Solution Manual for Intermediate Algebra Functions and Authentic Applications, 6th Edition

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SOLUTIONSMANUALDIACRITECHINTERMEDIATEALGEBRA:FUNCTIONS ANDAUTHENTICAPPLICATIONSSIXTHEDITIONJay LehmannCollege of San Mateo

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iContentsChapter 1Linear Equations and Linear Functions1.1Using Qualitative Graphs to Describe Situations11.2Graphing Linear Equations31.3Slope of a Line81.4Meaning of Slope for Equations, Graphs, and Tables121.5Finding Linear Equations171.6Functions23Chapter 1 Review Exercises25Chapter 1 Test29Chapter 2Modeling with Linear Functions2.1Using Lines to Model Data332.2Finding Equations of Linear Models352.3Function Notation and Making Predictions392.4Slope Is a Rate of Change46Chapter 2 Review Exercises51Chapter 2 Test55Chapter 3Systems of Linear Equations and Systems of LinearInequalities3.1Using Graphs and Tables to Solve Systems583.2Using Substitution and Elimination to Solve Systems633.3Using Systems to Model Data703.4Value, Interest, and Mixture Problems753.5Using Linear Inequalities in One Variable to Make Predictions803.6Linear Inequalities in Two Variables; Systems of Linear Inequalities85Chapter 3 Review Exercises90Chapter 3 Test98Cumulative Review of Chapters 1–3103Chapter 4Exponential Functions4.1Properties of Exponents1104.2Rational Exponents1144.3Graphing Exponential Functions1174.4Finding Equations of Exponential Functions1224.5Using Exponential Functions to Model Data128Chapter 4 Review Exercises135Chapter 4 Test139

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iiChapter 5Logarithmic Functions5.1Composite Functions1425.2Inverse Functions1475.3Logarithmic Functions1535.4Properties of Logarithms1565.5Using the Power Property with Exponential Models to Make Predictions1615.6More Properties of Logarithms1685.7Natural Logarithm172Chapter 5 Review Exercises177Chapter 5 Test184Cumulative Review of Chapters 1–5187Chapter 6Polynomial Functions6.1Adding and Subtracting Polynomial Expressions and Functions1946.2Multiplying Polynomial Expressions and Functions1966.3Dividing Polynomials: Long Division and Synthetic Division2016.4Factoring Trinomials of the Formx2+bx+c; Factoring Out the GCF2066.5Factoring Polynomials2086.6Factoring Special Binomials; A Factoring Strategy2106.7Using Factoring to Solve Polynomial Equations212Chapter 6 Review Exercises217Chapter 6 Test220Chapter 7Quadratic Functions7.1Graphing Quadratic Functions in Vertex Form2237.2Graphing Quadratic Functions in Standard Form2287.3Using the Square Root Property to Solve Quadratic Equations2377.4Solving Quadratic Equations by Completing the Square2437.5Using the Quadratic Formula to Solve Quadratic Equations2497.6Solving Systems of Linear Equations in Three Variables;Finding Quadratic Functions2597.7Finding Quadratic Models2667.8Modeling with Quadratic Functions270Chapter 7 Review Exercises275Chapter 7 Test282Cumulative Review of Chapters 1–7287Chapter 8Rational Functions8.1Finding the Domains of Rational Functions and SimplifyingRational Expressions2978.2Multiplying and Dividing Rational Expressions; Converting Units3018.3Adding and Subtracting Rational Expressions3078.4Simplifying Complex Rational Expressions3128.5Solving Rational Equations3178.6Modeling with Rational Functions326

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iii8.7Variation331Chapter 8 Review Exercises335Chapter 8 Test343Chapter 9Radical Functions9.1Simplifying Radical Expressions3489.2Adding, Subtracting, and Multiplying Radical Expressions3519.3Rationalizing Denominators and Simplifying Quotients of Radical Expressions3559.4Graphing and Combining Square Root Functions3609.5Solving Radical Equations3659.6Modeling with Square Root Functions372Chapter 9 Review Exercises378Chapter 9 Test384Chapter 10Sequences and Series10.1Arithmetic Sequences38810.2Geometric Sequences39010.3Arithmetic Series39510.4Geometric Series398Chapter 10 Review Exercises401Chapter 10 Test403Cumulative Review of Chapters 1–10405Chapter 11Additional Topics11.1Absolute Value: Equations and Inequalities41711.2Performing Operations with Complex Numbers42111.3Pythagorean Theorem, Distance Formula, and Circles42511.4Ellipses and Hyperbolas43111.5Solving Nonlinear Systems of Equations440Appendix AReviewing Prerequisite MaterialA.1Plotting Points448A.2Identifying Types of Numbers448A.3Absolute Value448A.4Performing Operations with Real Numbers448A.5Exponents448A.6Order of Operations449A.7Constants, Variables, Expressions, and Equations449A.8Distributive Law449A.9Combining Like Terms450A.10Solving Linear Equations in One Variable450A.11Solving Equations in Two or More Variables451A.12Equivalent Expressions and Equivalent Equations452

