{"id":62628,"date":"2022-12-13T18:06:59","date_gmt":"2022-12-13T21:06:59","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=62628"},"modified":"2024-09-30T10:40:37","modified_gmt":"2024-09-30T13:40:37","slug":"inequacao-e-funcao-de-1o-grau-como-resolver","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/","title":{"rendered":"Inequa\u00e7\u00e3o e Fun\u00e7\u00e3o de 1\u00ba grau: como resolver?"},"content":{"rendered":"\n<p>Est\u00e1 estudando para os seletivos das<a href=\"https:\/\/militares.estrategia.com\/portal\/mundo-militar\/forcas-armadas\/quais-sao-as-forcas-armadas\/\" target=\"_blank\" > For\u00e7as Armadas<\/a>? Ent\u00e3o confira os resumos dos principais conte\u00fados de Matem\u00e1tica do Estrat\u00e9gia Militares! Neste texto trazemos para voc\u00ea <strong>como resolver inequa\u00e7\u00f5es envolvendo <\/strong><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/\" target=\"_blank\" ><strong>fun\u00e7\u00f5es de 1<sup>o<\/sup> grau!<\/strong><\/a><strong>&nbsp;<\/strong><\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado\t<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/#O-que-e-uma-inequacao\" >O que \u00e9 uma inequa\u00e7\u00e3o?&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/#Inequacoes-simultaneas\" >Inequa\u00e7\u00f5es simult\u00e2neas&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/#Inequacoes-produto\" >Inequa\u00e7\u00f5es-produto&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/#Inequacoes-quociente\" >Inequa\u00e7\u00f5es-quociente&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-e-uma-inequacao-nbsp\"><span class=\"ez-toc-section\" id=\"O-que-e-uma-inequacao\"><\/span>O que \u00e9 uma inequa\u00e7\u00e3o?&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Diferente de uma equa\u00e7\u00e3o,<strong> a inequa\u00e7\u00e3o \u00e9 uma express\u00e3o matem\u00e1tica que n\u00e3o possui uma igualdade<\/strong>. Em vez disso, a inequa\u00e7\u00e3o apresenta <strong>sinais de desigualdade,<\/strong> como:&nbsp;&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>menor que (&lt;);&nbsp;<\/li>\n\n\n\n<li>maior que (&gt;);&nbsp;<\/li>\n\n\n\n<li>menor ou igual (\u2264); e&nbsp;<\/li>\n\n\n\n<li>maior ou igual (\u2265).&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>Elas s\u00e3o resolvidas de forma semelhante \u00e0 equa\u00e7\u00e3o, por\u00e9m as<strong> respostas podem ser infinitas solu\u00e7\u00f5es, organizadas em um conjunto<\/strong>. Ou seja, a resposta n\u00e3o \u00e9 uma s\u00f3, mas sim um intervalo onde ela pode ser encontrada. Agora, confira alguns poss\u00edveis casos de inequa\u00e7\u00f5es!<\/p>\n\n\n<div id=\"new-form\">\n\t<form id=\"forms_layout\">\n\t<input type=\"hidden\" id=\"chave_de\" name=\"chave_de\" value=\"01_EMIL_LEADS_TOTAIS\" \/>\n\t\t<input type=\"hidden\" id=\"mid\" name=\"mid\" value=\"515009017\" \/>\n\t\t<input type=\"hidden\" id=\"Objetivo\" name=\"Objetivo\" value=\"Militares\" \/>\n\t\t<input type=\"hidden\" id=\"Estado_de_Origem_do_IP\" name=\"Estado_de_Origem_do_IP\"  value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Cidade_de_Origem_do_IP\" name=\"Cidade_de_Origem_do_IP\" value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Modo_de_entrada\" name=\"Modo_de_entrada\" value=\"Portal\" \/>\n\t\t<input type=\"hidden\" id=\"Pagina_Origem\" name=\"Pagina_Origem\" value=\"\" \/>\n\n\t\t<div class=\"texto\">\n\t\t\t<h3 class=\"news\">Inscreva-se em nossa newsletter!