Solution Manual for Elementary and Intermediate Algebra, 4th Edition

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SOLUTIONSMANUALCTRIMBLE&ASSOCIATESELEMENTARY&INTERMEDIATEALGEBRAFOURTHEDITIONMichael Sullivan, IIIJoliet Junior CollegeKatherine R. StruveColumbus State Community CollegeJanet MazzarellaSouthwestern Collegewith contributions byJessica BernardsPortland Community CollegeWendy FreshPortland Community College

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ContentsChapter 11Chapter 250Chapter 3124Chapter 4202Chapter 5267Chapter 6315Getting Ready for Intermediate Algebra:A Review of Chapters 1−6362Chapter 7365Getting Ready for Intermediate Algebra:A Review of Chapters 1−7443Chapter 8448Chapter 9530Chapter 10606Chapter 11747Chapter 12812Chapter 13894Appendix A938Appendix B941Appendix C949Appendix E994

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1Chapter 1Section 1.21.2 Quick Checks1.A natural number is prime if its only factors are1 and itself.2.In the statement 6⋅8 = 48, 6 and 8 are calledfactors and 48 is called the product.3.1226223··The prime factorization of 12 is 2⋅2⋅3.4.120215222··· 35·22430The prime factorization of 120 is 2⋅2⋅2⋅3⋅5.5.31 is a prime number.6.117/ \9 13/ \\3⋅3⋅13The prime factorization of 117 is 3⋅3⋅13.7.The least common multiple of two or morenatural numbers is the smallest number that is amultiple of each of the numbers.8.6 = 2 · 38 = 2 ·· 2 · 22 · 3 · 2 · 2The LCM is 2⋅3⋅2⋅2 = 24.9.45 = 3 · 3 · 572 = 3 · 3 ·2 · 2 · 23 · 3 · 5 · 2 · 2 · 2The LCM is 2⋅2⋅2⋅3⋅3⋅5 = 360.10.14 = 2 · 79 =3·32·7·3·3The LCM is 2⋅7⋅3⋅3 = 126.11.18 = 2 ·3 · 330 = 2 ·3 ·52 · 2 · 3 · 3 · 512 = 2 · 2 · 3The LCM is 2⋅2⋅3⋅3⋅5 = 180.12.In the fraction712 ,7 is called the numerator and12 is called the denominator.13.Fractions which represent the same portion of awhole are called equivalent fractions.14.Multiply the numerator and denominator of12by 5.11 5522 510⋅==⋅15.Multiply the numerator and denominator of58by 6.55 63088 648⋅==⋅16.The least common denominator is the leastcommon multiple of the denominators of a groupof fractions.17.The denominators are 4 and 6.4 = 2 · 26 = 2 ·3LCD = 2 · 2 · 3 = 1211 3344 312⋅==⋅55 21066 212⋅==⋅

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra218.The denominators are 20 and 16.20 = 2 · 2 · 516 = 2 · 2 ·2 · 2LCD = 2 · 2 · 5 · 2 · 2 = 8099 4362020 480⋅==⋅1111 5551616 580⋅==⋅19.A fraction is written in lowest terms if thenumerator and denominator share no commonfactor other than 1.20.453 3 53 3 59802 2 2 2 52 2 2 2 516⋅⋅⋅⋅===⋅⋅⋅⋅⋅⋅⋅⋅21.42 2493 39⋅==⋅22.162 2 2 222222562 2 2 722277⋅⋅⋅⋅⋅⋅===⋅⋅⋅⋅⋅⋅23.The 1 is two places to the right of the decimal;this is the hundredths place.24.The 2 is one place to the right of the decimal;this the tenths place.25.The 8 is four places to the left of the decimal;this is the thousands place.26.The 9 is three places to the right of the decimal;this is the thousandths place.27.The number 1 is in the tenths place. The numberto its right is 7. Since 7 is greater than 5, weround to 0.2.28.The number 3 is in the hundredths place. Thenumber to its right is 2. Since 2 is less than 5, weround to 0.93.29.The number 9 is in the hundredths place. Thenumber to its right is 6. Since 6 is greater than 5,we round to 1.40.30.The number 0 is in the hundredths place. Thenumber to its right is 4. Since 4 is less than 5, weround to 690.00.31.The number 9 is in the tenths place. The numberto its right is 8. Since 8 is greater than 5, weround to 60.0.32.0 45 2 02 00..20 45.=33.0 8336 5 0004 8201820182..50 836.=34.1 3758 11 00083 02 4605640400..111 3758.=35.0 4285717 3 0000002 820146056403550491073..30 4285717.=

