Chapter Real Numbers

Transcription

1 Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex: 5,.7, 3 4,, 3... Irrtiol Numbers: rel umbers tht re ot rtiol umbers; otermitig d orepetig decimls. ex:, Rel Numbers: ll umbers tht c be represeted by poit o the umber lie; the set of rtiol umbers d irrtiol umbers. Absolute Vlue: of rel umber,, is the distce betwee d zero o the umber lie. For y rel umber, if d if Progress Check 6 Evlute ech expressio.. 5 c. Additio Multiplictio Commuttive Property b b b b Associtive Property b cb c bc b c Idetity Property Iverse Property Distributive Property b c b c b c c bc Orderig Rel Numbers less th less th or equl to greter th greter th or equl to Addig Rel Numbers Like Sigs: dd the two umbers (bsolute vlues) & keep the commo sig. Ulike Sigs: subtrct the two umbers (bsolute vlues) & keep the sig of the lrger (bsolute vlue) umber. Subtrctig Rel Numbers We defie subtrctio i terms of dditio. If d b re y rel umbers, the b b Multiplictio & Divisio of Rel Numbers Sme Sigs: multiply (or divide) the two umbers d the sig of the product (or quotiet) is positive. Differet Sigs: multiply (or divide) the two umbers d the sig of the product (or quotiet) is egtive.

2 Sig Rule A product of ozero fctors is positive if the umber of egtive fctors is eve. The product is egtive if the umber of egtive fctors is odd. Fctor: is ech umber (or vrible) i product. 5, 5 is fctor d is fctor ex: ex: x, is fctor d x is fctor Powers:,..., is the expoet or power d is clled the bse. fctors Progress Check Multiply (or evlute) c. 6 Multiplictio d Divisio Ivolvig the Number The product of y umber d zero is zero, so Divisio ivolvig. divided by y ozero umber is ex: or. Divisio by is udefied ex: udefied or udefied Reciprocls: whe the product of two umbers is, the the umbers re clled reciprocls of ech other. ex: reciprocl of 3 ex: reciprocl of Defiitio of Divisio If d b re rel umbers with b, the b b Order of Opertios (P E MD AS). Perform ll opertios withi groupig symbols, such s pretheses, first. If there is more th oe symbol of groupig, simplify the iermost symbol of groupig first, d simplify the umertor d deomitor of frctio seprtely.. Evlute powers of umber. 3. Multiply or divide workig from left to right. 4. Add or subtrct workig left to right. Progress Check 3 Evlute 856 Progress Check 4 Evlute o grphig clcultor

3 Chpter. - Algebric Expressios d Geometric Formuls Vrible: is symbol tht my be replced by differet umbers i prticulr problem. ex: y 7, y is the vrible or 9z, z is the vrible Costt: is symbol tht represets the sme umber throughout prticulr problem. ex: y 7, 7 is the costt Algebric Expressio: is expressio tht combies vribles d costts usig the opertios of rithmetic. ex: 4x 3x 8 or gt Progress Check Evlute ech expressio, give tht x, y 3, d z 4. y xz 3 Subscripts: re smll umbers to the bottom-right of vrible which re used to deote vrious vlues of the sme vrible. y y ex: m (slope formul) x x Progress Check Usig the slope formul bove evlute m whe x 3, y 4, x 7, d y. Terms: prts of lgebric expressio seprted by dditio or subtrctio. ex: x 5x 6x 4x 9, there re five terms i this expressio Like Terms: terms tht hve ideticl vrible fctors. ex: x 5 x 6 x 4 x 9 Coefficiets: is the costt (umber) multiplied with the vrible i term. ex: x 5x 6x 4x 9, -, 5, 6, -4 re the coefficiets of the terms respectively Progress Check 4 Simplify by combiig like terms if possible. d. x 6x 6x x Progress Check 5 Remove the symbols of groupig d combie like terms (or simplify). 5x 4x x 3

4 Chpter.3 - Iteger Expoets LAWS of INTEGER EXPONENTS Let d b be y ozero rel umbers, d m d be y itegers. m m m m b b b b m m b b b b b m b m Progress Check Simplify by lws of expoets. 5. 4x 9x 5 4xy xy 4 4 Progress Check 3 Evlute ech expressio.. 5x 5 c. 4.7 Progress Check 4 Evlute ech expressio c. 5 3 Progress Check 6 Simplify d write the result usig oly positive expoets y y 5 4 c. 3 x 5 4 d. 6y e. 3 x y 3 xy Progress Check 8 Simplify the give expressios.. x x 3 3 c. x x 3 4