<\/h3>\n\t\t\t<p>Receba not\u00edcias sobre os mais importantes concursos para as For\u00e7as Armadas brasileiras e informa\u00e7\u00f5es sobre o mundo militar!<\/p>\n\t\t<\/div>\n\t\t<div class=\"campos\">\n\t\t\t<div class=\"first_name labels\">\n\t\t\t\t<input type=\"text\" class=\"input_labels\" id=\"first_name\" name=\"Nome_Completo\" placeholder=\"Nome\" required \/>\n\t\t\t<\/div>\n\t\t\t<div class=\"email labels\">\n\t\t\t\t<input type=\"email\" class=\"input_labels\" id=\"email\" name=\"Email\" placeholder=\"E-mail\" required \/>\n\t\t\t<\/div>\n\t\t\t<button type=\"submit\" class=\"button_layout\">\n\t\t\t\t<div id=\"lblBotao\">Inscreva-se<\/div>\n\t\t\t\t<span id=\"divSpiner\" class=\"loader\" role=\"status\" style=\"display: none;\"><\/span>\t\n\t\t\t<\/button>\n\t\t<\/div><!-- \/campos -->\n\t\t\n\t\t<label class=\"container_checkbox\">Voc\u00ea concorda com a nossa\n\t\t\t<a class=\"politics_privacy_ahref\" target=\"_blank\" href=\"https:\/\/gratis.estrategiaconcursos.com.br\/politica-de-privacidade\/\">Pol\u00edtica de Privacidade<\/a> e aceita receber informa\u00e7\u00f5es adicionais do Estrat\u00e9gia Educacional.\n\t\t\t<input id=\"checkbox_layout\" type=\"checkbox\" checked=\"checked\" required \/>\n\t\t\t<span class=\"checkbox\"><\/span>\n\t\t<\/label>\n\t\t<div id=\"msg\"><\/div>\n\t\t<div id=\"error\" class=\"error\">\n\t\t<\/div>\n\t<\/form>\n<\/div><!-- \/new-form -->\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-inequacoes-simultaneas-nbsp\"><span class=\"ez-toc-section\" id=\"Inequacoes-simultaneas\"><\/span>Inequa\u00e7\u00f5es simult\u00e2neas&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p><strong>Uma inequa\u00e7\u00e3o simult\u00e2nea \u00e9 aquela que coloca mais de uma inequa\u00e7\u00e3o numa mesma express\u00e3o. <\/strong>Sejam \ud835\udc53, \ud835\udc54, \u210e: \ud835\udc34 \u2192 \u211d, tr\u00eas fun\u00e7\u00f5es na vari\u00e1vel \ud835\udc65. As inequa\u00e7\u00f5es simult\u00e2neas s\u00e3o da forma:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) &lt; \ud835\udc54(\ud835\udc65) &lt; \u210e(\ud835\udc65)<\/p>\n\n\n\n<p>Para resolver esse tipo de inequa\u00e7\u00e3o, devemos separ\u00e1-la em duas:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) &lt; \ud835\udc54(\ud835\udc65)<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc54(\ud835\udc65) &lt; \u210e(\ud835\udc65)<\/p>\n\n\n\n<p>Se \ud835\udc46<sub>1<\/sub> \u00e9 a solu\u00e7\u00e3o de \ud835\udc53(\ud835\udc65) &lt; \ud835\udc54(\ud835\udc65) e \ud835\udc46<sub>2<\/sub> \u00e9 a solu\u00e7\u00e3o de \ud835\udc54(\ud835\udc65) &lt; \u210e(\ud835\udc65), a solu\u00e7\u00e3o da inequa\u00e7\u00e3o ser\u00e1 dada por:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46 = \ud835\udc46<sub>1<\/sub> \u2229 \ud835\udc46<sub>2<\/sub><\/p>\n\n\n\n<p>Que tal ver na pr\u00e1tica? Resolva a seguinte inequa\u00e7\u00e3o definida em \u211d:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 3 &lt; \u2212\ud835\udc65 + 3 &lt; 2\ud835\udc65 + 4<\/p>\n\n\n\n<p>Temos inequa\u00e7\u00f5es simult\u00e2neas. Vamos dividi-las em duas inequa\u00e7\u00f5es e resolv\u00ea-las:<\/p>\n\n\n\n<p>I) \ud835\udc65 \u2212 3 &lt; \u2212\ud835\udc65 + 3<\/p>\n\n\n\n<p>II) \u2212\ud835\udc65 + 3 &lt; 2\ud835\udc65 + 4<\/p>\n\n\n\n<p>Resolvendo analiticamente:&nbsp;<\/p>\n\n\n\n<p>I) \ud835\udc65 \u2212 3 &lt; \u2212\ud835\udc65 + 3<\/p>\n\n\n\n<p class=\"has-text-align-center\">2\ud835\udc65 &lt; 6 \u21d2 \ud835\udc65 &lt; 3<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46<sub>1<\/sub> = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 &lt; 3}<\/p>\n\n\n\n<p>II) \u2212\ud835\udc65 + 3 &lt; 2\ud835\udc65 + 4&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/3tjb-GJcxfwp40rg3JnBExGFRAo5eYcxd47pU-apv7axKNUssJYhb_iXapoIzWr514xN6gm4EXoBwlTxcQwlPm50b6oIKgAHAxwbl46912-dPxOO5b6O_1lH8DMYniEozWAzjZeN5HmEw-3hEKTT8j76na0KitwYwTz1o0pgvqW6aPV1jnfo4xlZXsJx\" alt=\"\" style=\"width:213px;height:87px\"\/><\/figure><\/div>\n\n\n<p>A solu\u00e7\u00e3o \u00e9 dada pela intersec\u00e7\u00e3o de S<sub>1<\/sub> e S<sub>2<\/sub>:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/EsmMQMDHkpNbobkUedJxt5wyuGjSjGjqRiTjo6__VMrAFZLE6d-vGHyUHIlsVKfhIsF_xAjBYCMJTkAmFgyzwhvQVhAUc3UyWJktNdr-lruYBr7FeHwKaW-FZKjro_KIFWKLO73tzZ34f1jvYxiuc7yeJkJfmQgv77meim7NV6LFkXBkRBL0RobwUNXOJA\" alt=\"\" style=\"width:-92px;height:-13px\"\/><\/figure><\/div>\n\n\n<p>Para facilitar a visualiza\u00e7\u00e3o da resposta, podemos representar as solu\u00e7\u00f5es no eixo <em>x<\/em> de um gr\u00e1fico:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/OkF3tKAApzj3CwvCgkP9dHKltIETon415LgKLJ4-bm4sMxSEYTqhYDWXfqMqo2GUR5dcZU1pKJdzHGHJpNBHD632tgAMi2NTweQeDUYM4HEAj8NqF7-CZyn7bA4qK80dLHfEwdf0jG8ktvgvaVj6FQnZk2jFwN6BxAgMjBM7S39riGmB7AhaBi4vVt1J-A\" alt=\"\"\/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"h-inequacoes-produto-nbsp\"><span class=\"ez-toc-section\" id=\"Inequacoes-produto\"><\/span>Inequa\u00e7\u00f5es-produto&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Agora, considere \ud835\udc53, \ud835\udc54: \ud835\udc34 \u2192 \u211d, duas fun\u00e7\u00f5es na vari\u00e1vel \ud835\udc65.<strong> Se multiplicarmos as fun\u00e7\u00f5es podemos ter quatro casos de inequa\u00e7\u00f5es-produto.<\/strong> Observe:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) \u2219 \ud835\udc54(\ud835\udc65) &lt; 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) \u2219 \ud835\udc54(\ud835\udc65) \u2264 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) \u2219 \ud835\udc54(\ud835\udc65) &gt; 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) \u2219 \ud835\udc54(\ud835\udc65) \u2265 0<\/p>\n\n\n\n<p>A resolu\u00e7\u00e3o de cada uma dessas inequa\u00e7\u00f5es segue a mesma ideia. Assim, vamos resolver&nbsp; \ud835\udc53(\ud835\udc65) \u2219 \ud835\udc54(\ud835\udc65) &gt; 0.&nbsp;<\/p>\n\n\n\n<p>Quando resolvemos inequa\u00e7\u00f5es-produto,<strong> devemos nos atentar ao sinal de cada fun\u00e7\u00e3o envolvida.<\/strong> Nesse caso, o produto das duas fun\u00e7\u00f5es <strong>deve resultar em um n\u00famero positivo<\/strong> e, para isso acontecer, as duas devem possuir o mesmo sinal. As possibilidades s\u00e3o:&nbsp;<\/p>\n\n\n\n<p>I) \ud835\udc53(\ud835\udc65) &gt; 0 e \ud835\udc54(\ud835\udc65) &gt; 0; ou&nbsp;<\/p>\n\n\n\n<p>II) \ud835\udc53(\ud835\udc65) &lt; 0 e \ud835\udc54(\ud835\udc65) &lt; 0&nbsp;<\/p>\n\n\n\n<p>A solu\u00e7\u00e3o ser\u00e1 dada pela<strong> uni\u00e3o da solu\u00e7\u00e3o desses dois casos.