Revision Notes for Intermediate Algebra, 13th Edition

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INSTRUCTOR’SRESOURCEMANUALDAVIDATWOODRochester Community and Technical CollegeINTERMEDIATEALGEBRATHIRTEENTHEDITIONMargaret L. LialAmerican River CollegeJohn HornsbyUniversity of New OrleansTerry McGinnisLECTURE NOTES

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Contents____________________________iii__________________________________Mini-Lectures by Section………………………………………………….......................3Mini-Lecture Graph Answers…………………………………………..………77

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MINI-LECTURES

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Mini-Lecture R.13INTERMEDIATE ALGEBRAFractions, Decimals, and PercentsLearning Objectives:1.Write fractions in lowest terms.2.Convert between improper fractions and mixed numbers.3.Perform operations with fractions.4.Write decimals as fractions.5.Perform operations with decimals.6.Write fractions as decimals.7.Write percents as decimals and decimals as percents.8.Writer percents as fractions and fractions as percents.Examples:2.Simplify each fraction.a)510b)1664c)4277d)88903.Change each improper fraction to a mixed number or a whole number, or change each mixed number to animproper fraction.a)85b)344c)32 4d)26 94.Find each product and write in lowest terms.a)539 15b)5962c)4 189d)1131245.Find each quotient and write in lowest terms.a)53915b)5962c)5362424d)538346.Find each sum or difference and write in lowest terms.a)341414b)2133c)2148d)5768e)791230f)2213246g)31811010h)311210847.Write the following decimal numbers as fractions. Do not write in lowest terms.a)0.42b)1.285c)0.93758.Add or subtract as indicated..a)50.586b)45.333.306c)4.11.032.129.Multiply or divide as indicated.a)5.8643.2b)1687.1423.4c)7.863752.25d)57.4100010.Write each fraction as a decimal.a)245b)79c)38

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INTERMEDIATE ALGEBRA4Mini-Lecture R.111.Write each percent as a decimal.a)53%b)255%c)34%412.Write each decimal as a percent.a)0.428b)3.21c)0.30513.Write each percent as a fraction. Give answers in lowest terms.a)7%b)185%c)12.4%14.Write each fraction as a percent.a)35b)13c)92Teaching Notes:Encourage students to simplify fractions by dividing numerator and denominator by the same number andby factoring into primes and dividing out common factors.Some students try to multiply/divide whole number parts together, and then multiply/divide fractional partstogether when working with mixed numbers.Some students add/subtract the denominators when adding/subtracting fractions.Many students are confused by percents that are less than 1%. It may be helpful to have the studentsdistinguish between 0.2 and 0.2%.Many students are confused by percents that are greater than 100%. It may be helpful to show that100% =1Mention to students that another way to convert a fraction to a percent is to first convert the fraction to adecimal and then convert the decimal to a percent.Answers: 1a) prime, b) composite,5 13, c) prime, d) composite,2 2 2 3 3 3; 2a)12, b)14, c)611, d)4445; 3a)351, b)128, c)114, d)569; 4a)19, b)154or343, c) 8, d)358or384; 5a)259or792, b)527, c)536, d)421; 6a)12,b)13, c)58, d)4124or17241, e)5360, f)1112, g)259, h)182;7a)42100, b)12851000, c)937510, 000; 8a) 4.414, b) 11.994, c) 5.19,9a) 1.8325, b) 72.1, c)3.495, d) 0.0574; 10a) 4.8, b)0.7, c) 0.375; 11a) 0.53, b) 2.55, c) 0.475; 12a) 42.8%,b) 321%, c) 30.5%; 13a)7100, b)171 20, c)31250, 14a) 60%, b)33.3%, c) 450%

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Mini-Lecture R.25INTERMEDIATE ALGEBRABasic Concepts from AlgebraLearning Objectives:1.Write sets using set notation.2.Use number lines.3.Classify numbers4.Find additive inverses.5.Use absolute value.6.Use inequality symbols.Examples:1.List all numbers from the set310,5,2, 0,5, 104that are:a)natural numbersb)whole numbersc)integersd)rational numberse)irrational numbersf)real numbers2.Graph each of the numbers on a number line:115, 5,4, 1,32.3.Select the lesser number in each pair.a)–3, 8b)41,53c)10, –124.For each number, find the additive inverse.a)6b)–5c)2.75.Simplify.a)3b)6 c)13166.Decide whether the statement is true or false.a)511 b)35 c)71 7.Write the following in interval notation and graph the interval.a)|5xxb)|8xx c)|13xxTeaching Notes:Many students need to be reminded of set notation.Remind students that integers are rational numbers; any integer can be written as the ratio of itself and 1.Decimal numbers that terminate or repeat in a fixed block are examples of rational numbers; ask students togive examples of both.The decimal form of an irrational number neither terminates nor repeats.The number line is a good way to illustrate opposite numbers/additive inversesAnswers: 1a) 10, b) 0, 10, c) –10, –5, 0, 10, d) –10, –5,32 4, 0, 10, e)5, f) –10, –5,32 4, 0,5, 10;2); 3a) –3, b)45, c) –12; 4a) –6, b) 5, c) –2.7; 5a) 3, b) –6, c) 3;6a) true, b) true, c) false; 7a),5b)8,c)1,3

