed to print n copies np of a booklet that is p pages long, you can use the formula r, where r is the 00 number of reams needed. How many reams of paper must you buy to print copies of a -page booklet? np Substitute n and p into the formula r. 00 ()() r 00, You cannot buy 8. reams of paper. You will need to buy 9 reams to print copies. Exercises For Exercises, use the following information. For a science experiment, Sarah counts the number of breaths needed for her to blow up a beach ball. She will then find the volume of the beach ball in cubic centimeters and divide by the number of breaths to find the average volume of air per breath.. Her beach ball has a radius of 9 inches. First she converts the radius to centimeters using the formula.i, where is a length in centimeters and I is the same length in inches. How many centimeters are there in 9 inches?.8 cm. The volume of a sphere is given by the formula V r, where V is the volume of the sphere and r is its radius. What is the volume of the beach ball in cubic centimeters? (Use. for.) 0,0 cm. Sarah takes 0 breaths to blow up the beach ball. What is the average volume of air per breath? about 0 cm. person s basal metabolic rate (or MR) is the number of calories needed to support his or her bodily functions for one day. The MR of an 80-year-old man is given by the formula MR w (0.0)()w, where w is the man s weight in pounds. What is the MR of an 80-year-old man who weighs 0 pounds? 9 calories Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

66 nswers (Lesson -) - Skills Practice Expressions and Formulas Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Find the value of each expression ( 8) () 9. ( ) ( ). [ 9 0()].. (8 ) 8. [() 8 ] 8 Evaluate each expression if r, s, t, v 0, and w. 9. r s 0 0. st rs 8. w(s r). s r v. (s). s r wt v t. (r w). s t rv. w[t (t r)] 8. 0 s 9. 9r (s w )t 0 0. s v r. TEMPERTURE The formula K gives the temperature in kelvins (K) for a given temperature in degrees elsius. What is the temperature in kelvins when the temperature is degrees elsius? 8 K. TEMPERTURE The formula 9 (F ) gives the temperature in degrees elsius for a given temperature in degrees Fahrenheit. What is the temperature in degrees elsius when the temperature is 8 degrees Fahrenheit? 0 - Practice (verage) Expressions and Formulas Find the value of each expression.. ( ) 0. ( ). (). [0 ( )] ( ) () 8. ( ) ()(8) ()(0) 8. 8 { [ ( )]} 8. [( ) ( 8)] 9. [ ] 0. [ ( )] 8( ) ( 8).. ( ) ( 9) 9 Evaluate each expression if a, b 8, c, d, and e.. ab d. (c d)b 8 ab. d d(b c). c ac. (b de)e 8. ac b de 0 9. b[a (c d) ac ] 0 0. c e d. 9bc e. ab (d c) 9. TEMPERTURE The formula F gives the temperature in degrees Fahrenheit for a given temperature in degrees elsius. What is the temperature in degrees Fahrenheit when the temperature is 0 degrees elsius? 0 F. PHYSIS The formula h 0t t gives the height h in feet of an object t seconds after it is shot upward from Earth s surface with an initial velocity of 0 feet per second. What will the height of the object be after seconds? ft. GRIULTURE Faith owns an organic apple orchard. From her experience the last few seasons, she has developed the formula P 0x 0.0x 0 to predict her profit P in dollars this season if her trees produce x bushels of apples. What is Faith s predicted profit this season if her orchard produces 00 bushels of apples? $80 Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

67 nswers (Lesson -) Lesson - - Reading to Learn Mathematics Expressions and Formulas Pre-ctivity How are formulas used by nurses? Read the introduction to Lesson - at the top of page in your textbook. V d Nurses use the formula F to control the flow rate for IVs. Name t the quantity that each of the variables in this formula represents and the units in which each is measured. F represents the flow rate and is measured in per minute. drops V represents the volume of solution and is measured in milliliters d represents the drop factor and is measured in per milliliter. drops t represents time and is measured in minutes. Reading the Lesson Write the expression that a nurse would use to calculate the flow rate of an IV if a doctor orders 0 milliliters of IV saline to be given over 8 hours, with a drop factor of 0 drops per milliliter. Do not find the value of this expression There is a customary order for grouping symbols. rackets are used outside of parentheses. races are used outside of brackets. Identify the innermost expression(s) in each of the following expressions. a. [( ) 8] ( ) b. 9 [(8 ) (0 )] (8 ) and (0 ) c. { [8 ( ) ]} ( 00) ( ). Read the following instructions. Then use grouping symbols to show how the instructions can be put in the form of a mathematical expression. Multiply the difference of and by the sum of 9 and. dd the result to 0. Then divide what you get by. [( )(9 ) 0]. Why is it important for everyone to use the same order of operations for evaluating expressions? Sample answer: If everyone did not use the same order of operations, different people might get different answers. Helping You Remember. Think of a phrase or sentence to help you remember the order of operations. Sample answer: Please excuse my dear unt Sally. (parentheses; exponents; multiplication and division; addition and subtraction) Glencoe/McGraw-Hill Glencoe lgebra - Enrichment Significant Digits ll measurements are approximations. The significant digits of an approximate number are those which indicate the results of a measurement. For example, the mass of an object, measured to the nearest gram, is 0 grams. The measurement 0 g has significant digits. The mass of the same object, measured to the nearest 00 g, is 00 g. The measurement 00 g has one significant digit.. Nonzero digits and zeros between significant digits are significant. For example, the measurement 9.0 m has significant digits, 9, 0,, and.. Zeros at the end of a decimal fraction are significant. The measurement 0.00 mm has significant digits, and 0.. Underlined zeros in whole numbers are significant. The measurement 0,00 0 km has significant digits,, 0,, 0, and 0. In general, a computation involving multiplication or division of measurements cannot be more accurate than the least accurate measurement in the computation. Thus, the result of computation involving multiplication or division of measurements should be rounded to the number of significant digits in the least accurate measurement. Example The mass of quarters is 0 g. Find the mass of one quarter. mass of quarter 0 g 0 has significant digits. does not represent a measurement..8 g Round the result to significant digits. Why? Write the number of significant digits for each measurement m. 0.0 cm. 0.0 mm mg. 0,000 km. 0,00 0 km g g Solve. Round each result to the correct number of significant digits. 9. m. m 0.,00 0 ft 0 ft.. cm m cm..0 mm. 908 yd m.0 m. mm 80 yd 0 m Glencoe/McGraw-Hill Glencoe lgebra. Glencoe/McGraw-Hill Glencoe lgebra

