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# Introductory Algebra MTH 60

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This 33 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 60 at Portland Community College taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/224642/mth-60-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

14 MTH 60 Simonds class Key Concepts Multiplication and division of signed numbers Dealing with exponents and negative signs MartinGave sections and practice problems 17 all odds no calculator and division of signed numbers The product or quotient of two numbers with the same sign is positive The product or quotient of two numbers with opposite signs is negative Example 1 Find each product or quotient ems 7 wise 1 L 98 39 L 2 HM 7 e atan dug WHY c El J lt76gtlt4gt I I 1 anw39w lt73gtlt71gtlt76gtlt72gtlt7sgt 3 L l9 lt3gtlt71gtlt6gtlt2gtlt73gt 3 W t 393 tux w quotquotquot quot JUK J J H39J l oi 7 a l 135015 n Va39kV39J Jos It O F 3010 J Page 1 of 4 24 MTH 60 Simonds class When The number of negaTive facTors in The producT or quoTienT is WW The simplified producT or quoTienT is F I ll id c When The number of negaTive facTors in The producT or quoTienT is 00 The simplified producT or quoTienT is A s X l g Examgle 2 Find each of The following and sTaTe whaT you observe i 391 L L El L 4 74 Examgle 3 Find each of The following and sTaTe whaT you observe ILL ennuie 9L y is Mn 4quot 5 l 7 x lgtlt the opposite of 2 negative 1 times 2 Page 2 of 4 34 MTH 60 Simonds class An important new fact related to order of Unless a negative sign is in parentheses exponents come before negation in order of operations Example 4 Find each of the following Write sentences that describe each original expression lt73gt2 Ll 3 J i Em A 5 713z 39L 397 3 lt3 l 732 mquot 1 a 3239 rise OffiJiJ n 0L 3 lZzl Evaluate 772 62 when x77 Ukvn 1 I I gt gt 7quot4 7L L M 4 1 7 0 Page 3 of 4 Group work arithmetic of signed numbers 1 N w 0 N 1 EvaluaTe xyx when 263 and y74 x when x77 78 8 EvaluaTe x DeTermine wheTher or noT 731 is a soluTion To The equaTion 74t 7122 Find The value of 7321277575251 16712715726717 44 MTH 60 Simonds class No Calculator 3tt33 1 30 39 If x 2 which of The following evaluaTe To 4 x2 7x2 7x2 If x 7 2 which of The following evaluaTe To 4 x2 7x2 7x2 remember no calculafor Find each sumdifferenceproducTquoTienT and reduce The resulT remember no calculafor 3 19 7 b 8782 a 7 14 A e 32 7372 4 14 21 WhaT fracTion of The pie is missing 6 C 9 9 l E d 2 l 2 7 39 4 2 13Z h 2511 12 8 2 WhaT is The perimeTer of The given objecT foeet 3feet 2feet 4feet 4feet 10 Is 73 a soluTion To The equaTion 6 27x x 3 11 EvaluaTe 572273772 Page 4 of 4 MTH 60 Discovery Name Scatter plot Matchbox cars In this activity you will investigate the relationship between the distance a car rolls from the end of a ramp and the height of the ramp Each member of the group will complete the tables and plots Part I Data Collection Each group will need the following o a toy matchbox car 0 a piece ofcardboard o a ruler o a measuring tape 0 a calculator o textbooks notebooks other objects for varying the height of the ramp The experiment 1 Construct a ramp of a certain height maximum of 4 inches using textbooks notebooks or some other objects 2 Measure the height of the ramp in inches Record the height in the table below 3 Roll the matchbox