{"id":20951,"date":"2021-04-16T12:32:41","date_gmt":"2021-04-16T15:32:41","guid":{"rendered":"https:\/\/beduka.com\/blog\/?p=20951"},"modified":"2021-11-09T13:36:43","modified_gmt":"2021-11-09T16:36:43","slug":"exercicios-de-equacao-modular","status":"publish","type":"post","link":"https:\/\/beduka.com\/blog\/exercicios\/exercicios-de-equacao-modular\/","title":{"rendered":"Os 6 Melhores Exerc\u00edcios de Equa\u00e7\u00e3o Modular com Gabarito"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\"><p><strong>Equa\u00e7\u00f5es modulares s\u00e3o equa\u00e7\u00f5es que possuem termos em m\u00f3dulo, seja no primeiro termo ou no segundo. Resolver problemas envolvendo m\u00f3dulo exige que usemos a defini\u00e7\u00e3o de m\u00f3dulo, o que implica em dividir a equa\u00e7\u00e3o em dois poss\u00edveis casos. Entenda ambos os casos lendo esse artigo e fazendo os exerc\u00edcios sobre equa\u00e7\u00e3o modular.&nbsp;<\/strong><\/p><\/blockquote>\n\n\n\n<p><strong>Quer seguir diretamente para alguma parte espec\u00edfica?<\/strong> Clique em algum dos t\u00f3picos abaixo:<\/p>\n\n\n\n<ul><li><a href=\"#m\u00f3dulo\">O que \u00e9 um m\u00f3dulo?<\/a><\/li><li><a href=\"#equa\u00e7\u00e3o-modular\">Como resolver uma equa\u00e7\u00e3o modular?<\/a><\/li><li><a href=\"#exerc\u00edcios-sobre-equa\u00e7\u00e3o-modular\">Exerc\u00edcios de Equa\u00e7\u00e3o Modular;<\/a><\/li><li><a href=\"#gabarito-dos-exerc\u00edcios-de-equa\u00e7\u00e3o-modular\">Gabarito dos exerc\u00edcios de Equa\u00e7\u00e3o Modular.<\/a><\/li><\/ul>\n\n\n\n<p>Quando voc\u00ea terminar as quest\u00f5es de equa\u00e7\u00e3o modular<strong>,<\/strong> coloque em pr\u00e1tica todo seu conhecimento com <a href=\"https:\/\/beduka.com\/simulado-enem\/#\/\" target=\"_blank\" rel=\"noreferrer noopener\">O Melhor Simulado Enem do Brasil.<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"m\u00f3dulo\">O que \u00e9 um m\u00f3dulo?<\/h2>\n\n\n\n<p><strong>Um m\u00f3dulo \u00e9 a dist\u00e2ncia de um n\u00famero at\u00e9 o zero<\/strong> e para represent\u00e1-lo usamos a simbologia <strong>|<\/strong><strong><em>n<\/em><\/strong><strong>|<\/strong>. Como \u00e9 uma dist\u00e2ncia, o valor do m\u00f3dulo sempre ser\u00e1 positivo e para calcul\u00e1-lo dividimos em <strong>dois casos poss\u00edveis<\/strong>:&nbsp;<\/p>\n\n\n\n<p><strong>1- |<\/strong><strong><em>n<\/em><\/strong><strong>| =<\/strong> n \u2192 n &gt; 0<\/p>\n\n\n\n<p><strong>2- |<\/strong><strong><em>n<\/em><\/strong><strong>| =<\/strong> -n \u2192 n &lt; 0<\/p>\n\n\n\n<p>Assim, pode-se dizer que no 1\u00ba caso, <strong>|<\/strong><strong><em>n<\/em><\/strong><strong>| <\/strong>\u00e9 igual a<strong> n <\/strong>se este for igual ou maior que zero e, no 2\u00ba caso, <strong>|<\/strong><strong><em>n<\/em><\/strong><strong>| <\/strong>\u00e9 igual ao oposto de n se este for menor que zero.