68 nswers (Lesson -) - Study Guide and Intervention Properties of Real Numbers Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Real Numbers ll real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and integers. R real numbers {all rationals and irrationals} Q rational numbers {all numbers that can be represented in the form, where m and n are integers and n is not equal to 0} m n I irrational numbers {all nonterminating, nonrepeating decimals} N natural numbers {,,,,,,, 8, 9, } W whole numbers {0,,,,,,,, 8, } Z integers {,,,, 0,,,, } Example Name the sets of numbers to which each number belongs. a. rationals (Q), reals (R) b. naturals (N), wholes (W), integers (Z), rationals (Q), reals (R) Exercises Name the sets of numbers to which each number belongs.. Q, R. 8 Z, Q, R. 0 W, Z, Q, R Q, R. N, W, Z, Q, R. Q, R. 9 Q, R 8.. Q, R 9. I, R 0. N, W, Z, Q, R.. Q, R. N, W, Z, Q, R. Z, Q, R. I, R 8.. Q, R. Q, R. I, R 8.. Q, R 9. 89,000 N, W, Z, Q, R Q, R - Study Guide and Intervention (continued) Properties of Real Numbers Properties of Real Numbers Real Number Properties For any real numbers a, b, and c Property ddition Multiplication ommutative a b b a a b b a ssociative (a b) c a (b c) (a b) c a (b c) Identity a 0 a 0 a a a a Inverse a ( a) 0 ( a) a If a is not zero, then a a a a. Distributive a(b c) ab ac and (b c)a ba ca Example Simplify 9x y y 0.9x. 9x y y 0.9x 9x ( 0.9x) y y ommutative Property ( ) (9 ( 0.9))x ( )y Distributive Property 8.x y Simplify. Exercises Simplify each expression.. 8(a b) (b a). 0s 8t t s. (j k j k) 0a s t k j a b. 0(g h) (g h).. 8(.r.s) (.r.s) 80g h a b 0.r 9.s. (0 p) ( p) 8..j 8.9k.k 0.9j 9..(x ) (0.x) p.k.j.x 0. 9(e f) 0.(e f )..m( 8.). p r r p.e 9f 8.m p r. (0g 80h) 0(0h g). ( c) ( 8c) 0g 0h 0 0c. (.x) (.x ). (8 n n) 0.x 9 0 n. ( j ) j( ) 8. 0(a b) 0(b a) j 90a 0b Glencoe/McGraw-Hill 8 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

69 nswers (Lesson -) Lesson - - Skills Practice Properties of Real Numbers Name the sets of numbers to which each number belongs.. N, W, Z, Q, R. Z, Q, R. 0.8 Q, R. N, W, Z, Q, R. 9 Z, Q, R. 0 I, R Name the property illustrated by each equation.. x x 8. a 0 a omm. ( ) dd. Iden. 9. (r w) r w 0. r (r r) (r r) r Distributive ssoc. ( ). y y. x() x Mult. Inv. Mult. Iden.. 0.[(0.)] [0.()]0.. (0b b) b (b 0b) b ssoc. ( ) omm. ( ) Name the additive inverse and multiplicative inverse for each number..,..., 0.8., 8., Simplify each expression. 9. x x x 0. x y z y x z 0. (g h) g 0h g h. a a a a a a. (m z) (m z) m 8z. x y (x y z) x z. ( v) (v ) 8 v. (d ) (8 0d) 0d Glencoe/McGraw-Hill 9 Glencoe lgebra - Practice (verage) Properties of Real Numbers Name the sets of numbers to which each number belongs N, W, Z, Q, R I, R I, R W, Z, Q,R. Q, R. Z, Q, R. Z, Q, R 8..8 Q, R Name the property illustrated by each equation. 9. x (y x) x (x y) 0. x (9x 8) (x 9x) 8 omm. ( ) ssoc. ( ). (x y) (x y). n n ( )n Mult. Iden. Distributive. (x)y ( )(xy). x y x y. ( )y 0y ssoc. ( ) omm. ( ) dd. Inv.. y y. (x y) x y 8. n 0 n Mult. Inv. Distributive dd. Iden. Name the additive inverse and multiplicative inverse for each number ,. 0..., 0..,., Simplify each expression.. x y x y x. a b a b a b. 8x y ( y) 8x y. c c (c c) c. (r 0s) (s r) r 8s 8. (0a ) (8 a) a 9. ( x y) ( x y) x y 0. 8x y y (x y). TRVEL Olivia drives her car at 0 miles per hour for t hours. Ian drives his car at 0 miles per hour for (t ) hours. Write a simplified expression for the sum of the distances traveled by the two cars. (0t 00) mi. NUMER THEORY Use the properties of real numbers to tell whether the following a. Explain your reasoning. statement is true or false: If a b, it follows that a b b false; counterexample: Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