car from the top of the ramp When the car stops measure the distance the car rolled from the bottom of the ramp Record this measurement in the table below 4 With the ramp at the same height roll the car two more times and measure the distance the car rolled Record these measurements in the table below 5 Average the three distances you measured for the ramp at this height Record the average distance the car rolled in the table below 6 Repeat the above ve steps for four other ramp heights height of ramp distance car travels gtrial 1t1ial 2 trail 31 WN Part 11 Data Analysis In the table below record the height and average distance from your data collection table Write the values of h and d as an ordered pair in the last column We call h height the independent variable and d distance the dependent variable The distance the car travels depends on the height of the ramp height average distance ordered pairs Make a scatter plot of the ordered pairs hd on the grid below 2 3 h height of ramp inches Using the scatter plot estimate how far the car will travel if the height of the ramp is 45 inches iii Using the scatter plot estimate how high the ramp needs to be for the car to roll 85 inches 777 How far should the car travel if the ramp is 0 inches high iii How would this be represented on the scatter plot MTH 60 Simonds class Key Concepts Working with formulas and geometry MartinGaye Qraotioe Qroblems 25 1 3 9 17 19 25 27 29 31 35 37 39 41 55 56 26 1 71 odd Working with formulas Examgle 1 The formula D rt can be used To deTerrnine The disTance Traveled D when an objecT moves aT a consTanT speed r for a lengTh of Time 1 DeTermine how far you drive if you drive aT a consTanT speed of 62 mph for 2hr LA gt L W MM Include units while 39 39 Arm Mama NM quotI39I b lt61 MM MII J Z AH 55 FZIt J DeTermine The speed aT which a Joaquin was walking fTsec if he walked aT a consTanT speed for 5 rninuTes and Trekked 1020 feeT during Those 5 minuTes Kb QLL39 Lac v M k M 37 7quot Iquot J quotl 010g r39 m5 0193U quotM S S H SolveTheforrnulaDrt fort 30L PLAL r ii r V ao l z Us ll r 3quot 445 4 A f J quotL a v3 u 5 Helloquot 3 4 J 37 3quot Page 1 of 6 l NIVW aD MTH 60 Simonds class Examgle 2 1 The formula A Ebh can be used To deTerrnine The area of a TriangleA whose base is a lengTh of b and heighT is a lengTh of 11 Find The area of The Triangle in Figure 1 Define your variable Include units while making calculations A 1 A in L Mr NC 1 H I D moLJVquotJ quot b r ux llt M A 2N T I 41 Si 71 mil s d4 xu r ohquot I L Find The heighT of The Triangle in Figure 2 if you know ThaT The Triangle39s area is 65 cm2 round The heighT To The nearesT 100 Define your variable Include units while making calculations L L L L L 617quot 390 t K q gh b17cm MPLJJrL 4 M w A a 77 mm Cu H icnll 16m L L Tl Luann 439 Solve The formula A bh for The variable 11 1 tquot A 1L 2 A X39x T J 2 13 LL 1 a 2A LL T Page 2 of 6 MYH an e Smmnds dass mm The fuhmum V7rr2h can be used m fwd The vumme uf a Ngm ewemar eyhheeh whmh Ts mmh speak fur a eah Afy Temsueaean hu ds 355ml ufsudawhmh S eqwmehnu 3555m3 A um Hm asudaccm 5 When 5 The heTghv unhe c n7 Ruuhe hm quots radms Ts exacHy 313cm v 0 quot v L 4 k 5 L Ryan 9 IL 4 L an 1le a perfeu eyhheeh and v The heTghv m The heahes 10 Dehhe yuwahahTeT Theme UNIS WW2 makmg ca m atmns n 4n ufmtk 1 1 3 V 7T3llcr FTUYE szszzm 3m Trumm H as amouaK WIT yuueahhhwm