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions336.62 330 6102 55.⋅===⋅37.170 17100.=38.6255 1250 6251000.⋅==8 125⋅58=39.The word percent means parts per hundred, so35% means 35 parts out of 100 parts or35100 .40.12323230 23100100%%.%=⋅==41.11110 01100100%%.%=⋅==42.172 472 472 40 724100100.. %. %.%=⋅==43.11271271271 27100100%%.%=⋅==44.To convert a decimal to a percent, multiply thedecimal by1001% .45.1000 150 15151%..%=⋅=46.1000 80 8801%..%=⋅=47.1001 3721 372137 21%... %=⋅=48.1000 0040 0040 41%... %=⋅=1.2 Exercises50.933·The prime factorization of 9 is 3⋅3.52.100425225··· 5The prime factorization of 100 is 2⋅2⋅5⋅5.54.3557·The prime factorization of 35 is 5⋅7.56.5469233··· 3The prime factorization of 54 is 2⋅3⋅3⋅3.58.70235257··The prime factorization of 70 is 2⋅5⋅7.60.79 is a prime number.62.315935335··· 7The prime factorization of 315 is 3⋅3⋅5⋅7.64.10 = 2 · 514 = 2 ·72 · 5 · 7The LCM is 2⋅5⋅7 = 70.66.42 = 2·18 = 2·3·32·3·3·73·7The LCM of 42 and 18 is 2⋅3⋅3⋅7 = 126.68.55 =5 · 116 = 2 · 32 · 3 · 5 · 11The LCM of 55 and 6 is 2⋅3⋅5⋅11 = 330.70.8 = 2·2·260 =2·2·2·3·52·2·3·5The LCM of 8 and 60 is 2⋅2⋅2⋅3⋅5 = 120.

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra472.15 =3 · 520 = 2 · 2 ·52 · 2 · 3 · 3 · 59 =3 · 3The LCM of 9, 15, and 20 is2⋅2⋅3⋅3⋅5 = 180.74.18 =2 · 3 · 320 = 2 · 2 ·52 · 2 · 3 · 3 · 54 = 2 · 2The LCM of 4, 18, and 20 is2⋅2⋅3⋅3⋅5 = 180.76.44 31255 315⋅==⋅78.55 2101414 228⋅==⋅80.4 104041 1010⋅==⋅82.4 = 2 · 212 = 2 · 2 · 3LCD = 2 · 2 · 3 = 1233 3944 312⋅==⋅The equivalent fractions are912and512 .84.4 = 2 · 29 =3 · 3LCD = 2 · 2 · 3 · 3 = 3611 9944 936⋅==⋅22 4899 436⋅==⋅The equivalent fractions are936and836 .86.12 = 2 · 2 · 315 =3 · 5LCD = 2 · 2 · 3 · 5 = 6055 5251212 560⋅==⋅77 4281515 460⋅==⋅The equivalent fractions are2560and2860 .88.4 = 2 · 26 = 2 ·3LCD = 2 · 2 · 3 · 5 = 6010 = 2 ·577 6421010 660⋅==⋅11 151544 1560⋅==⋅55 105066 1060⋅==⋅The equivalent fractions are42156060,,and5060 .90.93 33 33153 53 55⋅⋅===⋅⋅92.813 3 3 33 3 339362 2 3 32 2 334⋅⋅⋅⋅⋅⋅===⋅⋅⋅⋅⋅⋅94.93 3331273 3 33 333⋅⋅===⋅⋅⋅⋅96.362 2 3 322 3 3942 222⋅⋅⋅⋅⋅⋅===⋅⋅98.The 8 is three places to the left of the decimal;this is the hundreds place.100.The 7 is one place to the right of the decimal;this is the tenths place.102.The 9 is one place to the left of the decimal; thisis the ones place.104.The number 8 is in the hundredths place. Thenumber to the right of 8 is 4. Since 4 is less than5, round to 7298.08.