70 nswers (Lesson -) - Reading to Learn Mathematics Properties of Real Numbers Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Pre-ctivity How is the Distributive Property useful in calculating store savings? Read the introduction to Lesson - at the top of page in your textbook. Why are all of the amounts listed on the register slip at the top of page followed by negative signs? Sample answer: The amount of each coupon is subtracted from the total amount of purchases so that you save money by using coupons. Describe two ways of calculating the amount of money you saved by using coupons if your register slip is the one shown on page. Sample answer: dd all the individual coupon amounts or add the amounts for the scanned coupons and multiply the sum by. Reading the Lesson. Refer to the Key oncepts box on page. The numbers. and both involve decimals that go on forever. Explain why one of these numbers is rational and the other is irrational. Sample answer:.. is a repeating decimal because there is a block of digits,, that repeats forever, so this number is rational. The number is a non-repeating decimal because, although the digits follow a pattern, there is no block of digits that repeats. So this number is an irrational number.. Write the ssociative Property of ddition in symbols. Then illustrate this property by finding the sum 8 in two different ways. (a b) c a (b c); Sample answer: ( 8) 0 ; (8 ). onsider the equations (a b) c a (b c) and (a b) c c (a b). One of the equations uses the ssociative Property of Multiplication and one uses the ommutative Property of Multiplication. How can you tell which property is being used in each equation? The first equation uses the ssociative Property of Multiplication. The quantities a, b, and c are used in the same order, but they are grouped differently on the two sides of the equation. The second equation uses the quantities in different orders on the two sides of the equation. So the second equation uses the ommutative Property of Multiplication. Helping You Remember. How can the meanings of the words commuter and association help you to remember the difference between the commutative and associative properties? Sample answer: commuter is someone who travels back and forth to work or another place, and the commutative property says you can switch the order when two numbers that are being added or multiplied. n association is a group of people who are connected or united, and the associative property says that you can switch the grouping when three numbers are added or multiplied. - Enrichment Properties of a Group set of numbers forms a group with respect to an operation if for that operation the set has () the losure Property, () the ssociative Property, () a member which is an identity, and () an inverse for each member of the set. Example Does the set {0,,,, } form a group with respect to addition? losure Property: For all numbers in the set, is a b in the set? 0, and is in the set; 0, and is in the set; and so on. The set has closure for addition. ssociative Property: For all numbers in the set, does a (b c) (a b) c? 0 ( ) (0 ) ; ( ) ( ) ; and so on. The set is associative for addition. Identity: Is there some number, i, in the set such that i a a a i for all a? 0 0; 0 0; and so on. The identity for addition is 0. Inverse: Does each number, a, have an inverse, a, such that a a a a i? The integer inverse of is since 0, and 0 is the identity for addition. ut the set does not contain. Therefore, there is no inverse for. The set is not a group with respect to addition because only three of the four properties hold. Example Is the set {, } a group with respect to multiplication? losure Property: ( )( ) ; ( )() ; ()( ) ; ()() The set has closure for multiplication. ssociative Property: ( )[( )( )] ( )() ; and so on The set is associative for multiplication. Identity: ( ) ; () The identity for multiplication is. Inverse: is the inverse of since ( )( ), and is the identity. is the inverse of since ()(), and is the identity. Each member has an inverse. The set {, } is a group with respect to multiplication because all four properties hold. Tell whether the set forms a group with respect to the given operation.. {integers}, addition yes. {integers}, multiplication no.,,,, addition no. {multiples of }, multiplication no. {x, x, x, x, } addition no. {,,, }, multiplication no. {irrational numbers}, addition no 8. {rational numbers}, addition yes Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