saiundEFka A 117 1T k 0quot kg L LL Jin L J p ILIJMI 3 Cd N L Examg eA The fuhmum fur The aheaumvmpezme see gure 4 Ts Ab bzh Suhe The fuhmma fur hmmbeh A i A L H w Qn3 M 39 l A U3 HQ A 51 1A Wk Rama 4 Auapezmd Ts a A smed hgme wheve a 2as1 uhe paw u smes Ts pave m The L 4 l k L L Engths unhe WEI payauex swdes ave caHed b and A the umahee hemeeh these swdes TscaHed h L Pagea nus 34y37 3 27 2 3 I ll 7X MTH 60 Simonds class Examgle 4 continued The formula for The area of a Trapezoid is A b1 172 h Solve The formula for A i L1Ll J A TilML L LLJL 5 L LLL 3A LL LuLl LLL LVL Lll lh Lv39quot 39LIg E K RA E LR L I L 1H Examgle 5 The surface area of a righT circular cylinder is given by The formula S Z rh Z rz Solve The formula for h 5 39TrL QTFrLlrrrL2nrL 5 9 er 2T1 le 5 39JlTKK 1frli i rL L L afquot I Page 4 of 6 MTH 60 Simonds class Example 6 In The Euclidean world The degree measuremenTs of The Three angles of a Triangle always sum To 180 Find The degree measuremenTs of each angle in an isosceles Triangle if The Two congruenT angles are each 150 larger Than The non congruenT angle Drawa picture define a variable with unit and label each angle with an expression that represents its measure Write an equation about the angle sum solve that equation I Check your solution in the context of the problem I State your conclusion including units l Lg39L X rtl39dgn l 1 43 MLaJUrc 0 1 bquot39 39jV ltn Ajlt X lXiS39l IS l T 7 3x 30 quot 3x 3 w 3o 3X 753 3x 19 3 s Ixtfb ILL J jrcg Jurgmgx l J 9439 L I j 6quot 5 0 LS s o Cr Nquot am l C Page 5 of 6 MTH 60 Simonds class Example 7 The perimeTer of The shape shown in Figure 5 is 18 cm The lengTh of each sTraighT side is 80 The lengTh of a squiggly side Find The lengTh of each side Define a variable with unit and label each side of Figure with an expression that represents its length Write an equation about the perimeter solve that equation I Check your solution in the context of the problem I State your conclusion including units l Ll i39 VLfI Jquot L q 4 1 D a JvllJ A i w j lion Tch IKCA 209 13 f 39 J gen 5 15 lqw C Em Sgtml LL Ian 7H gm sum m m LL 55 falsity 395 LI39 cm Page 6 of 6 Z39x UKIA S I03M dNL W2 LP LuluC WW fvg lt39 L 5 Arts J uu quot ltf11 L 39 S7 at H CF w M quotquot S39IPIOJ39V quot AL Cukc L Cv 39 CWquot quot39 VJ quotJ L1 LN L39 a quot MTH 60 Simonds class Key Concepts Ordered pairsThe Rectangular Coordinate system Solutions to equations with 2 variables Linear equations in two variables nterce ts Horizontal and Vertical ines MartinGave sections and practice problems 31 1 19 odd 23 51 odd 55 57 32 1 43 odd 47 53 55 33 1 10 all 11 63 odd Definitions A solution to an equation with two variables x and y is an ordered pair ab where a and b are real numbers with the property that if x a and y b the equation is true The number a is called the xcoordinate of the ordered pair and the number 17 is called theycoordinate ofthe ordered pair Example 1 What are the x and y coordinates of the ordered pair 972 Is the ordered pair a solution to the equation 3x 4y 10 x TL X CHIJL Ja I f and I kc jvLII39J Rrf39 if L J39 WE 3 k lj ox Example 2 What is the ordered pair with axcoordinate of 5 that satisfies the equation y 6x 4 Lu 7 6 L X 2 j xH LL IJU39RH J Page 1 of 8 MTH 60 Simonds class Example 3 WhaT is The ordered pair wiTh a ycoordinaTe of 5 ThaT saTisfies The equaTion y 6x 4 w jCgtc Q C6xf 2 