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions5106.The number 3 is in the ones place. The numberto the right of 3 is 5. Since 5 is 5 or more, roundto 544.108.The number 9 is in the hundredths place. Thenumber to the right of 9 is 8. Since 8 is greaterthan 5, round to 683.10.110.The number 9 is in the tenths place. The numberto the right of 9 is 9. Since 9 is greater than 5,round to 38.0.112.0 754 3 002820200..30 754.=114.0 2229 2 0001 8201820182..20 222or 0 29.....=116.0 3437532 11 000009 61 401 28120962402241601600..110 3437532.=118.0 46153813 6 0000005 280782013706550391101046..60 46153813.=120.1 1450 57 00507 0502 002 000..571 1450.=122.251 250 25100.⋅==425⋅14=124.42220 4102 55.⋅===⋅126.3582 1791790 35810002 500500.⋅===⋅128.15959590 59100100%%.%=⋅==130.18 258 258 250 0825100100..%.%.%=⋅==132.10 50 50 50 005100100.. %. %.%=⋅==

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra6134.1000 50 5501%..%=⋅=136.1000 3490 34934 91%... %=⋅=138.100111001%%=⋅=140.1 878 15 0087 06 460564..158rounded to the nearest tenth is 1.9.142.1 2857 9 00072 01 4605640355..97rounded to the nearest hundredth is 1.29.144.0 580631 18 000015 52 502 4820018614..1831rounded to the nearest thousandth is 0.581.146.The number of laps is the LCM of 21, 18, and15.18 = 2 · 3 · 315 =3 ·5LCM = 2 · 3 · 3 · 5 · 7 = 63021 =3 ·7The drivers would finish at the same time after630 laps.148.Find the LCM of 14 and 20.14 = 2 ·720 = 2 · 2 · 5LCM = 2 · 2 · 5 · 7 = 140Pamela and Geoff are in Columbus every140 days.150.450 of 7504502 3 3 5 523 3 5537502 3 5 5 523 5 555⋅⋅⋅⋅⋅⋅⋅⋅===⋅⋅⋅⋅⋅⋅⋅⋅The fraction of students that are enrolled in 15 ormore semester hours is35 .152.(a)64032201100⋅=5520⋅3255=The fraction of adult women who stated thatthey believe in ghosts is3255 .(b)0 581855 32 000027 54 504 401005545044010..JJJM320 58155.=(c)1000 58158 181858 181%..%.%⋅=≈

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions7154.0 69565115 80 0000069 011 0010 3565057575069060057525..801000 69570 695769 571151%...%≈=⋅=156.(a)480 47521011000 470147 521.%..%=⋅=47.52% were white ash.(b)510 50501011000 505050 501.%..%=⋅=50.50% were green ash.(c)20 01981011000 01981 981.%..%=⋅=1.98% were blue ash.158.0 4511 5 004 460555..50 4511.=1000 454545 451%..%⋅=45.45% is saturated fat.Section 1.3Are You Prepared for This Section?P1.0 6258 5 0004 8201640400..50 6258.=P2.0 818111 9 00008 82011908820119..90 818111....=or0 81.1.3 Quick Checks1.The first four positive odd numbers are 1, 3, 5,and 7. IfOrepresents this set, thenO= {1, 3, 5, 7}.2.The states whose names begin with the letter Aare Alabama, Alaska, Arizona, and Arkansas. IfArepresents this set, thenA= {Alabama, Alaska, Arizona, Arkansas}.3.There are no states whose names begin with theletter Z. IfZrepresents this set, thenZ= { } or∅.4.Every integer is a rational number. True5.Real numbers that can be represented with aterminating or repeating decimal are calledrational numbers.6.12 is the only natural number.7.0 and 12 are the whole numbers.