71 nswers (Lesson -) Lesson - - Study Guide and Intervention Solving Equations Verbal Expressions to lgebraic Expressions The chart suggests some ways to help you translate word expressions into algebraic expressions. ny letter can be used to represent a number that is not known. Word Expression Operation and, plus, sum, increased by, more than addition minus, difference, decreased by, less than subtraction times, product, of (as in of a number) multiplication divided by, quotient division Write an algebraic expression to represent 8 less than the quotient of a number and. n Example Example 8 Write a verbal sentence to represent (n ). Six times the difference of a number and two is equal to. Exercises Write an algebraic expression to represent each verbal expression.. the sum of six times a number and n. four times the sum of a number and (n ). less than fifteen times a number n 9n. the difference of nine times a number and the quotient of and the same number n. the sum of 00 and four times a number 00 n. the product of and the sum of and a number ( n). four times the square of a number increased by five times the same number n n 8. more than the product of and a number n Write a verbal sentence to represent each equation. Sample answers are given. 9. n 9 The difference of three times a number and is equal to (n n ) n Twice the sum of the cube of a number and three times the square of the number is equal to four times the number.. n n n 8 The quotient of five times a number and the sum of the number and is equal to the difference of the number and 8. Glencoe/McGraw-Hill Glencoe lgebra - Study Guide and Intervention (continued) Solving Equations Properties of Equality You can solve equations by using addition, subtraction, multiplication, or division. ddition and Subtraction For any real numbers a, b, and c, if a b, Properties of Equality then a c b c and a c b c. Multiplication and Division For any real numbers a, b, and c, if a b, a b Properties of Equality then a c b c and, if c is not zero, c c. Example Solve 00 8x 0. Example 00 8x x x 0 x x y 00 x y x 00 x y 00 x y (00 x) Solve x y 00 for y. y 0 x Exercises Solve each equation. heck your solution.. s. 9 a 8. t t. m. x 8. 8 (z ). 0.b x x 9. ( k) 0k 0. 0 y n 98 n 8.. p 8... n 0 n r. x 0 x 9 Solve each equation or formula for the specified variable. a c s. a b c, for b b. t 0, for t t h pq 8. h g, for g g 9., for p r p d f 0. xy x, for x x., for f f d y k. (j k) 08, for j j 8..s t, for s s t s 0 r q m 0n. m 0, for m m. x y 0, for y y x 0 n n Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 8 Glencoe lgebra

72 nswers (Lesson -) - Skills Practice Solving Equations Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Write an algebraic expression to represent each verbal expression.. times a number, increased by. 8 less than times a number n n 8. times the sum of a number and. the product of and a number, divided by 9 n (n ) 9. times the difference of and a number ( n). the product of and the square of a number n Write a verbal expression to represent each equation. 0. Sample answers are given.. n x The difference of a number The sum of 8 and times a and 8 is. number is. 9. b b 0. y y Three added to the square of number divided by is the a number is the number. difference of and twice the number. Name the property illustrated by each statement.. If a 0.b, and 0.b 0, then a 0.. If d f, then d f. Transitive ( ) Subtraction ( ). If x, then x.. If (8 )r 0, then r 0. Symmetric ( ) Substitution ( ) Solve each equation. heck your solution.. m 8. x x. t t 8. b b 9. x x 0. v 0 a. a..n 0.8n n Solve each equation or formula for the specified variable. I. I prt, for p p rt. y x, for x x y 8 x y., for y y x. r rh, for h h r r - Practice (verage) Solving Equations Write an algebraic expression to represent each verbal expression.. more than the quotient of a number and. the sum of two consecutive integers y n (n ). times the sum of a number and. less than twice the square of a number (m ) y Write a verbal expression to represent each equation. 8. Sample answers. x. y y are given. The difference of and twice a Three times a number is times number is. the cube of the number.. c (c ) 8. m (m ) The quotient Three times a number is twice the of a number and is times the difference of the number and. sum of twice the number and. Name the property illustrated by each statement. 9. If t, then t. 0. If 8(q ), then (q ). Symmetric ( ) Division ( ). If h, then h 0.. If m, then m. Subtraction ( ) Multiplication ( ) Solve each equation. heck your solution.. 8 r. 9 n 9. n 8. s..r x 9x 9. ( v) v 0. y (y ) Solve each equation or formula for the specified variable.. E mc E d, for m m c. c, for d d. h vt gt h gt, for v v. E Iw U, for I I t c (E U ) w Define a variable, write an equation, and solve the problem.. GEOMETRY The length of a rectangle is twice the width. Find the width if the perimeter is 0 centimeters. w width; (w) w 0; 0 cm. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $ each. The total cost of the rented clubs and the green fees for each person was $. What was the cost of the green fees for each person? g green fees per person; () g ; $ Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 9 Glencoe lgebra

73 nswers (Lesson -) Lesson - - Reading to Learn Mathematics Solving Equations Pre-ctivity How can you find the most effective level of intensity for your workout? Read the introduction to Lesson - at the top of page 0 in your textbook. To find your target heart rate, what two pieces of information must you supply? age () and desired intensity level (I ) Write an equation that shows how to calculate your target heart rate. (0 ) I P or P (0 ) I Reading the Lesson. a. How are algebraic expressions and equations alike? Sample answer: oth contain variables, constants, and operation signs. b. How are algebraic expressions and equations different? Sample answer: Equations contain equal signs; expressions do not. c. How are algebraic expressions and equations related? Sample answer: n equation is a statement that says that two algebraic expressions are equal. Read the following problem and then write an equation that you could use to solve it. Do not actually solve the equation. In your equation, let m be the number of miles driven.. When Louisa rented a moving truck, she agreed to pay $8 per day plus $0. per mile. If she kept the truck for days and the rental charges (without tax) were $., how many miles did Louisa drive the truck? (8) 0.m. Helping You Remember. How can the words reflection and symmetry help you remember and distinguish between the reflexive and symmetric properties of equality? Think about how these words are used in everyday life or in geometry. Sample answer: When you look at your reflection, you are looking at yourself. The reflexive property says that every number is equal to itself. In geometry, symmetry with respect to a line means that the parts of a figure on the two sides of a line are identical. The symmetric property of equality allows you to interchange the two sides of an equation. The equal sign is like the line of symmetry. Glencoe/McGraw-Hill Glencoe lgebra - Enrichment Venn Diagrams Relationships among sets can be shown using Venn diagrams. Study the diagrams below. The circles represent sets and, which are subsets of set S. S S S The union of and consists of all elements in either or. The intersection of and consists of all elements in both and. The complement of consists of all elements not in. You can combine the operations of union, intersection, and finding the complement. Example Shade the region ( ). ( ) means the complement of the intersection of and. First find the intersection of and. Then find its complement. S Draw a Venn diagram and shade the region indicated. See students diagrams ( ). Draw a Venn diagram and three overlapping circles. Then shade the region indicated. See students diagrams.. ( ) 8. ( ) 9. ( ) 0. ( ). Is the union operation associative? yes. Is the intersection operation associative? yes Glencoe/McGraw-Hill 8 Glencoe lgebra Glencoe/McGraw-Hill 0 Glencoe lgebra