I 6x lLlt Joiil kn if lt LL Example 4 WriTe The missing values inTo Table 1 so ThaT each implied ordered pair is a squTion To The equaTion x y 4 HOT The four ordered pairs onTo Figure 1 Table1 x 4 0 I 0 L L 2 15 S Figure 1 Example 5 The sum of The The coordinaTes of each poinT on The line in Figure 2 is always The same number WhaT is This consTanT sum ch r J ft Table 2 0quot 39 391 point ordered pair x y mm u Pr 3 H JDIJ Kn Id39 3 O Jquot 2 1 1 CI 3quot o L Figure2 l Page 2 of 8 MTH B rswmunds mass m E 2rd pews are shown m mm a 5mm m ordered paw assomamd Wm each pow and where m m coordmavaganz m pow hes assume that both coordmatzs of each pow are mergers wmch pow a o 39 37 o x x k P s asomnon to the equamon 3x 7 2y Coordinates Locaiion ixAn mlr 021 our Gum oIT Q JMMJV 6 75 74 73 72 71 Page 3 ma Xy Lurk JV pl MTH 60 Simonds39 class Examgle 7 NOT The poinTs A734 37373 and C52 onTo Figure 4 and find The area A of The resulTanT Triangle Assume ThaT The scale on each axis is in cenTimeTers An 1 H 401W 2 j OWL Figure4 t We o l AABL 395 Z ml Examgle 8 AT each poinT on The line in Figure 5 Twice The x coordinaTe minus The y coordinaTe is always The same number WhaT is This commor e ncti iJ LL J 1 4 Jn 1L 2 Y J Table 5 point ordered pair 2x 7 y A I r 3 8 03 2 c C I 2 D quotll 5 393 Figure 5 Page 4 of 8 MTH 60 Simonds39 class Example 9 3 Complete Table 6 with solutions to the equation y 75x Then graph the solutions and show that they are TableB39 y73x 39 7 2 6 75 74 73 72 71 Definition An equation that can be written in the form Ax By C not both A and B zero is called a linear equation of x and y The graph of all ofthe solutions to a linear equation with two variables is a straight line when graphed in the rectangular coordinate plane Three or more points in the plane are collinearlie on a common line if and only ifthey all satisfy a common linear equation Example 10 Complete Table 7 with four solutions to the equation x 2y 6 Then graph the solutions and show that they are collinear Table7 x2y6 x 6 75 74 73 72 71 l l ll L 1 MN L Figure 7 Page 5 of 8 MTH 60 Simonds39 class 1 CompleTe Table 8 wiTh four solLITions To The equaTion 3x7 2y 6 Then graph The solLITions and show ThaT They are collinear Kquot lv w lIJWL l 17 lL 6 75 74 73 72 71 1 cl Lm quot H I a FigureB l L Definition When a line or a curve is drawn in the xy plane any point on the line or curve that also lies on the y axis is called ayintemept and any point on the line or curve that also lies on the x axis is called an xintercept xintercept x0 yintercept 0y Example 12 STaTe all of The inTercest of The curve shown in Figure 9 39m 9 JAWfU 1 l 0 o l o oi 2 Figure 9 3 Muqlwlu L K 5 n MTH 60 Simonds39 class Examgle 13 Find The inTercest of The line wiTh eqLIuTion 3x 7 5y 7 20 x inLf f 39 i39ltfkn 1 3n L I 39 Ld x o 3110 Ku j l quot gt X L3 7 9 3 f lt J L 39xrinquot39rILsrJ l39 ltL 3I 0 539s i l I jur r 0 4i Examgle 14 HOT The line 2x 4y78 onTo Figure 10 ufTer firsT finding The inTercest of The line Find a Third poinT on your ploTTed line and show ThaT iT also suTisfies The eqLIuTion K idZ io 394 39 9 Ji Amu Tm ii LII V IL l l l 21 MPH Y IL c s Figure 10 Page 7 of 8 MTH 60 Simonds class Examgle 15 Plot onto Figure 11 several points in the Jayplane that satisfy the equation x74 What do you observe What are the intercepts ofthe resultant curve Table 11 x74 Figure 11 In QJinmsioJ 4 