18:@0.036727:0.974518:0.057230:0.974518:0.057230:0.956333:0.036727:0.956333:0.000000:0.000000 Números reales y funciones:@0.073089:0.043661:0.318030:0.043661:0.318030:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Recordemos algunas de las propiedades de las desigualdades::@0.073089:0.136836:0.523873:0.136836:0.523873:0.120402:0.073089:0.120402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Teorema:@0.464764:0.180942:0.535251:0.180942:0.535251:0.163225:0.464764:0.163225:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Sean:@0.082576:0.214339:0.118131:0.214339:0.118131:0.197905:0.082576:0.197905:0.000000:0.000000:0.000000:0.000000 a, b, c :@0.118028:0.214339:0.181717:0.214339:0.181717:0.197933:0.118028:0.197933:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [:@0.163163:0.214201:0.178496:0.214201:0.178496:0.200152:0.163163:0.200152:0.000000 R:@0.181643:0.213479:0.194933:0.213479:0.194933:0.200582:0.181643:0.200582:0.000000 . Se verifican las siguientes propiedades::@0.194841:0.214339:0.483469:0.214339:0.483469:0.197905:0.194841:0.197905:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a. Si a b y c 0, entonces ac bc.:@0.082576:0.245275:0.365722:0.245275:0.365722:0.228842:0.082576:0.228842:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.141180:0.244721:0.156513:0.244721:0.156513:0.231005:0.141180:0.231005:0.000000 .:@0.198445:0.244721:0.213778:0.244721:0.213778:0.231005:0.198445:0.231005:0.000000 ,:@0.325307:0.244721:0.340641:0.244721:0.340641:0.231005:0.325307:0.231005:0.000000 b. Si a b y c 0, entonces ac bc.:@0.082576:0.276213:0.365722:0.276213:0.365722:0.259779:0.082576:0.259779:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.141180:0.275658:0.156513:0.275658:0.156513:0.261942:0.141180:0.261942:0.000000 ,:@0.198445:0.275658:0.213778:0.275658:0.213778:0.261942:0.198445:0.261942:0.000000 .:@0.325307:0.275658:0.340641:0.275658:0.340641:0.261942:0.325307:0.261942:0.000000 c. Si a 0 y b 0, entonces ab 0.:@0.082576:0.307149:0.359342:0.307149:0.359342:0.290715:0.082576:0.290715:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.141180:0.306594:0.156513:0.306594:0.156513:0.292879:0.141180:0.292879:0.000000 .:@0.199218:0.306594:0.214551:0.306594:0.214551:0.292879:0.199218:0.292879:0.000000 ,:@0.328032:0.306594:0.343365:0.306594:0.343365:0.292879:0.328032:0.292879:0.000000 d. Si a 0 y b 0, entonces ab 0.:@0.082576:0.338086:0.359342:0.338086:0.359342:0.321653:0.082576:0.321653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.141180:0.337532:0.156513:0.337532:0.156513:0.323816:0.141180:0.323816:0.000000 ,:@0.199218:0.337532:0.214551:0.337532:0.214551:0.323816:0.199218:0.323816:0.000000 .:@0.328032:0.337532:0.343365:0.337532:0.343365:0.323816:0.328032:0.323816:0.000000 e. Si a 0, entonces a 0.:@0.082576:0.369024:0.296421:0.369024:0.296421:0.352590:0.082576:0.352590:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Þ:@0.141180:0.368469:0.156513:0.368469:0.156513:0.354753:0.141180:0.354753:0.000000 2:@0.255964:0.362960:0.261201:0.362960:0.261201:0.353379:0.255964:0.353379:0.000000 .:@0.265110:0.368469:0.280444:0.368469:0.280444:0.354753:0.265110:0.354753:0.000000 Sean :@0.509485:0.245278:0.548998:0.245278:0.548998:0.228844:0.509485:0.228844:0.000000:0.000000:0.000000:0.000000:0.000000 a, b:@0.548895:0.245278:0.573837:0.245278:0.573837:0.228872:0.548895:0.228872:0.000000:0.000000:0.000000:0.000000 :@0.573800:0.245278:0.577831:0.245278:0.577831:0.228844:0.573800:0.228844:0.000000 [:@0.577758:0.245139:0.593091:0.245139:0.593091:0.231091:0.577758:0.231091:0.000000 :@0.592999:0.245278:0.596312:0.245278:0.596312:0.228872:0.592999:0.228872:0.000000 R:@0.596239:0.244418:0.609529:0.244418:0.609529:0.231521:0.596239:0.231521:0.000000 . :@0.609437:0.245278:0.616671:0.245278:0.616671:0.228844:0.609437:0.228844:0.000000:0.000000 a. :@0.509487:0.276213:0.524931:0.276213:0.524931:0.259779:0.509487:0.259779:0.000000:0.000000:0.000000 a :@0.539688:0.276213:0.552187:0.276213:0.552187:0.259807:0.539688:0.259807:0.000000:0.000000 .:@0.552131:0.275658:0.567465:0.275658:0.567465:0.261942:0.552131:0.261942:0.000000 0 :@0.567373:0.276213:0.604276:0.276213:0.604276:0.259779:0.567373:0.259779:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.604202:0.276213:0.613461:0.276213:0.613461:0.259807:0.604202:0.259807:0.000000 –1:@0.613387:0.270149:0.623947:0.270149:0.623947:0.260568:0.613387:0.260568:0.000000:0.000000 0.:@0.623896:0.276213:0.659165:0.276213:0.659165:0.259779:0.623896:0.259779:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.627854:0.275658:0.643187:0.275658:0.643187:0.261942:0.627854:0.261942:0.000000 b. :@0.509487:0.307149:0.526385:0.307149:0.526385:0.290715:0.509487:0.290715:0.000000:0.000000:0.000000 a :@0.539688:0.307149:0.552187:0.307149:0.552187:0.290743:0.539688:0.290743:0.000000:0.000000 ,:@0.552131:0.306594:0.567465:0.306594:0.567465:0.292879:0.552131:0.292879:0.000000 0 :@0.567373:0.307149:0.604276:0.307149:0.604276:0.290715:0.567373:0.290715:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.604202:0.307149:0.613461:0.307149:0.613461:0.290743:0.604202:0.290743:0.000000 –1:@0.613387:0.301085:0.623947:0.301085:0.623947:0.291504:0.613387:0.291504:0.000000:0.000000 0.:@0.623896:0.307149:0.659165:0.307149:0.659165:0.290715:0.623896:0.290715:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.627854:0.306594:0.643187:0.306594:0.643187:0.292879:0.627854:0.292879:0.000000 c. 0 :@0.509487:0.338086:0.571809:0.338086:0.571809:0.321653:0.509487:0.321653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.552536:0.337532:0.567870:0.337532:0.567870:0.323816:0.552536:0.323816:0.000000 a < b:@0.571735:0.338086:0.607556:0.338086:0.607556:0.321680:0.571735:0.321680:0.000000:0.000000:0.000000:0.000000:0.000000 0 :@0.607519:0.338086:0.663617:0.338086:0.663617:0.321653:0.607519:0.321653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ,:@0.644345:0.337532:0.659678:0.337532:0.659678:0.323816:0.644345:0.323816:0.000000 b:@0.663544:0.338086:0.672803:0.338086:0.672803:0.321680:0.663544:0.321680:0.000000 –1:@0.672727:0.332022:0.683287:0.332022:0.683287:0.322442:0.672727:0.322442:0.000000:0.000000 :@0.683235:0.338086:0.706465:0.338086:0.706465:0.321653:0.683235:0.321653:0.000000:0.000000:0.000000 ,:@0.687192:0.337532:0.702525:0.337532:0.702525:0.323816:0.687192:0.323816:0.000000 a:@0.706391:0.338086:0.715650:0.338086:0.715650:0.321680:0.706391:0.321680:0.000000 –1:@0.715569:0.332022:0.726129:0.332022:0.726129:0.322442:0.715569:0.322442:0.000000:0.000000 .:@0.726078:0.338086:0.729281:0.338086:0.729281:0.321653:0.726078:0.321653:0.000000 Finalmente, revisemos el concepto de valor absoluto y las propiedades.:@0.073081:0.411116:0.588311:0.411116:0.588311:0.394683:0.073081:0.394683:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Valor absoluto:@0.073081:0.443845:0.182339:0.443845:0.182339:0.427176:0.073081:0.427176:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 El valor absoluto de un número real es su magnitud o sea la distancia sobre la recta numérica entre el punto que :@0.073081:0.476574:0.931150:0.476574:0.931150:0.460140:0.073081:0.460140:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 representa y el origen.:@0.073081:0.493216:0.233848:0.493216:0.233848:0.476782:0.073081:0.476782:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Definición:@0.073081:0.525945:0.151049:0.525945:0.151049:0.509275:0.073081:0.509275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si es un número real entonces, el valor absoluto de es :@0.073081:0.563389:0.527832:0.563389:0.527832:0.546955:0.073081:0.546955:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.089261:0.563389:0.096716:0.563389:0.096716:0.546983:0.089261:0.546983:0.000000 x:@0.456265:0.563389:0.463720:0.563389:0.463720:0.546983:0.456265:0.546983:0.000000 | | 5:@0.487171:0.562834:0.523893:0.562834:0.523893:0.549119:0.487171:0.549119:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.492196:0.563389:0.499651:0.563389:0.499651:0.546983:0.492196:0.546983:0.000000 x :@0.534962:0.555323:0.562297:0.555323:0.562297:0.538917:0.534962:0.538917:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 si :@0.562297:0.555638:0.578382:0.555638:0.578382:0.537710:0.562297:0.537710:0.000000:0.000000:0.000000 x :@0.578382:0.555638:0.590129:0.555638:0.590129:0.537741:0.578382:0.537741:0.000000:0.000000 $ :@0.590129:0.555033:0.611876:0.555033:0.611876:0.540070:0.590129:0.540070:0.000000:0.000000 0:@0.611876:0.555638:0.621676:0.555638:0.621676:0.537710:0.611876:0.537710:0.000000 2:@0.534968:0.573188:0.551695:0.573188:0.551695:0.558225:0.534968:0.558225:0.000000 x :@0.551690:0.573478:0.562458:0.573478:0.562458:0.557072:0.551690:0.557072:0.000000:0.000000 si :@0.562458:0.573793:0.578542:0.573793:0.578542:0.555865:0.562458:0.555865:0.000000:0.000000:0.000000 x :@0.578542:0.573793:0.590290:0.573793:0.590290:0.555895:0.578542:0.555895:0.000000:0.000000 , :@0.590290:0.573188:0.612037:0.573188:0.612037:0.558225:0.590290:0.558225:0.000000:0.000000 0:@0.612037:0.573793:0.621836:0.573793:0.621836:0.555865:0.612037:0.555865:0.000000 5:@0.527827:0.571215:0.532285:0.571215:0.532285:0.540055:0.527827:0.540055:0.000000 :@0.621072:0.563387:0.625103:0.563387:0.625103:0.546953:0.621072:0.546953:0.000000 |2 | 5 :@0.104106:0.594697:0.168665:0.594697:0.168665:0.582228:0.104106:0.582228:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.122697:0.595202:0.138193:0.595202:0.138193:0.580917:0.122697:0.580917:0.000000:0.000000:0.000000 7:@0.156316:0.595202:0.164482:0.595202:0.164482:0.580917:0.156316:0.580917:0.000000 | | 5 :@0.442047:0.595706:0.493654:0.595706:0.493654:0.583237:0.442047:0.583237:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2 :@0.446699:0.596210:0.463181:0.596210:0.463181:0.581926:0.446699:0.581926:0.000000:0.000000:0.000000 2:@0.481304:0.596210:0.489470:0.596210:0.489470:0.581926:0.481304:0.581926:0.000000 2:@0.081120:0.624257:0.093665:0.624257:0.093665:0.613035:0.081120:0.613035:0.000000 8:@0.093665:0.624711:0.101015:0.624711:0.101015:0.611855:0.093665:0.611855:0.000000 2:@0.118741:0.624257:0.131286:0.624257:0.131286:0.613035:0.118741:0.613035:0.000000 7:@0.131286:0.624711:0.138636:0.624711:0.138636:0.611855:0.131286:0.611855:0.000000 2:@0.156362:0.624257:0.168907:0.624257:0.168907:0.613035:0.156362:0.613035:0.000000 6:@0.168907:0.624711:0.176257:0.624711:0.176257:0.611855:0.168907:0.611855:0.000000 2:@0.193080:0.624257:0.205625:0.624257:0.205625:0.613035:0.193080:0.613035:0.000000 5:@0.205625:0.624711:0.212974:0.624711:0.212974:0.611855:0.205625:0.611855:0.000000 2:@0.230701:0.624257:0.243246:0.624257:0.243246:0.613035:0.230701:0.613035:0.000000 