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Chapter 1Linear Equations and Linear FunctionsHomework 1.12.a.(c)b.(b)c.(a)d.(d)4.The more pencils there are to buy, the morethe total cost. Total costcis the responsevariable and the number of pencilsnis theexplanatory variable.6.The greater the rate water is added to a pool,the less time it takes to fill the pool. Thenumber of hours to fill a pooltis the responsevariable and the rate water is addedris theexplanatory variable.8.The older the car in years, the greater theannual cost of repairs. The number of annualcost of repairscis the response variable andthe age of a carais the explanatory variable.10.The hotter the temperature, the more people goto the beach. The number of people at thebeachnis the response variable and thetemperature at the beachFis the explanatoryvariable.12.The greater your annual income, the morefederal taxes you pay. The federal taxesTisthe response variable and the person’s annualincomeIis the explanatory variable.14.The response variable iss, the runner’s speed,and the explanatory variable ist, time after shebegan her run. The graph shows her speedincreases to a certain point and then decreasesto a stop whens= 0. The rest of her time isspent walking at a moderate but slightlyincreasing speed.16.Heighthis the response variable and time,tisthe explanatory variable. The height of thetennis ball decreases as it drops. The heightthen increases after the ball bounces fromh=0. This pattern is repeated with the maximumheight decreasing after each bounce until theball stops ath= 0.18.P, the percentage of smokers, is the responsevariable andt, time in years since 2010, is theexplanatory variable. This graph shows that thepercentage of smokers approximately steadilydecreases as time increases from 2010.20.The foreign born percentage, p,is the responsevariable and time since 1910,t,is theexplanatory variable. The graph shows that thepercentage of the U.S. population that isforeign born decreased from 1910 to 1970 andthen increased over time.22.VolumeVis the response variable and timetisthe explanatory variable. The graph shows thevolume of air in a person’s lungs alternatelyincreases and decreases as air is taken in andout over time.

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2ISM:Intermediate Algebra24.AngleAis the response variable and minutestis the explanatory variable. The graph showsthat as the number of minutes past midnightincreases, the angle (in degrees) between thehour hand and the minute hand of a clockincreases.26.The response variable isn(species inexistence) and the explanatory variable isd(amount of deforestation). The graph showsthat the number of species in existencedecreases steadily due to increaseddeforestation.28.Speed,s, is the response variable and currentin miles per hour,c, is the explanatoryvariable.a.The graph shows that a speedboat travelsfaster downstream as the river currentincreases.b.The graph shows that a speedboat travelsslower upstream as the “opposing” rivercurrent increases.30.The percentage of times a person hits thebull’s-eyepis the response variable and thedistance in feet from the dartboarddis theexplanatory variable. The graph shows that asthe distance in feet increases, the percentage ofbull’s-eye hits decreases.32.The number of hours to paint a houseTis theresponse variable and the number of people onthe painting crewNis the explanatoryvariable. The graph shows that as the numberof people on the crew increases, the number ofhours to paint decreases.34.The driving time in hoursTis the responsevariable and the speed in miles per hour theperson drivesSis the explanatory variable.The graph shows that as the speed in miles perhour increases, the driving time decreases.36.PressurePis the response variable and volumeVis the explanatory variable. The graph showsthat pressure in the balloon is greater when thevolume is lower.38.An increasing curve can have at most twointercepts where it crosses each axis.

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Chapter 1:Linear Equations and Linear Functions340.First, determine what the response variable isand place it on the vertical axis. Then place theexplanatory variable on the horizontal axis.There is no need for scaling on these axes.Determine from the given situation how todescribe the relationship between the twovariables. Use this relationship to see if anincreasing or decreasing line or curve isnecessary. Note where a point of intersectionis needed on one or both axes.Homework 1.22.4.6.8.10.12.14.16.0420204445yxxyxy=−==

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4ISM:Intermediate Algebra18.10520510205524xyyxyx−=−−+=−−=−20.306120630126652xyyxyx+−=−+== −+22.3327339333yxyxyx+−=−+== −+24.64172423231123yxyxyxyxyx−−=−−−=−−−=−−= −+26.()()2341264424102225yxyxyxyx−=+−=++==+28.52(31)23(2)562236623544844421xyyxxyyxyyxxyxyx−−= −−−−+= −−+−+= −−+−−+=−−=−