is L And In ulk x iuts hv f ad no y h rutvf39 Examgle 16 What is an equation for the line in Figure 12 What are the intercepts of the line V rk39Ltl x K M L r L I lq lly 35k I ll 6 75 74 73 72 71 Figure 12 Page 8 of 8 MTH 60 Mr Simonds class Key Concepts Slopeintercept form of the equation of a line MartinGave sections and practice problems 351 51 odd 53 58all J Jquot L 9 Examgle l rL h l The line 3x 2y 2 is shown in Figure 1 0 Find Two poinTs on The line and verify ThaT They in facT saTisfy The equaTion 3x 2y 2 039 Find The slope andy inTercepT of The line 0 Solve The equaTion 3x2y 2 Tory STaTe The connecTion beTween This new equaTion and The slope and y inTercepT of The line CL K50 30 20 3 2 Jump Ll L 30 La L l r 1 L xwyl 539 Figure1 XLL quot LIL 35 M VK39Y39 3 mx 37305 1 L ll MWW 1005 I 10 l SIIIJL 7 Fquot jmx5 3L j Mm o I C39 3X1j L 31 2x L L 234j393x11 3x 1 3gtlt 1 1 3 KP 1n a L y x H MTH 60 Mr Simonds class Example2 The line 4x 7 3y 6 is shown in Figure 2 Find Two poinTs on The line and verify ThaT They in facT saTisfy The equaTion 4x 7 3y 6 a b Find The slope andy inTercepT of The line c Solve The equaTion 4x7 3y 6 Tory STaTe The connecTion beTween This new equaTion and The slope and y inTercepT of The line ILL K3 L Mai 304 39 Z Lch L 7 we 3lt n 04 L19 l a Figure2 m 0 3 l a 3 13 3 34 0 4 13 k Jlorlt TX kbl l u 5 fkgf39f Page 2 of8 MTH 60 Mr Simonds class Definition and most awesome fact The equation y mx b is called the slopeintercept form of a linear equation The equation of any nonvertical line can be written in this form When the equation is written in this form the number m is the slope of the line and the point 0 b is the yintercept of the line Examgle 3 STaTe The slope and yinTercepT of The line wiTh y igx 3 Graph The line onTo Figure 3 1 S 3 03 is is 4 a 2 1 m L 3931 2 5 RY 3 a v2 s Ls ling 3 I 14 L Examgle 4 STaTe The slope and yinTercepT of The line wiTh equaTion y 7 4 Graph The line onTo Figure 4 6 75 74 73 72 71 Figure 4 Page 3 of8 C Rem quot31 7 3 3 q F 041 MTH 60 Mr Simonds class Examgle 5 ST e and yinTercepT of The line wiTh equuTion 3x75y4 Graph Theline onTo Figure 5 3 q r Y 3 39I j Figure5 j 239 F m 3 39 Y 1 M J gtL1 o Ally yquot 5 39 Examgle 6 STuTe The slope and yinTercepT of The line wiTh equuTionx 7 5 Graph The line onTo Figure 6 x I I I Lon 5039 fquot 7 quot Ltr3 y 3quot u JcILJJIV Jr vimquotJ no yru17 Figure 6 Page 4 of8 MTH 60 Mr Simonds class Examgle 7 0 Find an equaTion for The line in Figure 7 b Find an equaTion for The line parallel To The given line ThaT passes Through The poinT jm l 39l Lhhl Slop l Mil Ynxb g law twin IA 1 34 1 Figurev LL 34 4 if lLe rwdlvl IIALP K t 0 39 quot X39I Mme 8f j Find an equaTion for The line shown in Figure 8 M i 7 l39 3 r I A L0 2 kg rola3 5 o L Figures LgL l quotA l 2 C N 2quot j 7XTI 3 L 7 03 7rquot l ii r H r I 1 efugfu u l l L l e H WMv Page 5 of8 MTH 60 Mr Simonds class Examgle 9 Find an equaTion for The line ThaT passes Through The poinTs 723 and 771 Page 6 of 8 MTH 60 Mr Simonds class Examgle 10 Find an equaTion for The line ThaT is parallel To The line wiTh equaTion 6y x 2 and passes Through The poinT 122 Page 7 of 8 MTH 60 Mr Simonds class Examgle 11 Find an equaTion for The line ThaT is perpendicular To The line in Figure 9 and passes Through The poinT 11 is 75 r4 73 72 71 Figure 9 Page 8 of8 MTH 60 Group work key from week 3 lecture notes Problem 1 