4:@0.243246:0.624711:0.250596:0.624711:0.250596:0.611855:0.243246:0.611855:0.000000 2:@0.268322:0.624257:0.280867:0.624257:0.280867:0.613035:0.268322:0.613035:0.000000 3:@0.280867:0.624711:0.288217:0.624711:0.288217:0.611855:0.280867:0.611855:0.000000 2:@0.304813:0.624257:0.317359:0.624257:0.317359:0.613035:0.304813:0.613035:0.000000 2:@0.317359:0.624711:0.324708:0.624711:0.324708:0.611855:0.317359:0.611855:0.000000 2:@0.342434:0.624257:0.354980:0.624257:0.354980:0.613035:0.342434:0.613035:0.000000 1:@0.354980:0.624711:0.362329:0.624711:0.362329:0.611855:0.354980:0.611855:0.000000 0:@0.388610:0.624711:0.395959:0.624711:0.395959:0.611855:0.388610:0.611855:0.000000 1:@0.425945:0.624711:0.433294:0.624711:0.433294:0.611855:0.425945:0.611855:0.000000 2:@0.463280:0.624711:0.470629:0.624711:0.470629:0.611855:0.463280:0.611855:0.000000 3:@0.500615:0.624711:0.507964:0.624711:0.507964:0.611855:0.500615:0.611855:0.000000 4:@0.537949:0.624711:0.545299:0.624711:0.545299:0.611855:0.537949:0.611855:0.000000 5:@0.575284:0.624711:0.582634:0.624711:0.582634:0.611855:0.575284:0.611855:0.000000 6:@0.612619:0.624711:0.619969:0.624711:0.619969:0.611855:0.612619:0.611855:0.000000 7:@0.649954:0.624711:0.657304:0.624711:0.657304:0.611855:0.649954:0.611855:0.000000 8:@0.687289:0.624711:0.694639:0.624711:0.694639:0.611855:0.687289:0.611855:0.000000 Propiedad de valor absoluto:@0.073089:0.662830:0.286981:0.662830:0.286981:0.646161:0.073089:0.646161:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Sea 0:@0.073089:0.695554:0.202363:0.695554:0.202363:0.678885:0.073089:0.678885:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.103958:0.695554:0.112075:0.695554:0.112075:0.678871:0.103958:0.678871:0.000000 [:@0.115830:0.695416:0.131164:0.695416:0.131164:0.681367:0.115830:0.681367:0.000000 R:@0.134900:0.694694:0.148190:0.694694:0.148190:0.681797:0.134900:0.681797:0.000000 | | $:@0.151927:0.695000:0.189183:0.695000:0.189183:0.681284:0.151927:0.681284:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.156952:0.695554:0.165070:0.695554:0.165070:0.678871:0.156952:0.678871:0.000000 | | 5 |2 |:@0.073089:0.727728:0.147546:0.727728:0.147546:0.714013:0.073089:0.714013:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.078114:0.728283:0.085569:0.728283:0.085569:0.711877:0.078114:0.711877:0.000000 x:@0.135048:0.728283:0.142503:0.728283:0.142503:0.711877:0.135048:0.711877:0.000000 | | 5 | || |:@0.073070:0.760457:0.160394:0.760457:0.160394:0.746742:0.073070:0.746742:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.078096:0.761012:0.096190:0.761012:0.096190:0.744606:0.078096:0.744606:0.000000:0.000000:0.000000 x y:@0.130446:0.761012:0.155351:0.761012:0.155351:0.744606:0.130446:0.744606:0.000000:0.000000:0.000000 5 :@0.073844:0.806030:0.122109:0.806030:0.122109:0.792314:0.073844:0.792314:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.081741:0.793832:0.089196:0.793832:0.089196:0.777426:0.081741:0.777426:0.000000 y:@0.081723:0.811375:0.089196:0.811375:0.089196:0.794969:0.081723:0.794969:0.000000 | |:@0.125584:0.794472:0.143108:0.794472:0.143108:0.780757:0.125584:0.780757:0.000000:0.000000:0.000000 x:@0.130609:0.795027:0.138064:0.795027:0.138064:0.778621:0.130609:0.778621:0.000000 | |:@0.125566:0.812016:0.143108:0.812016:0.143108:0.798300:0.125566:0.798300:0.000000:0.000000:0.000000 y:@0.130591:0.812570:0.138064:0.812570:0.138064:0.796164:0.130591:0.796164:0.000000 y :@0.146568:0.806579:0.162583:0.806579:0.162583:0.790145:0.146568:0.790145:0.000000:0.000000:0.000000 