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Chapter 1:Linear Equations and Linear Functions530.32.34.36.a.i.ii.iii.b.An equation of the formxa=is a verticalline passing through (a, 0).38.40.42.44.()236326262233xyyxxyx−=−= −+−+==−−

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6ISM:Intermediate Algebra(0,−2) is a point on the line.()()2362 032606666 truexy−=−−+46.a.b.Answers may vary. Example:01.8900.21.4040.40.9180.60.4320.80.054−xy48.352497513928224xxxxxxx−−=+−−+= −+−=−−= −50.4(21)578457131113131113tttttt−= −+−= −+==52.72(54 )3(83 )71083833868252252xxxxxxxxxx−−=−−−+=−+−+=−−== −54.4(6)5(3)3(1)4245153339332362218aaaaaaaaaa−++−=−−−+−=−−=−−=−−= −56.112343112121234343841144114xxxxx−=−=−===58.353148243531884824651221871818718wwwwwwww−−=+−−=+−−=+−=−−= −60.6.5487.354.6699.031.88186.381.881.8899.14xxxx−+= −−−−=−−≈62.axbycbycaxbbcaxyb+=−=−=64.( )11xyaaxyaaaaxyayax+=+=+==−66.312yx= −−To find thex-intercept, lety= 0 and solve forx.03121234xxx= −−= −−=Thex-intercept is (−4, 0). To find they-intercept, letx= 0 and solve fory.()3 01201212y= −−=−= −They-intercept is (0,−12).68.5420xy−=To find thex-intercept, lety= 0 and solve forx.

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Chapter 1:Linear Equations and Linear Functions7()54 02050205204xxxx−=−===Thex-intercept is (4, 0). To find they-intercept, letx= 0 and solve fory.()5 042004204205yyyy−=−=−== −They-intercept is (0,−5).70.2yx= −To find thex-intercept, lety= 0 and solve forx.020xx= −=Thex-intercept is (0, 0). This is also they-intercept.72.2x= −Sincex=−2 is a vertical line, it neverintersects they-axis. Therefore, there is noy-intercept.Since the graph passes through (−2, 0), this isthex-intercept.74.axbyc+=To find thex-intercept, lety= 0 and solve forx.(0)axbcaxccxa+===Thex-intercept is, 0ca.To find they-intercept, letx= 0 and solve fory.(0)abycbyccyb+===They-intercept is0,cb.76.()a xbyc−=To find thex-intercept, lety= 0 and solve forx.()0a xbcaxccxa−⋅===Thex-intercept is, 0ca.To find they-intercept, letx= 0 and solve fory.()0abycabyccyab−=−== −They-intercept is0,cab−.78.()ayb cdx=+To find thex-intercept, lety= 0 and solve forx.()()00ab cdxbcbdxbdxbccxd=+=+−== −Thex-intercept is, 0cd−.To find they-intercept, letx= 0 and solve fory.()0ayb cdaybcbcya=+⋅==They-intercept is0,bca.80.ybxm−=To find thex-intercept, lety= 0 and solve forx.(0)bxmbxm−=−=Thex-intercept is, 0bm−.To find they-intercept, letx= 0 and solve fory.00ybmybyb−=−==They-intercept is (0,b).82.Answers may vary. Example:33011.503163−−−xy84.2x=

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8ISM:Intermediate Algebra86.1y=88.12y≈ −90.8x=92.6x= −94.1x≈ −96.a.b.To find they-intercept, letx= 0 and solvefory.()2 01001010y= −+=+=They-intercept is (0, 10). This means thatthere are 10 gallons of gas in the tank atthe time of refueling.c.To find thex-intercept, lety= 0 and solveforx.02102105xxx= −+==Thex-intercept is (5, 0). This means thatwhen 5 hours have passed since filling thetank, the tank will be empty.98.The graph of2251yxxx+=++is a line.When combining like terms, note that theequation becomes51yx=+. The2xtermcancels out. It is now an equation in the formymxb=+.100.Since (2, 11) is a point on the graph of theequation3ymx=+, (2, 11) satisfies theequation. Substitutex= 2 andy= 11 into3ymx=+and solve form.()1123824mmm=+==Therefore,m= 4 in3ymx=+.102.a.Points A and F satisfy the equationyaxb=+since these points lie on the linefor this equation.b.Points D and F satisfy the equationycxd=+since these points lie on theline for this equation.c.Point F satisfies both equations since it lieson both lines. It lies at the point where thelines intersect.d.Points B,C,andE do not satisfy eitherequation since they do not lie on either line.104.They-coordinate of thex-intercept is 0because thex-intercept is a point on thex-axis.All points on thex-axis have ay-coordinateequal to 0. Thex-coordinate of they-interceptis 0 because they-intercept is a point on they-axis. All points on they-axis have anx-coordinate equal to 0.106.No, every line does not have anx-intercept.Answers may vary. Example:Most horizontal lines do not havex-intercepts.For example, the line1y=. does not have anx-intercept.Homework 1.32.25000.31258000Am==31000.3269500Bm=≈Airplane B is making a steeper climb thanairplane A.4.The slope of the ski run from the top of themountain to the chairlift is10014004−=. If thevertical distance from the top of the mountainto the chairlift is 100 yards and the verticaldistance for the entire run is 415 yards, thenthe vertical distance from the chairlift to therestaurant is 415−100 = 315 yards. Also,since the horizontal distance from the top ofthe mountain to the chairlift is 400 yards andthe horizontal distance for the entire run is1300 yards, the horizontal distance from thechairlift to the restaurant is 1300−400 = 900yards. Therefore, the slope of the ski run fromthe chairlift to the restaurant is3153157or90090020−=.

