一元二次不等式.ppt

LinearInequalities 2002byShawnaHaider Therearetwokindsofnotationforgraphsofinequalities opencircleorfilledincirclenotationandintervalnotationbrackets Youshouldbefamiliarwithboth Bothofthesenumberlinesshowtheinequalityabove Theyarejustusingtwodifferentnotations Becausetheinequalityis greaterthanorequalto thesolutioncanequaltheendpoint Thatiswhythecircleisfilledin Withintervalnotationbrackets asquarebracketmeansitcanequaltheendpoint circlefilledin squaredendbracket Remember thesemeanthesamething justtwodifferentnotations Let slookatthetwodifferentnotationswithadifferentinequalitysign Sincethissays lessthan wemakethearrowgotheotherway Sinceitdoesn tsay orequalto thesolutioncannotequaltheendpoint Thatiswhythecircleisnotfilledin Withintervalnotationbrackets aroundedbracketmeansitcannotequaltheendpoint circlenotfilledin roundedendbracket Remember thesemeanthesamething justtwodifferentnotations CompoundInequalitiesLet sconsidera doubleinequality havingtwoinequalitysigns Ithinkoftheseasthe inbetweeners xisinbetweenthetwonumbers Thisisan and inequalitywhichmeansbothpartsmustbetrue Itsaysthatxisgreaterthan 2andxislessthanorequalto3 CompoundInequalitiesNowlet slookatanotherformofa doubleinequality havingtwoinequalitysigns Insteadof and theseare or problems Onepartortheotherpartmustbetrue butnotnecessarilyboth Eitherxislessthan 2orxisgreaterthanorequalto3 Inthiscasebothpartscannotbetrueatthesametimesinceanumbercan tbelessthan 2andalsogreaterthan3 Justlikegraphicallytherearetwodifferentnotations whenyouwriteyouranswersyoucanuseinequalitynotationorintervalnotation Againyoushouldbefamiliarwithboth 0 4 6 8 2 Let shavealookattheintervalnotation Forintervalnotationyoulistthesmallestxcanbe acomma andthenthelargestxcanbesosolutionsareanythingthatfallsbetweenthesmallestandlargest unbounded Thebracketbeforethe 1issquarebecausethisisgreaterthan orequalto solutioncanequaltheendpoint Thebracketaftertheinfinitysignisroundedbecausetheintervalgoesonforever unbounded andsinceinfinityisnotanumber itdoesn tequaltheendpoint thereisnoendpoint Let stryanotherone Thebracketsusedintheintervalnotationabovearethesameonesusedwhenyougraphthis Thismeanseverythingbetween 2and4butnotincluding 2 Let slookatanotherone Noticehowthebracketnotationforgraphingcorrespondstothebracketsinintervalnotation Rememberthatsquareis orequalto androundisuptobutnotequal Bytheinfinitysignitwillalwaysberoundbecauseitcan tequalinfinity thatisnotanumber Nowlet slookan or compoundinequality Therearetwointervalstolistwhenyoulistinintervalnotation Whenthesolutionconsistsofmorethanoneinterval wejointhemwithaunionsign PropertiesofInequalities Essentially allofthepropertiesthatyoulearnedtosolvelinearequationsapplytosolvinglinearinequalitieswiththeexceptionthatifyoumultiplyordividebyanegativeyoumustreversetheinequalitysign Sotosolveaninequalityjustdothesamestepsaswithanequalitytogetthevariablealonebutifintheprocessyoumultiplyor