<\/strong> Sendo \ud835\udc46<sub>1<\/sub>, a solu\u00e7\u00e3o do caso (I) e \ud835\udc46<sub>2<\/sub>, a solu\u00e7\u00e3o do caso (II), a solu\u00e7\u00e3o ser\u00e1 dada por:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46 = \ud835\udc46<sub>1<\/sub> \u222a \ud835\udc46<sub>2<\/sub><\/p>\n\n\n\n<p>Agora, veja na pr\u00e1tica ao resolver a seguinte inequa\u00e7\u00e3o definida em \u211d:<\/p>\n\n\n\n<p class=\"has-text-align-center\">(\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) &gt; 0<\/p>\n\n\n\n<p>Resolvendo algebricamente:<\/p>\n\n\n\n<p>I) \ud835\udc65 \u2212 1 &gt; 0 e \ud835\udc65 + 1 &gt; 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 1 &gt; 0 \u21d2 \ud835\udc65 &gt; 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 + 1 &gt; 0 \u21d2 \ud835\udc65 &gt; \u22121<\/p>\n\n\n\n<p>Assim, fazendo a intersec\u00e7\u00e3o das duas solu\u00e7\u00f5es, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46<sub>1<\/sub> = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 &gt; 1}<\/p>\n\n\n\n<p>II) \ud835\udc65 \u2212 1 &lt; 0 e \ud835\udc65 + 1 &lt; 0&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 1 &lt; 0 \u21d2 \ud835\udc65 &lt; 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 + 1 &lt; 0 \u21d2 \ud835\udc65 &lt; \u22121<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46<sub>2<\/sub> = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 &lt; \u22121}<\/p>\n\n\n\n<p>Portanto, a solu\u00e7\u00e3o \u00e9 dada pela uni\u00e3o dessas duas:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc46 = \ud835\udc46<sub>1<\/sub> \u222a \ud835\udc46<sub>2<\/sub> = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 &gt; 1 \ud835\udc5c\ud835\udc62 \ud835\udc65 &lt; \u22121}<\/p>\n\n\n\n<p>A solu\u00e7\u00e3o pode ser representada no eixo \ud835\udc65:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/ljXgEaBbjR9Fil1179RpAlQ2h6mnYZ6zp6aNbWBSGjfKFRzt5Jdn77QDoSvd_GlOj-vF80L2xVOpZ01B6w3b1H39CW6ClCPT_HPqW9uvGkPw5pSqRgg03x2It77gEmBb8BnvOeeNqCQ72phS8-F9l9DoVjYDhzT_7jC5OM6LgXcjTolpzsMq3AEwOs4wxg\" alt=\"\"\/><\/figure><\/div>\n\n\n<h3 class=\"wp-block-heading\" id=\"h-resolucao-pelo-estudo-do-sinal\">Resolu\u00e7\u00e3o pelo estudo do sinal<\/h3>\n\n\n\n<p>Al\u00e9m do m\u00e9todo acima,<strong> existe um modo de resolver diretamente a inequa\u00e7\u00e3o-produto apenas com o estudo do sinal das fun\u00e7\u00f5es envolvidas<\/strong>. Veja:<\/p>\n\n\n\n<p class=\"has-text-align-center\">(\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) &gt; 0<\/p>\n\n\n\n<p>Vamos estudar o sinal de cada fun\u00e7\u00e3o:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 1 = 0 \u21d2 \ud835\udc65 = 1<\/p>\n\n\n\n<p>\ud835\udc53(\ud835\udc65) = \ud835\udc65 \u2212 1 \u00e9 uma fun\u00e7\u00e3o crescente, ent\u00e3o representando o seu sinal no eixo \ud835\udc65, temos:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/q5mYAGh_p-zIMseI1Z8c0hR1t2Flrw3LQld_Wvst51Nv6BS1SiTwPaLSHHgnDopEmekhxGsYx6U0lR9lJ4C3nC2WxidKOdZ6AdURHhPf3Hf60caEEL0gI78eHMsUQpdwveisnwwB7QZkj0eoyZ72-WCPKxJTM-Jw9h4EL-RLX3l8lVZA5VvJc5_T1gB6qw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>N\u00e3o \u00e9 necess\u00e1rio desenhar a reta no eixo \ud835\udc65 para encontrar o sinal da fun\u00e7\u00e3o. Isso foi feito no exemplo apenas para lembrar que quando temos uma fun\u00e7\u00e3o crescente, <strong>os n\u00fameros \u00e0 direita da raiz da fun\u00e7\u00e3o resultam em n\u00fameros positivos e \u00e0 esquerda resultam em negativos.