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra88.−5, 12, and 0 are the integers.9.115 12 2 76 05 ,,,.,,−and184are the rationalnumbers.10.2.737737773... is the only irrational number.11.All the numbers listed are real numbers.12.The point on the real number line whosecoordinate is 0 is called the origin.13.–2–101–3432–20 123 3.514.The symbols <, >,≤,≥are called inequalitysymbols.15.2 < 9 because 2 lies to the left of 9 on a realnumber line.16.−5 <−3 because−5 lies to the left of−3 on a realnumber line.17.4152>because48510=and15210 .=Having 8parts out of 10 is more than having 5 parts out of10. Also,40 85.=and10 52.=and 0.8 lies tothe right of 0.5 on a real number line.18.40 57.>because40 5714287.=and0 571428.lies to the right of 0.5 on a real number line.19.420315=20.4534−< −21.The distance from zero to a point on a realnumber line whose coordinate isais called theabsolute value ofa.22.|−15| = 15 because the distance from 0 to−15 ona real number line is 15.23.3344−=because the distance from 0 to34−ona real number line is34 .24.−|−4| =−4 (|−4| = 4) because the distance from 0to−4 on a number line is 4.1.3 Exercises26.The set of natural numbers less than 25 isB= {1, 2, 3, 4, ..., 24}.28.The set of integers between−6 and 4 isC= {−5,−4,−3,−2,−1, 0, 1, 2, 3}.30.There are no whole numbers that are less than 0;F= { } or∅.32.3 and 0 are the whole numbers.34.−4, 3,132 ,−and 0 are the rational numbers.36.All the numbers listed are real numbers.38.−4.2,3 5. ,and515=are the rational numbers.40.515,=so55is the only integer.42.515,=so55is the only natural number.44.5402–0.51.5012–1–2–3446.0 lies to the right of−5 on a number line. Thestatement is false.48.−3 lies to the right of−5 on a number line. Thestatement is true.50.Since4 74 777.....,=4 74 7.. .<The statement isfalse.52.Since10 3333....,=10 333..>The statement isfalse.

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions954.Since−8 lies to the right of−8.5 on a number line,−8 >−8.5.56.Since50 41612.=and20 6663.,=52123 .<58.Since50 454511....,=50 4511..=60.Since30 754.=and30 65. ,=3345 .>62.|−8| = 8 because the distance from 0 to−8 on a real number line is 8.64.|7| = 7 because the distance from 0 to 7 on a real number line is 7.66.131399−=because the distance from 0 to139−on a real number line is139 .68.−|−3.2| =−3.270.(a)43–1.5–20123–1–3–4–54567|–4|0–153––288(b)From left to right on the number line, the order of the numbers is153 ,−−2,−1.5,43 ,−0, |−4|, 8.(c)i.1553,−= −−2, 0, |−4| = 4, 8 are integers.ii.All numbers listed are rational numbers.NaturalWholeIntegerRationalIrrationalReal72.0√√√√74.2√√76.4√√√√√78.6 45.√√80.Only decimal numbers that are terminating or repeating are rational. The decimal number 4.010010001... isirrational. The statement is false.82.0 is neither positive nor negative. The statement is false.84.The set of integers is a subset of the real numbers. The statement is true.86.Consider the number in the form842,=which is an integer. The statement is true.88.−1 is a negative or nonpositive integer. The statement is true.