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INTERMEDIATE ALGEBRA6Mini-Lecture R.3Operations on Real NumbersLearning Objectives:1.Add real numbers.2.Subtract real numbers.3.Find the distance between two points on a number line.4.Multiply real numbers.5.Find reciprocals and divide real numbers.Examples:1.Perform the operations indicated.a)–5 + (–3)b)5 + 3c)5 + (–3)d)–5 + 3e)3 – 5f)3 – (–5)g)–3 – 5h)–3 – (–5)i)(3)(5)j)(–3)(–5)k)(–3)(5)l)(3)(–5)2.Find the distance between each pair of points.a)7 and6b)3and93.Give the reciprocal of each number.a)12b)56c)12d)04.Find each quotient.a)84b)–8–4c)–84d)8–45.Perform each of the following operations, if possible.a)|–2.1 + (–7.3)|b)315520c)–212+324d)|5.4 – 9.2|e)0–4f)3.20g)1122h)4.24.28.4i)13.32 + 9.75 – 5.21j)384-Teaching Notes:Refer students to the charts forAdding,Subtracting,Multiplying, andDividing Real Numbers.Many students find working with fractions difficult. It may be useful to review finding the least commondenominator.Remind students that only nonzero numbers have inverses.The fraction is undefined when zero is “underneath”.Answers: 1a) –8, b) 8, c) 2, d) –2, e) –2, f) 8, g) –8, h) 2, i) 15, j) 15, k) –15, l) –15; 2a) 13 , b) 6; 3a) 1/12, b) 6/5,c) –2, d) zero does not have a reciprocal, or multiplicative inverse; 4a) 2, b) 2, c) –2, d) –2; 5a) 9.4, b) –4/5,c) –1/24, d) 3.8, e) 0, f) undefined, g) 0, h) 1, i) 17.86, j) –3/32

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Mini–Lecture R.47INTERMEDIATE ALGEBRAExponents, Roots, and Order of OperationsLearning Objectives:1.Use exponents.2.Find square roots.3.Use the order of operations.4.Evaluate algebraic expressions for given values of variables.Examples:1.Write in exponential form.a)(4)(4)(4)(4)(4)b)(–2)(–2)(–2)(–2)c)xx xyy2.Evaluate.a)25b)32c)43d)(–5)2e)–52f)(–2)3g)–23h)(–3)4i)212j)(–1.2)33.Find each square root, if it exists.a)4b)4c)4d)25e)0.25f)4936g)306h)1316164.Simplify using the order of operations.a)5(24)6b)273(24)c)3(961)49d)( 9)(36)( 2)( 9) e)23( 5)8 34(65) f)226936 5.Evaluate each expression for the values given.a)25 ;3xxx b)242;2,4yxxy Teaching Notes:In 1b) some students answer –24instead of (–2)4. Many students think these are equal.Many students do not understand that the square root symbol indicates a principal square root, which isnever negative.Order of Operations – introduce PEMDAS or Please Excuse My Dear Aunt Sally.Answers: 1a) 45, b) (–2)4, c) x3y2; 2a) 25, b) 8, c) 81, d) 25, e) –25, f) –8, g) –8, h) 81, i) ¼, j) –1.728; 3a) 2, b) –2,c) DNE, d) DNE, e) 0.5, f) –7/6, g) 6, h) –1/2; 4a) –4, b) –43, c) –7, d) –72, e) 13, f) –1/3; 5a) –6, b) 0