<\/strong><\/p>\n\n\n\n<p>Para acelerar a resolu\u00e7\u00e3o dos exerc\u00edcios, o mais simples \u00e9 memorizar esse fato e apenas montar a reta no eixo <em>x<\/em> da seguinte forma:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/FduEt4dWpVe87GfdrTtqfszCG5Px0oxfZdod9yiX0gXhjSkwoGaxf7GG_9rgf1YBt0KBg7EUz2EIfL2XOMtqpQ4N6Cx2PVUbSvmdHiRaGfY9jWE2qsyMwTNCuSg3JAhpmyRf7KOvv-mpsMOtef0YxCkas3Soy9QeCjVOm0JIa2jwsvXf4V1bDZrx6_yQpg\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Repare que representamos os valores de <em>x<\/em> acima do eixo e os valores de <em>x &#8211; 1<\/em>, abaixo do eixo.&nbsp;<\/p>\n\n\n\n<p>Agora, para a outra fun\u00e7\u00e3o \ud835\udc65 + 1 = 0 \u21d2 \ud835\udc65 = \u22121. Veja que \ud835\udc54(\ud835\udc65) = \ud835\udc65 + 1 \u00e9 uma fun\u00e7\u00e3o crescente, logo:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/oFdLM1Z038ccYZJoEGTxXE1q34SYe_JznoqWIpeKx_HsvZgE565-vW8aLptEZemeEvlHfdNIuWiWb27IlEHtzWHy7HnHAEchovjGNMwAf5NRDq1VLTVlJGplGUqwaE8SL6Cj4YiQSIVHQkxjqsaktbWcYJU3d46IgbNmuFj9LaaSUxzEa8iofXrhLr_mkw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Juntando os dois eixos com seus respectivos sinais, obtemos o sinal do produto (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1):&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/KGrpcZ8IZKy4aPA0g7A9Qmt-oWA4081FLS4_hTu34GJKXlSqKIwdWt7I7TBtIflTlTRFW0DPvvYMZzojfvC-uq5Xx0wKTy8AvpSAmW0skcjMEObUwP1Lph-xNABntDF0MVwZvbuqyetsfv7FYmvJQ0ccbvQYtaIIKQKBxykoWaY77JwCMRhrB2g95eszQA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Ficou confuso sobre como montamos o quadro acima? N\u00e3o se preocupe, esclarecemos para voc\u00ea!&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-como-montar-o-quadro-de-sinais-nbsp\">Como montar o quadro de sinais?&nbsp;<\/h3>\n\n\n\n<p>Quando queremos unir o resultado do estudo do sinal de cada fun\u00e7\u00e3o, devemos construir um quadro da seguinte forma:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/IFfAAYzLgOZAZd7ihYa9VANxaJrTXug50BijvF0QDgCsRc9KO8ce16wAaVjNbnwjWs3qtzgrn3Lr1gmg0VVg3guhJUPF-ZTalP1eJo0d7YjtO6ifxoEKbR9exTW0JpjbYeACvSpJLqOCsn95N83T0ppvS28iSjydfQedjtlbFvg-2_SuASbOYyz5OrqhyA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Para descobrir o sinal da fun\u00e7\u00e3o produto, combinamos o sinal das duas que a comp\u00f5em. <\/strong>Lembrando que negativo com negativo resulta em positivo e que negativo com positivo resulta em negativo, temos:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/bi-SkuEvq1IBp2poIPAwU58M4BNRcikkAoGm4fCv2bb6f94kxqrputSJdsZsioyny8E0duFLLOzbKXtveUV7K9kKnKmYgjWR8CeRtUUzNriEe7OunwPXXHKF3rC_p2iiwi6hlU6aTCkDPHmNiapfH--4jm5FvCaorItLRKYs2BgJ3aTtuVMW5ubORf3MMg\" alt=\"\"\/><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/0tXQ5K167cMPIpb3_RbjrgS5Fom41goh4wu5QN41AxcER43sKPhv6StCthtpcr7DqC6a4G8DSMlbFYPQhmZU9_h7uTfm3hkMD0CaUBVeCy0qX_2UwQJlSTxLooiCKIPocFrjs3dpOYypPvaEKDHdwmt12FHiVEv0AYbB4omkSMWw3xIAhotmaHll0w3U4g\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Com esse quadro, sabemos quais valores de <em>x <\/em>resultam em valores positivos ou negativos. Se quisermos (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) &gt; 0:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/lfK1dYcy7HlyGFeD4sFxomnEf_cqqpFfb4BL40jXtvWCBQj07nQC4jJGeel8iMpI-YWc9kXFdxZUGNW0FgTViVXHz1T1nD7FQ2jZO_8TtmK4zj_X1eNwTlkqnBu6W-v_S_X8S_7bN_jcXA0uDwtYNjOe97gYzk92MBoaZRiEO-ZzxBmdU1ow6Da8p01kOg\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Agora, se quisermos (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) &lt; 0:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/VNOGUnC7DQlA8W5L1UqnlHKcXdOy4hX3DRDxJcRv4_Hugtxi_r0iGtiiqNeqyrMxv1vqA-oSRQut1XGphdC-Xlu66jWbmBu0ARToyPvktjelHWGBbJhKSkz3fDYTGQIjznT3mVoHuHkDGmDoreysKhEJwyo9KPM51M30KsM-AxII7MdddLuTaS1zZla2qA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>No caso de (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) \u2265 0 ou (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) \u2264 0, basta trocar o c\u00edrculo aberto pelo c\u00edrculo fechado para indicar que \u22121 e 1 fazem parte da solu\u00e7\u00e3o.&nbsp;<\/p>\n\n\n\n<p>Assim, se (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) \u2265 0:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/GqTgwI5Vb-sa9PuxdvW3czN357eRY5q5CVjsK3-FHQW74qxNe9dYpUpQAvKoEuCemm1iEMdvEXzPY4JWc9e8mCmnFTgv7et1uJwxhXUvMirnIVUnOj_zoBbw4Pj83oXj_urvLAM55JbBfO3FzMnQ5S8DGGZtz1fFcjUg_kSHZWC3FhaE_UCEsX2Q58_RBg\" alt=\"\"\/><\/figure>\n\n\n\n<p>Se quisermos representar o quadro acima algebricamente, o equivalente seria \ud835\udc46 = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 \u2264 \u22121 \ud835\udc5c\ud835\udc62 \ud835\udc65 \u2265 1}.&nbsp;<\/p>\n\n\n\n<p>Agora, se (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) \u2264 0, ent\u00e3o:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/vr4EM3r_TuZGYr9o1IpfXmLJzFdAhczmSHaYhkKnNL0nr3IMufihY5CMGjGi9JXZeg6zplUBV5VtKEUJy9WH3-ILRrPHno0Ea-CJ6PWE2NGaqhq0k2smjYcgnhbWIgkedTrm3dPJKzk-tdfkyAVVcIzG6s-w1wyxVBNNU679Bp_zmOvCMdqbCf4vaJCDhw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Algebricamente, a solu\u00e7\u00e3o \u00e9 dada como \ud835\udc46 = {\ud835\udc65 \u2208 \u211d|\u22121 \u2264 \ud835\udc65 \u2264 1}.&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-multiplicidade-das-raizes-das-funcoes\">Multiplicidade das ra\u00edzes das fun\u00e7\u00f5es<\/h3>\n\n\n\n<p>Al\u00e9m do m\u00e9todo do quadro de sinais, <strong>existe uma forma mais r\u00e1pida de resolver a inequa\u00e7\u00e3o: usando a multiplicidade das ra\u00edzes das fun\u00e7\u00f5es envolvidas. <\/strong>Pense novamente na inequa\u00e7\u00e3o (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) &gt; 0. Para resolv\u00ea-la, precisamos encontrar as ra\u00edzes das fun\u00e7\u00f5es envolvidas \ud835\udc65 \u2212 1 e \ud835\udc65 + 1:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 1 = 0 \u21d2 \ud835\udc65 = 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 + 1 = 0 \u21d2 \ud835\udc65 = \u22121&nbsp;<\/p>\n\n\n\n<p>Representamos as ra\u00edzes na reta \ud835\udc65 levando em considera\u00e7\u00e3o se elas pertencem \u00e0 solu\u00e7\u00e3o. No caso, \u00b11 n\u00e3o pertence \u00e0 solu\u00e7\u00e3o devido ao sinal da desigualdade, logo representamos essas ra\u00edzes com o c\u00edrculo aberto:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/EzXx5XT1bIC6En1TGVnClZGgsEO-NJ3n_9lo_Mq-HIFBs-YO2kAYaGhLiMU-lRIwNzFHMgRZ4QfjLa8eCfBN3XsLhrE5hxz-Pr7lMuLgI81QvN8VTzCEfkTD_EkGpSWLqVeB_5c7TD0Opx5cksbbDueYQTZLshU3fy1pKQykk2YRgq01-QsJrqLbsVXD\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Para descobrir o sinal no intervalo, devemos arbitrar um valor para \ud835\udc65 e encontrar o sinal resultante.