&nbsp;<\/p>\n\n\n\n<p><strong>Exemplos:<\/strong><\/p>\n\n\n\n<p><strong>a) |3| = 3<\/strong><\/p>\n\n\n\n<p><strong>b) |-3| = -(-3) = 3<\/strong><\/p>\n\n\n\n<p><strong>c) |X + 3| = X + 3, se X + 3 <\/strong><strong>&gt;<\/strong><strong> 0<\/strong><\/p>\n\n\n\n<p><strong>OU<\/strong><\/p>\n\n\n\n<p><strong>|X + 3| = -(X + 3), se X + 3 &lt; 0<\/strong><\/p>\n\n\n\n<ul><li><strong>S\u00e3o mais de 200 resumos gratuitos para o ENEM<\/strong> no <a href=\"https:\/\/www.instagram.com\/beduka_oficial\/\" target=\"_blank\" rel=\"noreferrer noopener\">Instagram do Beduka<\/a>, aproveite!<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"equa\u00e7\u00e3o-modular\">Como resolver uma equa\u00e7\u00e3o modular?<\/h2>\n\n\n\n<p>Para resolver uma equa\u00e7\u00e3o modular \u00e9 obrigat\u00f3rio dividi-la nos dois casos e testar ambas as possibilidades. Al\u00e9m disso, voc\u00ea tamb\u00e9m precisa saber resolver <a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/fatoracao\/\" target=\"_blank\" rel=\"noreferrer noopener\">equa\u00e7\u00f5es polinomiais<\/a>.&nbsp;<\/p>\n\n\n\n<p><strong>Exemplos:&nbsp;&nbsp;&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>1- <\/strong><strong>|x \u2013 4| = 6<\/strong><\/p>\n\n\n\n<p>Para solucionar essa equa\u00e7\u00e3o temos duas possibilidades: <strong>X &#8211; 4 = 6 ou X &#8211; 4 = -6<\/strong>. Sendo assim, precisamos resolver as duas.&nbsp;<\/p>\n\n\n\n<p><strong>1\u00aa possibilidade:<\/strong><\/p>\n\n\n\n<p><strong>X &#8211; 4 = 6<\/strong><\/p>\n\n\n\n<p><strong>X = 6 + 4 = 10&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>2\u00aa possibilidade:<\/strong><\/p>\n\n\n\n<p><strong>X &#8211; 4 = -6<\/strong><\/p>\n\n\n\n<p><strong>X = -6 + 4 = -2&nbsp;<\/strong><\/p>\n\n\n\n<p>Assim, chegamos a duas solu\u00e7\u00f5es poss\u00edveis: <strong>S= {-2 , 10}<\/strong>.<\/p>\n\n\n\n<p>Se voc\u00ea substituir o X por 10 na primeira equa\u00e7\u00e3o e o X por -2 na segunda, ir\u00e1 verificar que ambas as equa\u00e7\u00f5es s\u00e3o verdadeiras. <strong>Usando esse m\u00e9todo voc\u00ea consegue resolver qualquer equa\u00e7\u00e3o modular facilmente<\/strong>.&nbsp;<\/p>\n\n\n\n<p><strong>2- |x + 4| = |3x \u2013 2|&nbsp;<\/strong><\/p>\n\n\n\n<p>Quando precisamos calcular uma igualdade entre dois m\u00f3dulos, tamb\u00e9m dividimos em dois casos, por\u00e9m de uma forma levemente distinta:&nbsp;<\/p>\n\n\n\n<p><strong>1\u00ba caso:<\/strong> Ambos os membros da equa\u00e7\u00e3o com o mesmo sinal&nbsp;<\/p>\n\n\n\n<p><strong>2\u00ba caso:<\/strong> Membros da equa\u00e7\u00e3o com sinais opostos.