74 nswers (Lesson -) - Study Guide and Intervention Solving bsolute Value Equations Glencoe/McGraw-Hill 9 Glencoe lgebra nswers Lesson - bsolute Value Expressions The absolute value of a number is the number of units it is from 0 on a number line. The symbol x is used to represent the absolute value of a number x. Words For any real number a, if a is positive or zero, the absolute value of a is a. bsolute Value If a is negative, the absolute value of a is the opposite of a. Symbols For any real number a, a a, if a 0, and a a, if a 0. Example Evaluate x Example Evaluate x y if x. if x and y. x 8 x y ( ) () 8 9 Exercises Evaluate each expression if w, x, y, and z.. x 8. z. w z. x w. x y z. x x. w x 8. wz xy 9. z yz 9 0. w z y. z z y 0. 0 xw. y z yz. wx x 8y. yz 0 9. w xy. x y y 8. xyz wxz 9. z z x x 0. 0x 0y. z 8w. yz w w. wz 8y 0. xz xz - Study Guide and Intervention (continued) Solving bsolute Value Equations bsolute Value Equations Use the definition of absolute value to solve equations containing absolute value expressions. For any real numbers a and b, where b 0, if a b then a b or a b. lways check your answers by substituting them into the original equation. Sometimes computed solutions are not actual solutions. Example Solve x. heck your solutions. ase a b x x x 0 x 0 HEK x (0) 0 There are two solutions, 0 and. ase a b x x x x HEK ( ) Exercises Solve each equation. heck your solutions.. x {, }. t 0 {, 9}. x { 0, 0}. m m {}. b 9. k {, 0}. n 8 n { } 8. 8 a a, 9. p p, 0. x x {, }. x. 0 x x 0 {, 0}. f f 0 {}. b b {, 9}. x x. x x {} Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

75 nswers (Lesson -) Lesson - - Skills Practice Solving bsolute Value Equations Evaluate each expression if w 0., x, y, and z 0.. w. 9y. 9y z. z 0. 0z. 8x y y x. z 8. x y 9. w. 0. w.. x y.. w Solve each equation. heck your solutions.. y {, }. a 0 {, } 8. k,. g 0 { }. 0 c { 9, } 8. x x 9 {, } 9. p 0. w {, }. x x 8 {, }. y {, 0}. n,. 8d d {, }. a,. k 0 9 Glencoe/McGraw-Hill Glencoe lgebra - Practice (verage) Solving bsolute Value Equations Evaluate each expression if a, b 8, c, and d... a. b. 0d a. c b. 0a. b 8b. a c 8. c a 9. 0.c 0.b. 0. d a.. a b b a. d b 9. Solve each equation. heck your solutions.. n { 9, }. x {, }. y 9 {, }. x { 9, }. u {, } 8. x 9. x 9 0. y 9.,.. 8 p p {}. w w { 8}. y 9 {, }. y,. s s { 8}. w {, }. r 0 {, } 8. d 9. WETHER thermometer comes with a guarantee that the stated temperature differs from the actual temperature by no more than. degrees Fahrenheit. Write and solve an equation to find the minimum and maximum actual temperatures when the thermometer states that the temperature is 8. degrees Fahrenheit. x 8..; or 8.9 x OPINION POLLS Public opinion polls reported in newspapers are usually given with a margin of error. For example, a poll with a margin of error of % is considered accurate to within plus or minus % of the actual value. poll with a stated margin of error of % predicts that candidate Tonwe will receive % of an upcoming vote. Write and solve an equation describing the minimum and maximum percent of the vote that candidate Tonwe is expected to receive. x or 8 x Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