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Chapter 1:Linear Equations and Linear Functions96.21218441154yymxx−−==== −−−−Sincemis negative 1, the line is decreasing.8.21216( 2)842863yymxx−− −==== −−−−Sincemis negative43, the line is decreasing.10.21216( 10)164314yymxx−− −==== −−−−−Sincemis negative 4, the line is decreasing.12.21213( 9)632642yymxx−−− −==== −−−−Sincemis negative32, the line is decreasing.14.212112( 2)1051784yymxx−−− −−====−−−−Sincemis positive54, the line is increasing.16.()()212165651112121yymxx−− −−−+−===== −−−− −−+Sincemis negative 1, the line is decreasing.18.2121100010011000100yymxx−−====−−Sincemis positive 1, the line is increasing.20.21211( 1)00358yymxx−−− −====−−−−Sincemis zero, the line is horizontal.22.2121189undefined440yymxx−−−−====−−Sincemis undefined, the line is vertical.24.()212103316062yymxx− −−==== −−−−−Sincemis negative12, the line is decreasing.26.21217.62.29.88.175.1( 3.9)1.2yymxx−−−−====−−− −−Sincemis positive 8.17, the line is increasing.28.212182.78( 66.66)149.4425.41( 11.26)14.1510.56yymxx−− −===−−− −−= −Sincemis negative 10.56, the line isdecreasing.30.The points (0, 4) and (−2, 1) lie on the line.()2121413022yymxx−−===−− −The slope of the line is32.32.Since21mm=, the lines are parallel.34.Since211mm= −, the lines are perpendicular.36.Since211mm= −, the lines are perpendicular.38.Since21mm=, the lines are parallel.40.Since1mand2mare undefined,1land2larevertical lines. Vertical lines are parallel.42.The two lines are not parallel, since theirslopes,13and310, are not equal.44.Answers may vary. Example:46.Answers may vary. Example:

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10ISM:Intermediate Algebra48.Answers may vary. Example:50.Answers may vary. Example:52.-4-224-4-224xyAnswers may vary. Example:Since2163−= −and1133= −−, the slopes arethe same. Both lines have the same slope andthe samey-intercept, so they must be the sameline.54.Answers may vary. Example:The lines have the same steepness.56.Answers may vary. Example:The points (0, 0) and (10,−1) are on the lineand can be used to find the slope.212110111001010yymxx−−−−==== −−−58.Answers may vary. Example:The points (0, 0) and (1, 10) are on the lineand can be used to find the slope.21211001010101yymxx−−====−−60.Answers may vary. Example:The line must be increasing and almosthorizontal.

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Chapter 1:Linear Equations and Linear Functions1162.Answers may vary. Example:The line must be decreasing and almost vertical.64.The student exchanged1ywith2yincalculating the slope. The correct slope isnegative instead of positive since212119887255yymxx−−−==== −−−.66.Answers may vary. Example:Three points that lie on the line are (3, 11),(0, 6), and (−9,−9) since the line equation is563yx=+.68.a.2121532725yymxx−−===−−b.121235222755yymxx−−−====−−−c.The results are the same. It does not matterwhich point is taken first.d.21122112212121212121212121212121111yyyyxxxxyyyyxxxxyyyyxxyyyyyyxxyy−−=−−−−−=⋅−−−−−=⋅−−−−=−−e.It does not matter which point is used firstand which point is used second. The slopewill be the same regardless of which pointis considered first.70.a.There are two possibilities for the othertwo vertices, since the figure is a squareand sides must have equal length. Thecoordinates for the vertices are (−3, 1) and(−3, 7). Another possibility is (9, 1) and(9, 7).b.One possibility is the point (0,−3).A second possibility is the point (6, 5).Another possibility is (−14,−1).72.Answers may vary. Example:74.Answers may vary.5 ftRoad with 5% grade100 ft

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