&nbsp;<\/p>\n\n\n\n<p><strong>1\u00aa possibilidade:<\/strong><\/p>\n\n\n\n<p>Vamos considerar ambos os lados positivos, mas se consider\u00e1ssemos ambos negativos o resultado seria o mesmo. Basicamente, nessa possibilidade, apenas retiramos o m\u00f3dulo e calculamos normalmente:&nbsp;<\/p>\n\n\n\n<p><strong>x + 4 <\/strong><strong>&gt;<\/strong><strong> 0 \u2192 |x + 4| = x + 4<\/strong><\/p>\n\n\n\n<p><strong>3x \u2013 2 <\/strong><strong>&gt;<\/strong><strong> 0 \u2192&nbsp; |3x \u2013 2| = 3x &#8211; 2<\/strong><\/p>\n\n\n\n<p><strong>x + 4 = 3x &#8211; 2<\/strong><\/p>\n\n\n\n<p><strong>x &#8211; 3x = -2 &#8211; 4<\/strong><\/p>\n\n\n\n<p><strong>-2x = -6<\/strong><\/p>\n\n\n\n<p><strong>x = -6\/-2 = 6\/2 = 3<\/strong><\/p>\n\n\n\n<p><strong>2\u00aa possibilidade:<\/strong><\/p>\n\n\n\n<p>Nessa 2\u00aa possibilidade, precisamos escolher um termo para ser positivo e o outro negativo. Vamos colocar o primeiro como positivo, por\u00e9m <strong>se escolh\u00eassemos o segundo<\/strong> <strong>n\u00e3o faria diferen\u00e7a<\/strong>.&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>|x + 4| \u2265 0 \u2192 |x + 4| = x + 4<\/strong><\/p>\n\n\n\n<p><strong>|3x \u2013 2| &lt; 0 \u2192 |3x \u20132| = \u2013 (3x \u2013 2)&nbsp;<\/strong><\/p>\n\n\n\n<p>Depois disso, basta resolver a equa\u00e7\u00e3o sem os m\u00f3dulos:&nbsp;<\/p>\n\n\n\n<p><strong>x + 4 = -(3x &#8211; 2)<\/strong><\/p>\n\n\n\n<p><strong>x + 4 = -3x + 2<\/strong><\/p>\n\n\n\n<p><strong>x + 3x = -4 + 2<\/strong><\/p>\n\n\n\n<p><strong>4x = -2<\/strong><\/p>\n\n\n\n<p><strong>x = -2\/4&nbsp; -1\/2<\/strong><\/p>\n\n\n\n<p>Assim, chegamos \u00e0s duas solu\u00e7\u00f5es poss\u00edveis: <strong>S={3, -1\/2}<\/strong><\/p>\n\n\n\n<p>Agora que voc\u00ea j\u00e1 entendeu como se calcula, \u00e9 a sua vez de fazer exerc\u00edcios de equa\u00e7\u00e3o modular. Consulte o gabarito quando acabar.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"exerc\u00edcios-sobre-equa\u00e7\u00e3o-modular\">Exerc\u00edcios sobre Equa\u00e7\u00e3o Modular com Gabarito<\/h2>\n\n\n\n<p>Esperamos que, com esse resumo, <strong>tudo tenha ficado mais claro para voc\u00ea.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Obrigado<\/strong> por ter lido at\u00e9 aqui!<\/p>\n\n\n\n<p>Baixe <strong>gratuitamente<\/strong> o <a href=\"https:\/\/beduka.com\/blog\/dicas\/enem-dicas\/plano-de-estudos-gratuito\/\" target=\"_blank\" rel=\"noreferrer noopener\">Plano de Estudos<\/a> do <strong>Beduka<\/strong> e tenha uma prepara\u00e7\u00e3o<strong> perfeita <\/strong>para o <strong>ENEM<\/strong>.<\/p>\n\n\n\n<p><strong>Quest\u00e3o 1- (<\/strong><a href=\"https:\/\/www2.ufjf.br\/ufjf\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>U.F. Juiz de Fora-MG<\/strong><\/a><strong>)<\/strong><\/p>\n\n\n\n<p>O n\u00famero de solu\u00e7\u00f5es negativas da equa\u00e7\u00e3o | 5x-6 | = x<sup>2<\/sup> \u00e9:<\/p>\n\n\n\n<p><strong>a)<\/strong> 0&nbsp; <strong>&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>b) <\/strong>1 <strong>&nbsp;&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>c) <\/strong>2&nbsp; <strong>&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>d) <\/strong>3&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>e) <\/strong>4<\/p>\n\n\n\n<p><strong>Quest\u00e3o 2- (<\/strong><a href=\"https:\/\/www.tuiuti.edu.br\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>U. Tuiuti \u2013 PR<\/strong><\/a><strong>)<\/strong><\/p>\n\n\n\n<p>As ra\u00edzes reais da equa\u00e7\u00e3o |xl <sup>2 <\/sup>+ |x| &#8211; 6 = 0 s\u00e3o tais que:<\/p>\n\n\n\n<p><strong>a) <\/strong>a soma delas \u00e9 \u2013 1.<\/p>\n\n\n\n<p><strong>b)<\/strong> o produto delas \u00e9 \u2013 6.<\/p>\n\n\n\n<p><strong>c) <\/strong>ambas s\u00e3o positivas.<\/p>\n\n\n\n<p><strong>d) <\/strong>o produto delas \u00e9 \u2013 4.<\/p>\n\n\n\n<p><strong>e)<\/strong> n.d.a.&nbsp;<\/p>\n\n\n\n<p><strong>Quest\u00e3o 3- (<\/strong><a href=\"https:\/\/www5.pucsp.br\/paginainicial\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>PUC \u2013 SP<\/strong><\/a><strong>) <\/strong>O conjunto solu\u00e7\u00e3o S da equa\u00e7\u00e3o |2x \u2013 1| = x \u2013 1 \u00e9:<\/p>\n\n\n\n<p><strong>a) <\/strong>S = {0, <sup>2<\/sup>\/<sub>3<\/sub>}<\/p>\n\n\n\n<p><strong>b) <\/strong>S = {0, <sup>1<\/sup>\/<sub>3<\/sub>}<\/p>\n\n\n\n<p><strong>c) <\/strong>S = \u00d8<\/p>\n\n\n\n<p><strong>d) <\/strong>S = {0, \u2013 1}<\/p>\n\n\n\n<p><strong>e) <\/strong>S = {0, <sup>4<\/sup>\/<sub>3<\/sub>}<\/p>\n\n\n\n<p><strong>Muito bem! <\/strong>Voc\u00ea chegou \u00e0 metade dos Exerc\u00edcios sobre Equa\u00e7\u00e3o Modular. Continue fazendo o restante.<\/p>\n\n\n\n<p><strong>Quest\u00e3o 4- (<\/strong><a href=\"https:\/\/www.uepa.br\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>UEPA<\/strong><\/a><strong>) <\/strong>O conjunto solu\u00e7\u00e3o da equa\u00e7\u00e3o |x|\u00b2 \u2013 2|x| \u2013 3 = 0 \u00e9 igual a:<\/p>\n\n\n\n<p><strong>a) <\/strong>S = {\u2013 1, 3}<\/p>\n\n\n\n<p><strong>b) <\/strong>S = {\u2013 3, 3}<\/p>\n\n\n\n<p><strong>c) <\/strong>S = {\u2013 1, 1}<\/p>\n\n\n\n<p><strong>d) <\/strong>S = {\u2013 3, 1}<\/p>\n\n\n\n<p><strong>e) <\/strong>S = {1, 3}<\/p>\n\n\n\n<p><strong>Quest\u00e3o 5- (<\/strong><a href=\"https:\/\/www.mackenzie.br\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Mackenzie \u2013 SP<\/strong><\/a><strong>) <\/strong>A soma dos valores de x que satisfazem a igualdade |x\u00b2 \u2013 x \u2013 2| = 2x + 2 \u00e9:<\/p>\n\n\n\n<p><strong>a) <\/strong>1<\/p>\n\n\n\n<p><strong>b) <\/strong>3<\/p>\n\n\n\n<p><strong>c) <\/strong>\u2013 2<\/p>\n\n\n\n<p><strong>d) <\/strong>2<\/p>\n\n\n\n<p><strong>e) <\/strong>\u2013 3<\/p>\n\n\n\n<p><strong>Quest\u00e3o 6- (<\/strong><a href=\"https:\/\/www2.ufjf.br\/ufjf\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>UFJF<\/strong><\/a><strong>) <\/strong>O n\u00famero de solu\u00e7\u00f5es negativas da equa\u00e7\u00e3o modular |5x \u2013 6| = x\u00b2 \u00e9:<\/p>\n\n\n\n<p><strong>a) <\/strong>0<\/p>\n\n\n\n<p><strong>b) <\/strong>1<\/p>\n\n\n\n<p><strong>c) <\/strong>2<\/p>\n\n\n\n<p><strong>d) <\/strong>3<\/p>\n\n\n\n<p><strong>e) <\/strong>4<\/p>\n\n\n\n<p><strong>Parab\u00e9ns,<\/strong> voc\u00ea fez todos os exerc\u00edcios sobre Equa\u00e7\u00e3o Modular. Confira agora o Gabarito:<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"gabarito-dos-exerc\u00edcios-de-equa\u00e7\u00e3o-modular\">Gabarito dos Exerc\u00edcios de Equa\u00e7\u00e3o Modular<\/h2>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 1 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b) <\/strong>1 <strong>&nbsp;&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 2 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong><strong>d) <\/strong><\/strong>o produto delas \u00e9 \u2013 4.<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 3 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>c) <\/strong>S = \u00d8<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 4 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b) <\/strong>S = {\u2013 3, 3}<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 5 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b) <\/strong>3<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 6 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong><strong>b) <\/strong><\/strong>1<\/p>\n\n\n\n<p>Estude para o Enem com o <a href=\"https:\/\/beduka.com\/simulado\" target=\"_blank\" rel=\"noreferrer noopener\">Simulado Beduka.<\/a> <strong>\u00c9 gratuito!<\/strong><\/p>\n\n\n\n<p><strong>Gostou dos nossos exerc\u00edcios de Equa\u00e7\u00e3o Modular? <\/strong>Compartilhe com os seus amigos e comente abaixo sobre as \u00e1reas que voc\u00ea deseja mais explica\u00e7\u00f5es.<\/p>\n\n\n\n<p><strong>Queremos te ajudar a encontrar a FACULDADE IDEAL! <\/strong>Logo abaixo, fa\u00e7a uma pesquisa por curso e cidade que te mostraremos todas as faculdades que podem te atender.<strong> <\/strong>Informamos a nota de corte, valor de mensalidade, nota do MEC, avalia\u00e7\u00e3o dos alunos, modalidades de ensino e muito mais.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Leia este artigo e aprenda de uma vez o que \u00e9 um m\u00f3dulo e como se calcular um. Temos exerc\u00edcios de equa\u00e7\u00e3o modular com gabarito para voc\u00ea praticar. <\/p>\n","protected":false},"author":1,"featured_media":20952,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0},"categories":[568,610],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.10 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Os 6 Melhores Exerc\u00edcios de Equa\u00e7\u00e3o Modular com Gabarito<\/title>\n<meta name=\"description\" content=\"Leia este artigo e aprenda de uma vez o que \u00e9 um m\u00f3dulo e como se calcular um. 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