76 nswers (Lesson -) - Reading to Learn Mathematics Solving bsolute Value Equations Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Pre-ctivity How can an absolute value equation describe the magnitude of an earthquake? Read the introduction to Lesson - at the top of page 8 in your textbook. What is a seismologist and what does magnitude of an earthquake mean? a scientist who studies earthquakes; a number from to 0 that tells how strong an earthquake is Why is an absolute value equation rather than an equation without absolute value used to find the extremes in the actual magnitude of an earthquake in relation to its measured value on the Richter scale? Sample answer: The actual magnitude can vary from the measured magnitude by up to 0. unit in either direction, so an absolute value equation is needed. If the magnitude of an earthquake is estimated to be.9 on the Richter scale, it might actually have a magnitude as low as. or as high as.. Reading the Lesson. Explain how a could represent a positive number. Give an example. Sample answer: If a is negative, then a is positive. Example: If a, then a ( ).. Explain why the absolute value of a number can never be negative. Sample answer: The absolute value is the number of units it is from 0 on the number line. The number of units is never negative.. What does the sentence b 0 mean? Sample answer: The number b is 0 or greater than 0.. What does the symbol mean as a solution set? Sample answer: If a solution set is, then there are no solutions. Helping You Remember. How can the number line model for absolute value that is shown on page 8 of your textbook help you remember that many absolute value equations have two solutions? Sample answer: The number line shows that for every positive number, there are two numbers that have that number as their absolute value. - Enrichment onsidering ll ases in bsolute Value Equations You have learned that absolute value equations with one set of absolute value symbols have two cases that must be considered. For example, x must be broken into x or (x ). For an equation with two sets of absolute value symbols, four cases must be considered. onsider the problem x x. First we must write the equations for the case where x 0 and where x 0. Here are the equations for these two cases: x x x (x ) Each of these equations also has two cases. y writing the equations for both cases of each equation above, you end up with the following four equations: x x x (x ) (x ) x x (x ) Solve each of these equations and check your solutions in the original equation, x x. The only solution to this equation is. Solve each absolute value equation. heck your solution.. x x x.. x 9 x x,. x x 0 x. x x x.. How many cases would there be for an absolute value equation containing three sets of absolute value symbols? 8. List each case and solve x x x. heck your solution. x x x (x ) x x x x (x ) (x ) x (x ) (x ) ( x ) x x ( x ) x (x ) ( x ) (x ) x ( x ) (x ) No solution Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

77 nswers (Lesson -) Lesson - - Study Guide and Intervention Solving Inequalities Solve Inequalities The following properties can be used to solve inequalities. ddition and Subtraction Properties for Inequalities Multiplication and Division Properties for Inequalities For any real numbers a, b, and c: For any real numbers a, b, and c, with c 0:. If a b, then a c b c and a c b c.. If c is positive and a b, then ac bc and.. If a b, then a c b c and a c b c. a b. If c is positive and a b, then ac bc and c c. a c b c a b. If c is negative and a b, then ac bc and c c. a b. If c is negative and a b, then ac bc and c c. These properties are also true for and. Example Example Solve x. Then graph the solution set on a number line. x x x The solution set is {x x }. Solve w. Then graph the solution set on a number line. w w w 8 w The solution set is (, ] Exercises Solve each inequality. Describe the solution set using set-builder or interval notation. Then graph the solution set on a number line.. (a 9) 8. (9z ) z. ( n) {a a } or (, ] {z z } or (, ) {n n } or (, ) k (k ). (b ). (m ) (m ) {k k } or (, ) {b b } or (, ) {m m } or (, ] x (x ) 8. (y ) y 9..d x x or, {y y 9} or (, 9) {d d } or (, ] Glencoe/McGraw-Hill Glencoe lgebra - Study Guide and Intervention (continued) Solving Inequalities Real-World Problems with Inequalities Many real-world problems involve inequalities. The chart below shows some common phrases that indicate inequalities. is less than is greater than is at most is at least is fewer than is more than is no more than is no less than is less than or equal to is greater than or equal to Example SPORTS The Vikings play games this year. t midseason, they have won games. How many of the remaining games must they win in order to win at least 80% of all their games this season? Let x be the number of remaining games that the Vikings must win. The total number of games they will have won by the end of the season is x. They want to win at least 80% of their games. Write an inequality with. x 0.8() x 0.8() x.8 Since they cannot win a fractional part of a game, the Vikings must win at least of the games remaining. Exercises. PRKING FEES The city parking lot charges $.0 for the first hour and $0. for each additional hour. If the most you want to pay for parking is $.0, solve the inequality.0 0.(x ).0 to determine for how many hours you can park your car. t most hours PLNNING For Exercises and, use the following information. Ethan is reading a 8-page book for a book report due on Monday. He has already read 80 pages. He wants to figure out how many pages per hour he needs to read in order to finish the book in less than hours Write an inequality to describe this situation. n or n Solve the inequality and interpret the solution. Ethan must read at least pages per hour in order to finish the book in less than hours. OWLING For Exercises and, use the following information. Four friends plan to spend Friday evening at the bowling alley. Three of the friends need to rent shoes for $.0 per person. string (game) of bowling costs $.0 per person. If the friends pool their $0, how many strings can they afford to bowl?. Write an equation to describe this situation. (.0) (.0)n 0. Solve the inequality and interpret the solution. The friends can bowl at most strings. Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