5 Chpter.4 - Products of Algebric Expressios Objectives: Use the distributive property to multiply lgebric expressios. Use the FOIL method to multiply two expressios tht ech coti two terms. Use specil product formuls to fid certi products. Progress Check Fid ech product. x 3x x 9. Progress Check Divide x x by 4x. Progress Check 3 Fid ech product. 3y 4 7y 3 5. Progress Check 4 Fid ech product. 4x 3x x 3. Specil Products Formuls. b b b. b b b 3. b b b Progress Check 7 Multiply usig the FOIL method d specil product formuls. x h c. 3y53y 5 Progress Check 9 Simplify x h x h 5

6 Chpter.5 Lier Equtios d Literl Equtios Progress Check Solve the equtio. 53y 8y Progress Check 3 Solve the equtio. x5 x6 6 9 Progress Check 4 Solve the equtio. x 3 x x Progress Check 5 Fid the vlue of r i the formul A P rt if A $6,4, P $4, d t 7yers. Progress Check 7 Solve S t gt for g. Progress Check 9 Solve 5xy 3 for the idicted letter.. For x For y Choose umber. Add three. Multiply by two. Add six. Divide by two. Subtrct your origil umber. Your result is six. 6

7 Chpter.6 Applictios of Lier Equtios To Solve Word Problem. Red the problem severl times.. Let vrible represet ukow qutity. 3. Set up equtio. 4. Solve the equtio. 5. Aswer the questio. 6. Check the swer. Progress Check The totl cost (icludig tx) of ew cr is $4,56. If the sles tx rte is 8 percet, how much is pid i txes? Geometric Problems For geometric figure problems use perimeter, re, d volume formuls from sectio. For trigle problems remember the sum of the gle mesures i trigle is 8 d right trigle cotis 9 gle For gle problems remember tht y two gles whose mesures dd up to 9 re clled complemetry d y two gles whose mesures dd up to 8 re clled supplemetry. Progress Check 3 I right trigle the mesure of oe of the cute gles is 36 greter th the other. Wht is the mesures if the lrger cute gle? Aul Iterest Problems Use I P r where P represets pricipl (mout ivested), r represets the ul iterest rte ( percetge which must be chged to deciml form), d I represets the mout of iterest ered i oe yer. Progress Check 4 How should $, ivestmet be split so tht the totl ul erigs re $8,, if oe portio is ivested t percet ul iterest d the rest t 6 percet? Proportio Problems A rtio is compriso of two qutities by divisio, d proportio is sttemet tht two rtios re equl. Progress Check 7 If idlig cr uses 35 oz. of gsolie i 5 miutes, how log to the erest miute must it idle to use gl of gs? Oe gllo is 8 oz. 7

8 Chpter.7 Lier Iequlities i Oe Vrible b is equivlet to b 7 is equivlet to 7 Progress Check Solve 3x 7 5. Express the solutio set grphiclly, i set ottio, d i itervl ottio. Progress Check 3 Solve 5x x. Express the solutio set grphiclly, i set ottio, d i itervl ottio. Progress Check 4 Solve 6x x. Express the solutio set grphiclly, i set ottio, d i itervl ottio. Progress Check 5 Solve xx 3. 8

9 Chpter.8 Compoud Iequlities Solutio of Ad Iequlities To solve compoud iequlity ivolvig d:. Solve seprtely ech iequlity i the compoud iequlity.. Fid the itersectio (oly wht belogs to both, or the overlp) of the solutio sets of the seprte iequlities. Progress Check 3 Solve x 4 d x 3x. Express the solutio set grphiclly d i itervl ottio. Solutio of Compct Form Iequlities To solve compoud iequlity i compct form the gol is to isolte the vrible i the middle member of the iequlity. Progress Check 4 Solve 8 6 3t 8. Express the solutio set grphiclly d i itervl ottio. Progress Check 5 I philosophy clss verge from 74.5 up to but ot icludig 79.5 results i grde of C+. If your first three grdes re 68, 83, d 75, fid ll possible grdes o the fourth exm tht would result i grde of C+. Solutio of Or Iequlities To solve compoud iequlity ivolvig or:. Solve seprtely ech iequlity i the compoud iequlity.. Fid the uio (everythig tht belogs to either or both) of the solutio sets of the seprte iequlities. Progress Check 6 Solve x 3 or x 3. Express the solutio set grphiclly d i itervl ottio. 9