<\/strong> Vamos ver o que ocorre quando \ud835\udc65 = 0:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 = 0 \u21d2 (\ud835\udc65 \u2212 1)(\ud835\udc65 + 1) = (0 \u2212 1)(0 + 1) = (\u22121)(1) = \u22121<\/p>\n\n\n\n<p>O sinal resultante \u00e9 negativo (\u22121) e o n\u00famero 0 est\u00e1 entre \u22121 e 1. Dessa forma, todos os n\u00fameros entre \u22121 e 1 s\u00e3o negativos:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/yi54Q0ZhV21PxZFqYItBngClOds2rEj2cuobjGS1-w7n_J-97lJ86IASbfwRnIyDxwHCVwRHWDoNbHdwZDi8BDwpxbRTc7RT6clnBtnesorAObTyNOGaOCb6Q_cNUzix8GoXD0fZYy8fHAwT7Fy5ema02jXvCO1MUP9FUiTiE9IfdKZUo-IyCHncuhvK\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Para encontrar o sinal do resto do intervalo, devemos verificar a multiplicidade das ra\u00edzes da fun\u00e7\u00e3o. <strong>Se a multiplicidade da raiz for \u00edmpar, o sinal do intervalo ao passar pela raiz \u00e9 trocado. Se a multiplicidade for par, o sinal do intervalo \u00e9 mantido.<\/strong><\/p>\n\n\n\n<p>No exemplo, a multiplicidade da raiz 1 \u00e9 \u00edmpar (multiplicidade 1), portanto o intervalo do lado direito da raiz 1 possui sinal oposto ao lado esquerdo:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/pOSW68QWWk9CTbAIJwrN7t86i6_VOHMZOPpUjcF2uTfgr95mBj73fnwuA6YKDqt7Eu3oqql-RRQi3MKPsjr9E8x4wEcNv-EfVEAe4IBeTiw5J_3her2JOzg-NVZZs9sUByzmj32oKrodRWrkB4xOMmtkG7_2hz7dC3tk7KDaWTLzfTCbQ6_PquWlmci1\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Da mesma forma para a raiz \u22121, por possuir multiplicidade \u00edmpar, trocamos seu sinal no lado esquerdo da raiz:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/YXH5qtWmSVCtpGZKUu-yqwlroCvKLYszxKvu3FEu6g6Vgun2RRKq3xJmOhjkuHrujFLFVR44VtbYIMbh5vIosjBXGQ8VDXaRblqeEr83XH-qOsyD4Hn0P3clhWn-RchwWcQZUe2sEx4ySI2hONbIYwHGNy0IlqkGyAWycgSo8Zd0yI_separSrYMWHpx\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Com o eixo completo, basta encontrar os valores que interessam:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/k7y8D9kxiEYp2E0vDNCinW-LOwBEEyK6HIsMiazHo865umw9nG7aL1afW289nfiUo4ZV8UnmcgSLJ3CPqIMWr_GB4IS9_sl2VnVyZ6OHogj9x6pAhrIuvQA15SHbVMPa_zVgJxiOHvwAcLShookx_ben7rux_UrbQGKRAUyCnCQKOzEboMDLxpAooSlp\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Algebricamente, podemos representar a solu\u00e7\u00e3o como \ud835\udc46 = {\ud835\udc65 \u2208 \u211d|\u22121 &lt; \ud835\udc65 &lt; 1}.&nbsp;<\/p>\n\n\n\n<p>Que tal ver um exemplo pr\u00e1tico para fixar o conte\u00fado? Considere a inequa\u00e7\u00e3o abaixo com ra\u00edzes de multiplicidade par:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">(\ud835\udc65 \u2212 1)<sup>4<\/sup>(2\ud835\udc65 + 3)(\ud835\udc65 \u2212 3) \u2265 0<\/p>\n\n\n\n<p>As ra\u00edzes dela s\u00e3o:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(\ud835\udc65 \u2212 1)<sup>4<\/sup> = 0 \u21d2 \ud835\udc65 = 1 multiplicidade 4 (par)<\/li>\n\n\n\n<li>2\ud835\udc65 + 3 = 0 \u21d2 \ud835\udc65 = <img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/cXAMLUZ_Ye-jCATaAHmOPVN9EA9v1gd-_AF1j4QztVUBNA5EY6vVy1az9QcPi9IzHC3I3w_7oeHbtsiGTEgxPrfeRCaflm85Zc70lemGoMK5irUFQ5n3TRIhrrlabtM8WrsvlsFUG0G2aJkljNrU6sE25wTl3dChIyegFRpYhx8am29t64ZWujkS96pY8g\" width=\"20\" height=\"28\"> multiplicidade 