78 nswers (Lesson -) - Skills Practice Solving Inequalities Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line. z. {z z 8} or (, 8]. a {a a } or (, ] q {q q } or (, ). 0 s s {s s } or (, ) 0 0. x 9 {x x } or [, ). b 9 {b b } or (, ] 0 0. z 9 z {z z } or (, ) 8. f 9 f {f f } or (, ) s 8 s {s s } or [, ) 0. t (t ) t t or, m 0.m m {m m }. (x ) x x or or (, ] 0 0,..y 0.8 {y y.}. x 9 x {x x } or (, ) or (., ) 0 0 Define a variable and write an inequality for each problem. Then solve.. Nineteen more than a number is less than. n 9 ; n. The difference of three times a number and is at least 8. n 8; n 8. One half of a number is more than less than the same number. n n ; n 8. Five less than the product of and a number is no more than twice that same number. n n; n - Practice (verage) Solving Inequalities Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line.. 8x 0 {x x } or [, ). u {u u } or (, ) r 0 {r r } or (, ]. s 9s {s s } or (, ) 0. 9x x 9 x x or,. (w ) 8 w w or,. 8u u 0 {u u } or [, ) x {x x 0.} or (, 0.) (r ) r {r r } 0. (x 8) (x ) {x x } or (, ) or (, ] x.. {x x } or [, ). q ( q) 0 q q or, 0 0. (w ) (w ). n (n ) (n ) {w w } or (, ) {n n } or (, ) 0 0 Define a variable and write an inequality for each problem. Then solve.. Twenty less than a number is more than twice the same number. n 0 n; n 0. Four times the sum of twice a number and is less than. times that same number. [n ( )].n; n.8. HOTELS The Lincoln s hotel room costs $90 a night. n additional 0% tax is added. Hotel parking is $ per day. The Lincoln s expect to spend $0 in tips during their stay. Solve the inequality 90x 90(0.)x x 0 00 to find how many nights the Lincoln s can stay at the hotel without exceeding total hotel costs of $00. nights 8. NKING Jan s account balance is $800. Of this, $0 is for rent. Jan wants to keep a balance of at least $00. Write and solve an inequality describing how much she can withdraw and still meet these conditions w 00; w $0 Glencoe/McGraw-Hill 8 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

79 nswers (Lesson -) Lesson - - Reading to Learn Mathematics Solving Inequalities Pre-ctivity How can inequalities be used to compare phone plans? Read the introduction to Lesson - at the top of page in your textbook. Write an inequality comparing the number of minutes per month included in the two phone plans or 00 0 Suppose that in one month you use 0 minutes of airtime on your wireless phone. Find your monthly cost with each plan. Plan : $ Plan : $ Which plan should you choose? Plan Reading the Lesson. There are several different ways to write or show inequalities. Write each of the following in interval notation. a. {x x } (, ) b. {x x } [, ) c. (, ] 0 d. (, ) 0. Show how you can write an inequality symbol followed by a number to describe each of the following situations. a. There are fewer than 00 students in the senior class. 00 b. student may enroll in no more than six courses each semester. c. To participate in a concert, you must be willing to attend at least ten rehearsals. 0 d. There is space for at most students in the high school band. Helping You Remember. One way to remember something is to explain it to another person. common student error in solving inequalities is forgetting to reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. Suppose that your classmate is having trouble remembering this rule. How could you explain this rule to your classmate? Sample answer: Draw a number line. Plot two positive numbers, for example, and 8. Then plot their additive inverses, and 8. Write an inequality that compares the positive numbers and one that compares the negative numbers. Notice that 8, but 8. The order changes when you multiply by. Glencoe/McGraw-Hill 9 Glencoe lgebra - Enrichment Equivalence Relations relation R on a set is an equivalence relation if it has the following properties. Reflexive Property For any element a of set, a R a. Symmetric Property For all elements a and b of set, if a R b, then b R a. Transitive Property For all elements a, b, and c of set, if a R b and b R c, then a R c. Equality on the set of all real numbers is reflexive, symmetric, and transitive. Therefore, it is an equivalence relation. In each of the following, a relation and a set are given. Write yes if the relation is an equivalence relation on the given set. If it is not, tell which of the properties it fails to exhibit.., {all numbers} no; reflexive, symmetric., {all triangles in a plane} yes. is the sister of, {all women in Tennessee} no; reflexive., {all numbers} no; symmetric. is a factor of, {all nonzero integers} no; symmetric., {all polygons in a plane} yes. is the spouse of, {all people in Roanoke, Virginia} no; reflexive, transitive 8., {all lines in a plane} no; reflexive, transitive 9. is a multiple of, {all integers} no; symmetric 0. is the square of, {all numbers} no; reflexive, symmetric, transitive., {all lines in a plane} no; reflexive. has the same color eyes as, {all members of the leveland Symphony Orchestra} yes. is the greatest integer not greater than, {all numbers} no; reflexive, symmetric, transitive. is the greatest integer not greater than, {all integers} yes Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