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INTERMEDIATE ALGEBRA8Mini-Lecture R.5Properties of Real NumbersLearning Objectives:1.Use the distributive property.2.Use the identity properties.3.Use the inverse properties.4.Use the commutative and associative properties.Examples:1.Name the property that justifies each statement.a)(–5) + 3 = 3 + (–5)b)(5 + 2) + 4 = 5 + (2 + 4)c)1 313d)7 + (–7) = 0e)4(5 + 2) = 45 + 42f)33155g)6.3 + 0 = 6.3h)5 (4 2)( 5 4) 2  i)43 = 342.Simplify.a)45xxb)112yyc)3912xxd)2.58.63.412.3yye)3 69yf)254kkg)7.38.12 3.20.4bbh)111243026xxyi)936ttTeaching Notes:Remind students that thecommutative propertydeals with theorderof the operands, while, theassociativepropertydeals with thegroupingof the operands.Discuss difference between theidentity propertiesand theinverse properties.Remind students that when we use the identity property the result is identical to the quantity we begin with.Remind students that parentheses do not always imply that thedistributive propertyis to be used (e.g.,35690abab, not3536aab).Answers: 1a) commutative property of addition, b) associative property of addition, c) inverse property ofmultiplication, d) inverse property of addition, e) distributive property of multiplication with respect to addition,f) identity property of multiplication, g) identity property of addition, h) associative property of multiplication, i)commutative property of multiplication; 2a)9x, b)111y, c)9x9, d)..5 9 y20 9, e)18 y27, f)3k4,g)..0 9b8 9, h)1xy26, i)4t15

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Mini-Lecture 1.19INTERMEDIATE ALGEBRALinear Equations in One VariableLearning Objectives:1.Distinguish between expressions and equations.2.Identify linear equations.3.Solve linear equations using the addition and multiplication properties of equality.4.Solve linear equations using the distributive property.5.Solve linear equations with fractions or decimals.6.Identify conditional equations, contradictions, and identities.Examples:1.Decide whether each of the following is an expression or an equation.a)41xb)410xc)234 25xxd)234 25xx2.Determine whether the given value is a solution to the equation.a)Isx= 3 a solution to4218x? Why or why not?b)Isx= 12 a solution to5106x? Why or why not?3.Solve. Check your solutions.a)122x b)1335x c)545xd)570xe)10517xf)6329xxg)141085xxh)625412xxxi)582(1)xx4.Solve. Check your solutions.a)182xb)5106xc)3535xd)45352xxe)23(2)13xf)0.10.50.40.9xxg)0.50.20.30.8xxh)0.4(31)1xi)0.4(3)20.4xx5.Classify each of the following equations as aconditional equation, acontradiction, or anidentity.a)32562xxxb)237xc)57394xxxTeaching Notes:Encourage students to check their solutions.Encourage students to simplify each side of the equation as a first step.Some students prefer to always end with the variable on the left, while others prefer to arrive at a positivecoefficient of the variable.Some students try to subtract the coefficient from the variable instead of eliminating it by dividing.Answers: 1a) expression, b) equation, c) equation, d) expression; 2a) no because 43+218, b) yes because(5/6)12=10; 3a) {10}, b) {–48},c) {9}, d) {–14}, e) {6/5},f) {–3}, g) {–2}, h) {2}, i) {5}; 4a) {16}, b) {–12}, c){–66/5}, d) {25/22}, e) {1}, f) {.4 6}, g) {5}, h) {0.5}, i); 5a) identity, b) conditional equation, c) contradiction

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INTERMEDIATE ALGEBRA10Mini-Lecture 1.2Formulas and PercentLearning Objectives:1.Solve a formula for a specified variable.2.Solve applied problems using formulas.3.Solve percent problems.4.Solve problems involving percent increase or decrease.5.Solve problems from the health care industry.Examples:1.Solve each formula for the specified variable.a)635xy; foryb)233yx ; forxc)Vlwh; forhd)()2hABb; forbe)2Arh; forrf)3(4)3axyaxy; forx2.Solve as indicated.a) Solve9325FCforC; then evaluate forF= 25. Round to the nearest tenthb) Solve213Vr hforh; then evaluate forV= 5.35,r= 2,3.14 . Round to the nearest hundredth3.Application problems.a)15 is what percent of 90?a)Suppose economists use0.6445.822CDas a model of the country’s economy, whereCandDare in billions of dollars. Solve the equation forD, and use this result to determine the disposableincomeDif the consumptionCis $9.44 billion. Round your answer to the nearest tenth of a billionb)Some doctors use the formulaND= 1.1Tto relateN(the number of appointments scheduled in one day),D(the duration of each appointment), andT(the total number of minutes the doctor can use to see patientsin one day). Solve the formula forD, and use this result to find the duration of each appointment if thedoctor has 6 hours available for appointments and must see 25 patients per day.c)A car salesman earned $6750 in commission on sales of automobiles amounting to $45,000. What is hisrate of commission?d)In July, there were 14,150 visitors to theDunes State Park. In August, there were 9644 visitors. Whatwas the percent increase/decrease in visitors from July to August?Teaching Notes:For questions like number 1, encourage students to underline or circle the specified variable.When calculating a percent increase or decrease, be sure to use the original number as the denominator.Answers: 1a) y=–2x+(5/3), b) x=(–3/2)y+(9/2), c)Vhlw, d)2 AbBh, e)Ar2h, f)4 yx9a;2a) C=(5/9)(F–32), C–3.9, b)3Vh2r, h1.28; 3a) 16.7%, b)..C5 822D0 644, D$5.6 billion, c) D=1.1T/N,D=15.84 min, d) 15%, e) 31.8% decrease