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra1090.Nonnegative integers are 0, 1, 2, ..., or the wholenumbers.92.Terminating or repeating decimals are rationalnumbers.94.There is no number that is both negative andpositive. This is the empty set or∅.96.The element f inZis not an element ofY. Thestatement is false.98.The element f inZis not an element ofX. Thestatement is false.100.A∩Bis the set of all elements common to bothAandB, soA∩B= {10, 11, 12}.102.B∩Cis the set of all elements common to bothBandC, soB∩C= {11, 12, 13, 14, 15}.104.A∪Cis the set of elements in eitherAorC, soA∪C= {7, 8, 9, 10, 11, 12, 13, 14, 15}.106.SinceYis a subset ofX, thenX∩Y=Y, soX∩Y= {60, 62, 64, ..., 80}.108.(a)Answers may vary. One possibility: Thereare subsets with no elements, subsets withone element, subsets with two elements, andsubsets with three elements.0:∅1: {a}, {b}, {c}2: {a,b}, {a,c}, {b,c}3: {a,b,c}(b)There are a total of 8 subsets.110.An irrational number is any number that cannotbe written as the quotient of two integers. Non-repeating, non-terminating decimals areirrational numbers. Numbers such asπand2and 8.98987987698765... are irrational numbers.Section 1.4Are You Prepared for This Section?P1.162 2 2 2222 22 24362 2 3 322 3 33 39⋅⋅⋅⋅⋅⋅⋅====⋅⋅⋅⋅⋅⋅⋅P2.|−5| = 5 because the distance from 0 to−5 on anumber line is 5.1.4 Quick Checks1.The answer to an addition problem is called thesum.2.1112131410986 steps rightStartEnd8 + 6 = 143.–6–5–4–3–7–8–95 steps leftStartEnd−3 + (−5) =−84.12320–1–24 steps rightStartEnd−1 + 4 = 35.12340–1–24 steps leftStartEnd3 + (−4) =−16.–6–5–4–3–7–8–94 steps rightStartEnd−8 + 4 =−47.1617181514133 steps leftStartEnd17 + (−3) = 148.The sum of two negative integers will benegative.9.|−11| = 11|7| = 7The signs of−11 and 7 are different, so wesubtract the absolute values: 11−7 = 4.The larger absolute value, 11, corresponds to anegative number in the original problem, so thesum is negative.−11 + 7 =−4

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions1110.|5| = 5|−8| = 8The signs of 5 and−8 are different, so wesubtract the absolute values: 8−5 = 3.The larger absolute value, 8, corresponds to anegative number in the original problem, so thesum is negative.5 + (−8) =−311.|−8| = 8|−16| = 16The signs of−8 and−16 are the same, so we addthe absolute values: 8 + 16 = 24.Both numbers in the original problem arenegative, so the sum is negative.−8 + (−16) =−2412.|94| = 94|−38| = 38The signs of 94 and−38 are different, so wesubtract the absolute values: 94−38 = 56.The larger absolute value, 94, corresponds to apositive number in the original problem, so thesum is positive.94 + (−38) = 5613.For any real numberaother than 0, there is areal number−a, called the additive inverse, oropposite, ofasuch thata+ (−a) =−a+a= 0.14.The additive inverse of 7 is−7 because7 + (−7) = 0.15.The additive inverse of−21 is−(−21) = 21because−21 + 21 = 0.16.The additive inverse of85−is8855⎛⎞−−=⎜⎟⎝⎠because88055.−+=17.The additive inverse of 5.75 is−5.75 because5.75 + (−5.75) = 0.18.The answer to a subtraction problem is called thedifference.19.The subtraction problem−3−10 is equivalent to−3 + (−10).20.59−(−21) = 59 + 21 = 8021.−32−146 =−32 + (−146) =−17822.−19−(−40) =−19 + 40 = 2123.81358135550()−+=+ −+= −+=24.274918274918761858()−−+= −+ −+= −+= −25.314833148317839312()()()− −−+=++ −+=+ −+=+=26.−825 + 375−(−735) + 265=−825 + 375 + 735 + 265=−450 + 735 + 265= 285 + 265= 55027.The product of two integers with the same sign ispositive.28.−3(7) =−2129.13(−4) =−5230.5⋅16 = 8031.−9(−12) = 10832.(−13)(−25) = 32533.The product of thirteen negative factors isnegative. True34.−3⋅9(−4) =−27(−4) = 10835.34561256606360()()()()()()−−−= −−−=−= −36.The reciprocal of 6 is16 .37.The reciprocal of−2 is12 .−38.In the division problem1863,=18 is called thedividend, 3 is called the divisor, and 6 is calledthe quotient.