80 nswers (Lesson -) - Study Guide and Intervention Solving ompound and bsolute Value Inequalities Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - ompound Inequalities compound inequality consists of two inequalities joined by the word and or the word or.to solve a compound inequality, you must solve each part separately. nd ompound Inequalities Example: x and x The graph is the intersection of solution sets of two inequalities. 0 Or ompound Inequalities Example: x or x The graph is the union of solution sets of two inequalities. 0 Example Example Solve x 9. Graph the solution set on a number line. x and x 9 8 x x x x x Solve y or y 9. Graph the solution set on a number line. y or y 9 y 9 or y 8 y or y Exercises Solve each inequality. Graph the solution set on a number line.. 0 x. a 8 or a {x x } {a a or a } x 0 0. k or 8k 9 {x x } {k k or k.} y. b 0 or b {y 9 y } {b b or b 8} w 8 8. d 9 or d {w w } {all real numbers} Study Guide and Intervention (continued) Solving ompound and bsolute Value Inequalities bsolute Value Inequalities Use the definition of absolute value to rewrite an absolute value inequality as a compound inequality. For all real numbers a and b, b 0, the following statements are true.. If a b, then b a b.. If a b, then a b or a b. These statements are also true for and. Example Example Solve x. Graph the solution set on a number line. y statement above, if x, then x or x. Subtracting from both sides of each inequality gives x or x. Solve x. Graph the solution set on a number line. y statement above, if x, then x. dding to all three parts of the inequality gives x. Dividing by gives x Exercises Solve each inequality. Graph the solution set on a number line.. x 8 x x. s {s s. or s.} c. {c c }. a 9 0 {a a 9 or a } f 9 {f f or f 0}. w 8 {w w.} x. 0 k {k k } 8. 0 {x x or x } b b b m 0 m m or m Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

81 nswers (Lesson -) Lesson - - Skills Practice Solving ompound and bsolute Value Inequalities Write an absolute value inequality for each of the following. Then graph the solution set on a number line.. all numbers greater than or equal to. all numbers less than and greater or less than or equal to n than n all numbers less than or greater. all numbers between and n than n Write an absolute value inequality for each graph.. n. n 0 0. n 8. n. 0 0 Solve each inequality. Graph the solution set on a number line. 9. c or c 0 {c c 0. y {y y } or c 0} x {x x }. a 8 or a {a a or a } x {x x }. w 0 or w all real numbers t {t t or t }. x {x x } 0 0. r {r r or r } 8. p n {n n } 0. h {h h or h } Glencoe/McGraw-Hill Glencoe lgebra - Practice (verage) Solving ompound and bsolute Value Inequalities Write an absolute value inequality for each of the following. Then graph the solution set on a number line.. all numbers greater than or less than n. all numbers between. and., including. and. n Write an absolute value inequality for each graph.. n 0. n Solve each inequality. Graph the solution set on a number line.. 8 y 0 {y y }. (x ) or x x {x x or x 8}. x or x {x x } 8. x 0 and x {x x } w w w or w 0. y {x x } x 8 {x x or x }. z z z x {x x 0}. x x all real numbers 0 0. b. n n n 0 0. RINFLL In 90% of the last 0 years, the rainfall at Shell each has varied no more than. inches from its mean value of inches. Write and solve an absolute value inequality to describe the rainfall in the other 0% of the last 0 years. r.; {r r. or r 0.} 8. MNUFTURING company s guidelines call for each can of soup produced not to vary from its stated volume of. fluid ounces by more than 0.08 ounces. Write and solve an absolute value inequality to describe acceptable can volumes. v. 0.08; {v. v.8} Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 8 Glencoe lgebra

82 nswers (Lesson -) - Reading to Learn Mathematics Solving ompound and bsolute Value Inequalities Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Pre-ctivity How are compound inequalities used in medicine? Read the introduction to Lesson - at the top of page 0 in your textbook. Five patients arrive at a medical laboratory at :0.M. for a glucose tolerance test. Each of them is asked when they last had something to eat or drink. Some of the patients are given the test and others are told that they must come back another day. Each of the patients is listed below with the times when they started to fast. (The P.M. times refer to the night before.) Which of the patients were accepted for the test? Ora :00.M. Juanita :0 P.M. Jason and Juanita Jason :0.M. Samir :00 P.M. Reading the Lesson. a. Write a compound inequality that says, x is greater than and x is less than or equal to. x b. Graph the inequality that you wrote in part a on a number line. 0. Use a compound inequality and set-builder notation to describe the following graph. {x x or x } 0. Write a statement equivalent to x that does not use the absolute value symbol. x or x. Write a statement equivalent to x 8 that does not use the absolute value symbol. 8 x 8 Helping You Remember. Many students have trouble knowing whether an absolute value inequality should be translated into an and or an or compound inequality. Describe a way to remember which of these applies to an absolute value inequality. lso describe how to recognize the difference from a number line graph. Sample answer: If the absolute value quantity is followed by a or symbol, the expression inside the absolute value bars must be between two numbers, so this becomes an and inequality. The number line graph will show a single interval between two numbers. If the absolute value quantity is followed by a or symbol, it becomes an or inequality, and the graph will show two disconnected intervals with arrows going in opposite directions. - Enrichment onjunctions and Disjunctions n absolute value inequality may be solved as a compound sentence. Example Solve x 0. x 0 means x 0 and x 0. Solve each inequality. x and x. Every solution for x 0 is a replacement for x that makes both x and x true. compound sentence that combines two statements by the word and is a conjunction. Example Solve x. x means x or x. Solve each inequality. x 8 or x x or x Every solution for the inequality is a replacement for x that makes either x or x true. compound sentence that combines two statements by the word or is a disjunction. Solve each inequality. Then write whether the solution is a conjunction or disjunction.. x. x 8 x or x ; disjunction x and x ; conjunction. x. x x and x ; conjunction x or x 0; disjunction. x x. x x and x ; conjunction x and x ; conjunction x 8. x. x or x ; disjunction x and x 8; conjunction 9. 8 x 0. x x or x 0; disjunction x and x ; conjunction Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 9 Glencoe lgebra