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Mini-Lecture 1.311INTERMEDIATE ALGEBRAApplications of Linear EquationsLearning Objectives:1.Translate from words to mathematical expressions.2.Write equations from given information.3.Distinguish between simplifying expressions and solving equations.4.Use the six steps in solving an applied problem.5.Solve percent problems.6.Solve investment problems.7.Solve mixture problems.Examples:1.Translate into a mathematical expression, then simplify.a)The sum of three consecutive integers if the first integer isxb)The perimeter of a rectangle with lengthxand widthx– 7c)The total amount of money (in cents) inxquarters, 5xdimes, and (3x– 1) nickels2.Use thesix-step approachto solving applied problems to solve each of the following.a)GeometryThe length of a rectangular room is 6 feet longer than twice the width. If the room’s perimeteris 168 feet, what are the room’s dimensions?b)RentalIt costs $40 plus $0.25 per mile to rent a truck from U-Rent. How many miles weretraveled if the final bill was $48.25?c)RevenueThe revenue of Company X quadrupled. Then it increased by $1.6 million. The presentrevenue is $24.4 million. What was the original revenue?d)Satellite PhoneShaun’s satellite phone provider charges its customers $15 per month plus 30 cents perminute of on-line usage. Shaun received a bill from the provider covering a month period and wascharged a total of $52.50. How many minutes did he spend using this phone service that period, to thenearest whole minute?e)Balloon AltitudeA hot air balloon spent several minutes ascending. It then stayed at a level altitude forthree times as long as it had ascended. It took 5 minutes less to descend than it had to ascend. The entiretrip took one hour and 30 minutes. For how long was the balloon at a level altitude?f)InvestmentJoe invested $12,000 in certificates of deposit paying 4%. How much additional moneyshould he invest in certificates paying 6% so that his total return for the investments will be $792?g)MixtureHow much water must be added to 10 quarts of an 8% concentrated detergent to reduce theconcentration to 5%?Teaching Notes:Have students create a list of key terms and associated mathematical operations.Some students need to see many of the number problems in order to understand how an English sentencecan be represented as an algebraic equation.Encourage students to draw and label diagrams when appropriate.Refer students to theSolving an Applied Problemand theTranslating from Words to MathematicalExpressionscharts in the textbook.Answers: 1a),xx1x23x3, b)2x2 x7,4x14c)()25x10 5x5 3x1,90x5; 2a) l=58ft, w=26 ft, b) 33 miles, c) $5.7 million,d) 125 minutes, e) 57 minutes, f) $5200, g) 6 quarts

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INTERMEDIATE ALGEBRA12Mini-Lecture 1.4Further Applications of Linear EquationsLearning Objectives:1.Solve problems about different denominations of money.2.Solve problems about uniform motion.3.Solve problems about angles.4.Solve problems about consecutive integers.Examples:1.Mary has a bank in which she has saved only dimes and quarters. She has 17 coins in the bank worth$2.45. How many coins of each type does she have?2.Angela has 37 coins, consisting of only dimes and quarters, worth $7.45. How many coins of each type doesshe have?3.A collection of 70 coins consisting of dimes, quarters, and half-dollars has a value of $17.75. There are threetimes as many quarters as dimes. Find the number of each kind of coin.4.Two cars are traveling on opposite directions on a highway. One car is driving 4 miles per hour faster than theother car. After 3 hours the cars are 414 miles apart. Find the speed of both cars.5.Lori starts jogging at 5 miles per hour. One half hour later, Debbie starts jogging on the same route at 7 milesper hour. How long will it take Debbie to catch Lori?6.One angle of a triangle measures 4 more degrees than three times the measure of the second angle. The thirdangle measures40. What are the measures of the two unknown angles of the triangle?7.Two angles are complementary if the sum of their measures is 90º. If the measure of the first angle isxº, andthe measure of the second angle is (3x– 2)º, find the measure of each angle.8.The sum of three consecutive positive integers is 135. Find the three integers.9.The sum of three consecutive odd integers is 69. Find the three integers.Teaching Notes:Encourage students to draw and label diagrams when appropriate.Some students need to see several examples of each type of word problem.Refer students to theSolving an Applied Problemchart in the previous section of the text..Answers: 1) 12 dimes and 5 quarters; 2) 12 dimes and 25 quarters; 3) 15 dimes, 45 quarters, and 10 half-dollars;4) 67 mph, 71 mph; 5) 1 hour 15 minutes; 6) 34º and 106º; 7) 23º and 67º, 8) 44, 45, 46 , 9) 21, 23, 25

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