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Chapter 1:Operations on Real Numbers and Algebraic ExpressionsISM: Elementary & Intermediate Algebra1239.The quotient of two negative numbers ispositive. True40.205 4545541 41 41⋅⋅==== −−− ⋅− ⋅−41.721 4 181 4 1818204 54 55−− ⋅⋅− ⋅⋅=== −⋅⋅42.976397637717()()()()−−−−÷ −===−−17()−9=1.4 Exercises44.6 + 4 = 1046.−4 + 12 = 848.7 + (−13) =−650.−13 + (−5) =−1852.−32 + 49 = 1754.−145 + (−68) =−21356.(−13) + 37 + (−22) = 24 + (−22) = 258.3446127212127207272()()−++ −+=+ −+=+=60.The additive inverse of−34 is−(−34) = 34because−34 + 34 = 0.62.The additive inverse of 7 is−7 because7 + (−7) = 0.64.35−23 = 35 + (−23) = 1266.12−19 + 12 + (−19) =−768.−15−9 =−15 + (−9) =−2470.14−(−18) = 14 + 18 = 3272.−15−(−15) =−15 + 15 = 074.0−18 = 0 + (−18) =−1876.46−(−25) = 46 + 25 = 7178.49−(−49) = 49 + 49 = 9880.7⋅9 = 6382.9(−7) =−6384.−21⋅0 = 086.(−22)(−5) = 11088.(−128)7 =−89690.−6⋅4⋅8 =−24⋅8 =−19292.−8(2)(−9) =−16(−9) = 14494.346112 6172172()()( )()( )()()−−−=−=−= −96.The reciprocal of 10 is110 .98.The reciprocal of−3 is13 .−100.The reciprocal of 2 is12 .102.364 94 9369491 91 9⋅⋅÷====⋅⋅104.976397717()()()−−−==−−17()−9=106.14424 62462461 61 6−−⋅−⋅=== −⋅⋅108.205 4545164 4444⋅⋅===⋅⋅110.12020 620620206611 611 61111⋅⋅==== −−−⋅−⋅−112.204802041234()()()−−−==−−34()−203=114.7(−15) =−105116.87−19 = 87 + (−19) = 68118.7⋅209 = 1463

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ISM: Elementary & Intermediate AlgebraChapter 1:Operations on Real Numbers and Algebraic Expressions13120.62615662626126()()()−−−==−−126()−6=122.103 + (−66) = 37124.42427 676712122 6262−⋅⋅====⋅⋅126.|−10| + (−62) = 10 + (−62) =−52128.|−193|−(−20) = 193−(−20) = 193 + 20 = 213130.The sum of 32 and−64 is written as32 + (−64) =−32.132.−85 subtracted from−16 is written as−16−(−85) =−16 + 85 = 69.134.32 multiplied by−8 is written as32⋅(−8) =−256.136.−40 divided by 100 is written as402 20220401001005 20−−⋅−⋅−÷===⋅520⋅25 .= −138.The product of−11 and−5 is written as−11(−5) = 55.140.The difference of−16 and−42 is written as−16−(−42) =−16 + 42 = 26142.−14°; the number is negative because thetemperature is below zero.144.+$125,000 or $125,000; the number is positivebecause there was a profit.146.+12,368 or 12,368; the number is positivebecause the population grew.148.15 + (−7) + 14 + 26 = 48The ball is on the 48-yard marker.150.399 + 839 + (−69) + (−78) + (−739) = 352She has $352 left for spending money.152.35,000 + 4290 + (−10,400) + (−2605) = 26,285The plane is flying at an elevation of 26,285 feet.154.14,495−(−280) = 14,495 + 280 = 14,775The difference is 14,775 feet.156.6 + (−4) = 26(−4) =−24The integers are 6 and−4.158.−15 + (−3) =−18−15(−3) = 45The integers are−15 and−3.160.Consider 5(−3). The statement 5(−3) means(−3) + (−3) + (−3) + (−3) + (−3) =−15. Weconclude that the product of a positive numberand a negative number is negative.Section 1.5Are You Prepared for This Section?P1.12 = 2⋅2⋅316 = 2⋅2⋅2⋅2LCD = 2⋅2⋅2⋅2⋅3 = 48The LCD of512and316is 48.P2.30 = 5⋅64464 6245565 630⋅=⋅==⋅P3.2.183 63 63305 65 65⋅⋅−= −= −= −⋅⋅3.246 4646641 41 41⋅⋅−= −= −= −= −⋅⋅1.5 Quick Checks1.393 927484 832⋅⋅==⋅2.5565 567157 151 5 7 87 3 51 57 873 583−−⋅⋅=⋅− ⋅⋅⋅=⋅⋅− ⋅⋅⋅=⋅⋅= −

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