Þ :@0.162491:0.806025:0.182334:0.806025:0.182334:0.792309:0.162491:0.792309:0.000000:0.000000 0:@0.182242:0.806579:0.191224:0.806579:0.191224:0.790145:0.182242:0.790145:0.000000 | | 5 | | :@0.659137:0.680684:0.722881:0.680684:0.722881:0.666968:0.659137:0.666968:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.664162:0.681239:0.671617:0.681239:0.671617:0.664832:0.664162:0.664832:0.000000 y:@0.705854:0.681239:0.713328:0.681239:0.713328:0.664832:0.705854:0.664832:0.000000 ↔ :@0.722789:0.680822:0.746479:0.680822:0.746479:0.663834:0.722789:0.663834:0.000000:0.000000 ( :@0.746387:0.681239:0.762604:0.681239:0.762604:0.664805:0.746387:0.664805:0.000000:0.000000:0.000000 x:@0.751191:0.681239:0.758646:0.681239:0.758646:0.664832:0.751191:0.664832:0.000000 5 :@0.762530:0.680684:0.782373:0.680684:0.782373:0.666968:0.762530:0.666968:0.000000:0.000000 y x y:@0.782281:0.681239:0.872845:0.681239:0.872845:0.664832:0.782281:0.664832:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ):@0.789681:0.681239:0.794559:0.681239:0.794559:0.664805:0.789681:0.664805:0.000000 ~:@0.797725:0.681821:0.813058:0.681821:0.813058:0.664167:0.797725:0.664167:0.000000 (:@0.816206:0.681239:0.821084:0.681239:0.821084:0.664805:0.816206:0.664805:0.000000 5 2:@0.831650:0.680684:0.865464:0.680684:0.865464:0.666968:0.831650:0.666968:0.000000:0.000000:0.000000 ):@0.872771:0.681239:0.877649:0.681239:0.877649:0.664805:0.872771:0.664805:0.000000 | | 5 :@0.659118:0.713413:0.711487:0.713413:0.711487:0.699697:0.659118:0.699697:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.664070:0.713967:0.671525:0.713967:0.671525:0.697561:0.664070:0.697561:0.000000 y:@0.699522:0.713967:0.706996:0.713967:0.706996:0.697561:0.699522:0.697561:0.000000 ↔ :@0.710861:0.713551:0.734478:0.713551:0.734478:0.696563:0.710861:0.696563:0.000000:0.000000 ( :@0.733870:0.713967:0.749995:0.713967:0.749995:0.697534:0.733870:0.697534:0.000000:0.000000:0.000000 y:@0.738601:0.713967:0.746074:0.713967:0.746074:0.697561:0.738601:0.697561:0.000000 $ :@0.749461:0.713413:0.769231:0.713413:0.769231:0.699697:0.749461:0.699697:0.000000:0.000000 0:@0.768623:0.713967:0.777606:0.713967:0.777606:0.697534:0.768623:0.697534:0.000000 x y :@0.777459:0.713967:0.844903:0.713967:0.844903:0.697561:0.777459:0.697561:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 `:@0.780330:0.714550:0.795663:0.714550:0.795663:0.696896:0.780330:0.696896:0.000000 (:@0.798369:0.713967:0.803247:0.713967:0.803247:0.697534:0.798369:0.697534:0.000000 5:@0.813316:0.713413:0.828649:0.713413:0.828649:0.699697:0.813316:0.699697:0.000000 ~:@0.844443:0.714550:0.859776:0.714550:0.859776:0.696896:0.844443:0.696896:0.000000 52:@0.859610:0.713413:0.910028:0.713413:0.910028:0.699697:0.859610:0.699697:0.000000:0.000000:0.000000:0.000000 x y:@0.863586:0.713967:0.917335:0.713967:0.917335:0.697561:0.863586:0.697561:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 )):@0.917225:0.713967:0.926843:0.713967:0.926843:0.697534:0.917225:0.697534:0.000000:0.000000 | | . :@0.659082:0.746142:0.712776:0.746142:0.712776:0.732426:0.659082:0.732426:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.664107:0.746696:0.671562:0.746696:0.671562:0.730290:0.664107:0.730290:0.000000 y:@0.700774:0.746696:0.708247:0.746696:0.708247:0.730290:0.700774:0.730290:0.000000 ↔ :@0.712684:0.746280:0.736374:0.746280:0.736374:0.729292:0.712684:0.729292:0.000000:0.000000 ( :@0.736282:0.746696:0.752498:0.746696:0.752498:0.730263:0.736282:0.730263:0.000000:0.000000:0.000000 x:@0.741086:0.746696:0.748541:0.746696:0.748541:0.730290:0.741086:0.730290:0.000000 . :@0.752425:0.746142:0.772268:0.746142:0.772268:0.732426:0.752425:0.732426:0.000000:0.000000 y x y:@0.772176:0.746696:0.853149:0.746696:0.853149:0.730290:0.772176:0.730290:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ~:@0.782834:0.747279:0.798167:0.747279:0.798167:0.729625:0.782834:0.729625:0.000000 , 2:@0.811954:0.746142:0.845768:0.746142:0.845768:0.732426:0.811954:0.732426:0.000000:0.000000:0.000000 ):@0.853076:0.746696:0.857954:0.746696:0.857954:0.730263:0.853076:0.730263:0.000000 | | , :@0.659063:0.778871:0.712757:0.778871:0.712757:0.765155:0.659063:0.765155:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.664088:0.779425:0.671543:0.779425:0.671543:0.763019:0.664088:0.763019:0.000000 y:@0.700756:0.779425:0.708229:0.779425:0.708229:0.763019:0.700756:0.763019:0.000000 ↔ :@0.712665:0.779009:0.736355:0.779009:0.736355:0.762021:0.712665:0.762021:0.000000:0.000000 $ 2 , ,:@0.736263:0.778871:0.833141:0.778871:0.833141:0.765155:0.736263:0.765155:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.756014:0.779425:0.760892:0.779425:0.760892:0.762992:0.756014:0.762992:0.000000 y:@0.776060:0.779425:0.783533:0.779425:0.783533:0.763019:0.776060:0.763019:0.000000 :@0.783459:0.779425:0.787491:0.779425:0.787491:0.762992:0.783459:0.762992:0.000000 x y:@0.807168:0.779425:0.843762:0.779425:0.843762:0.763019:0.807168:0.763019:0.000000:0.000000:0.000000:0.000000:0.000000 ):@0.843706:0.779425:0.848584:0.779425:0.848584:0.762992:0.843706:0.762992:0.000000 | | 5 :@0.659063:0.811600:0.706128:0.811581:0.706128:0.797865:0.659063:0.797884:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.664088:0.812154:0.671543:0.812154:0.671543:0.795748:0.664088:0.795748:0.000000 2:@0.676586:0.806071:0.681823:0.806071:0.681823:0.796491:0.676586:0.796491:0.000000 x :@0.706036:0.812135:0.716731:0.812135:0.716731:0.795729:0.706036:0.795729:0.000000:0.000000 2:@0.716672:0.806071:0.721909:0.806071:0.721909:0.796491:0.716672:0.796491:0.000000 1.2 Ecuaciones e inecuaciones con o sin valor absoluto en expresiones racionales:@0.073089:0.853377:0.704349:0.853377:0.704349:0.836430:0.073089:0.836430:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejercicios resueltos al aplicar las propiedades de valor absoluto y de las desigualdades: :@0.073089:0.886106:0.698442:0.886106:0.698442:0.869672:0.073089:0.869672:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1. Resolver la desigualdad :@0.073089:0.925278:0.285844:0.925278:0.285844:0.908608:0.073089:0.908608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.297825:0.935007:0.306849:0.935007:0.306849:0.916461:0.297825:0.916461:0.000000  1:@0.306849:0.935007:0.345091:0.935007:0.345091:0.916477:0.306849:0.916477:0.000000:0.000000:0.000000:0.000000 2:@0.317550:0.934390:0.334595:0.934390:0.334595:0.919144:0.317550:0.919144:0.000000 4  :@0.290111:0.917864:0.320333:0.917864:0.320333:0.899334:0.290111:0.899334:0.000000:0.000000:0.000000:0.000000 x:@0.300608:0.917864:0.309631:0.917864:0.309631:0.899319:0.300608:0.899319:0.000000 1 :@0.320327:0.917241:0.342487:0.917241:0.342487:0.901995:0.320327:0.901995:0.000000:0.000000 4:@0.342488:0.917858:0.352985:0.917858:0.352985:0.899328:0.342488:0.899328:0.000000 :@0.355783:0.925278:0.359685:0.925278:0.359685:0.908608:0.355783:0.908608:0.000000 # :@0.359608:0.924723:0.379451:0.924723:0.379451:0.911007:0.359608:0.911007:0.000000:0.000000 5.:@0.379359:0.925278:0.392539:0.925278:0.392539:0.908608:0.379359:0.908608:0.000000:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000