CompIeTer simplify each expression Make sure ThaT you presenT your work in The proper formaT NOTE I39m showing more sTeps Than you need To Show A IoT of The sTeps I39m showing are There To help you find your misTake if you don39T have The correcT simplificaTion Simplify 2x 5 2x 52x 25 2x 10 Simplify 3y 74y 2 3y 74y 23y 374y 2 3y 214y 2 3y4y 21 2 7y 23 Simplify 8d 6 d 8d 6 d8d d 6 7d 6 Simplify 0180x2 90x 14x 6x2 0180x2 90x 14x 6x2 8x2 9x 14x 6x2 8x2 6x2 9x 14x 14x2 23x Page 1 of 4 Problem 2 CompIeTe each senTence wiTh one of The wordsphr39asesnumber39snames below Pick The one ThaT makes The senTence Tr39ue commutative additive 0 numerator difference property Inverse distributive multiplicative 1 denominator positive property Inverse 4 assoc39atlve dd39t39l e opposite The Strokes property identity 87 order Of mut39pllcatlve ostrich negative operations identity a If you mulTipIy 87 wiTh iTs mulTipIicaTive inverse The r39esulT is 1 b 34 7 3 4 3 7 is an example of The disTr ibuTive properTy c 22 is negaTive d When reading aloud 6 The firsT minus sign is read as opposiTe and The second minus sign is read as negaTive e 6 9 8 7 8 7 6 9 is an example of The commuTaTive properTy of addiTion f If you mulTipIy 87 wiTh The addiTive idenTiTy The r39esulT is 0 g 12 7 9 19 9 is an example of order of operaTions h If The opposiTe of lxl is noT zer39o Then iT is definiTer negaTive i If you mulTipIy 87 wiTh The mulTipIicaTive idenTiTy The r39esulT is 87 j 8 5 5 8 0 is an 39 rrquot 39 of The 39 39 properTv of addiTion Page 2 of 4 Problem 3 Decide wheTher each sTaTemenT is True or false T No maTTer whaT values x y and z assume x yz xz yz F 7 4 87 4 8 F 4x3 and 3x4 are like Terms The variable x does noT have The same exponenT in The Two Terms T If you mulTiply a number by The mulTiplicaTive idenTiTy The resulT is always The original number T No maTTer whaT values x y and z assume x y z x y 2 F No maTTer whaT values x y and z assume x yl z x y 2 This sTaTemenT is True if The Three numbers are all posiTive andor zero If negaTive numbers geT involved Though This sTaTemenT is frequenle false For example see whaT happens ifyou leT x1 y 1and z 0 F If you add a number To iTs addiTive inverse The resulT is always The original number The sum of a number and iTs addiTive inverse is always 0 Problem 4 Name ThaT numberexpression NoTe These quesTions were prep work for whaT we will be discussing in class on Tuesday a WhaT number do you add To 4 so ThaT The resulT is 0 4 b WhaT number do you add To 83 so ThaT The resulT is 0 83 2 3 c WhaT number do you mulTiply wiTh a so ThaT The resulT is 1 E 1 d WhaT number do you mulTiply wiTh 6 so ThaT The resulT is 1 g e WhaT number do you subTracT from To 162 so ThaT The resulT is 0 162 f WhaT number do you divide 4322 by so ThaT The resulT is 1 4322 Page 3 of 4 WhaT would you subTr acT from 3x so ThaT The r39esulT is 0 3x WhaT would you add To x so ThaT The r39esulT is 0 x WhaT would you add To 254x2 so ThaT The r39esulT is 0 254xZ WhaT is The soluTion To The equaTion x 11 87 The soluTion is 76 WhaT is The soluTion To The equaTion 5x 35 The soluTion is 7 WhaT is The soluTion To The equaTion 10x 11 22x 5 This one is kind of Tough If you figured iT ouT gr eaT We39re going To learn a process on Tuesday ThaT will allow us To figure ouT ThaT The soluTion is Page 4 of 4

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