1 (\u00edmpar)<\/li>\n\n\n\n<li>\ud835\udc65 \u2212 3 = 0 \u21d2 \ud835\udc65 = 3 multiplicidade 1 (\u00edmpar)&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>As ra\u00edzes fazem parte da solu\u00e7\u00e3o, pois o sinal de desigualdade \u00e9 \u201c\u2265\u201d. Logo, as representamos na reta real com o c\u00edrculo fechado:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/ABTaN148adrKkrGEnfe7OjtMGSrGW1M9aQKjsJ4W315dB3mZgIxo8QL41esGH6MpgDUXTVTTmXCdkIHJapLmLfCKoT34aExG4ezjUDZDYujgKYwPMA48Nw6b2yBpzgr04mf_O1pFQOOvCm0UkmiZDstvz2Vwg1uIDawSuUQIlZQQ-xHVk0tQfsEOxfpbiw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Agora, vamos verificar o sinal quando x = 0:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">(0 \u2212 1)<sup>4<\/sup>(2(0) + 3)(0 \u2212 3) = (\u22121)4(3)(\u22123) = \u22129<\/p>\n\n\n\n<p>Logo, zero resulta em n\u00fameros negativos no intervalo:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/EDw9J5DMjUDGwW0eHzRQci6Gg-8mxKF8cOyPoalVO6-tgWqZ9I3fEKmmGpGl8-8j5VIeL0FrN0vj-m9jDTeG0KRJzMPAN9e0Vc-OQ8jOIYBpLNP8KYi34EPN5LGhBx7hSn9ld0v-BUs1xfSyiRdNXYa0at000d2f3ovtUEOZl7ljyakP46urzB90WNof3w\" alt=\"\" style=\"width:89px;height:49px\"\/><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/-RApifye_sIP3Ps5UIA5Ev7T4zOkZ8vyh8bHxpPUoYkz_B3lvzcGE38dr_wVd0ic_4pNPsTLyulWSr7tBvSMK5ZDWGfUZunuh1-1Gxo0ruJDrx79Glo4kVefdxejbJjV0gJ0hiYxV517oivxPU4RyYdYkadWTnmTls3dN5IMQlWBpY_YxLLJ_xCReao_Kw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>O 1 \u00e9 uma raiz de multiplicidade par, logo ao passarmos por essa raiz o sinal permanece igual:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/J2EwtwGtOn3QbYQtpoPgKV3jUeqqrAe1GVdUUUXJzWNJYGdlDZFyb9zEQfqMQt9j_KZSGYojauIGlVQVD-tf8QSTwxP4cM0yv7s1G8KKHPveKtRbgO_CO2v--KT6NGxgSod22UvwEFl9fNbq_xDgwsaY_sHSZbbjl1qqfBhmHE7gDjNxdn6bjemrPT9ODg\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>J\u00e1 o 3 possui multiplicidade \u00edmpar, ent\u00e3o devemos trocar o sinal:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/DebSkBgtUYDAhOSoBHdZA2-WVR7uuc5rkAU0MgQn2tL3M3xdHlFTaM-smtHsql-v09KN-em3KbjnshnBRoNMfrwWKv2a12CjM2x9UKCp3WBwsqi9K5XB_KZv2A-6ayJXw9QGWFNXFK9YwJi4eFZwozWCRVCIsKR516WsRHoRZbhD15sYc59rL7A5Tv4c1w\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Por fim, o \u22123\/2 possui multiplicidade \u00edmpar, logo trocamos o sinal:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/V64KI_XmDoono6QTywtfYlNQXO0IiJmEUt6lD6D_fKaszCy6-rsm5dm_znP1ZAI7TxYglPwZ2riT1bIHYwrfooQJ-4F78sURQ0rYI4Ndj392eirNnlTbI1epmnhdcPqLhKL0NnlPo5A0i2vNlw6kjv6l2OuWQQQnhMzCH1A4PQ6z0KaOJdAngq1VFw5vAA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Ap\u00f3s colocar todos os sinais, observamos os valores que a inequa\u00e7\u00e3o pede. No caso, s\u00e3o aqueles \u2265 0, com isso:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/_d96e-4RoKBh9WRu7eVmqI-xag6skNbO6-k1P-Oe-kj4jDguM0fG3NbqngZXOz6j7tidaWVURwVJhjCjYvT7SIYYFy-a7EEXyroip8mJ5AhOqAfzYizFMQEGe-VdIgn62fUaIPZiMUJWxSyPKe4oCSip0CplKJbYJf6qPF_0YqrQd9ZL2d7NPJR-7GqzvQ\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>A solu\u00e7\u00e3o alg\u00e9brica equivalente \u00e9 \ud835\udc46 = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 &lt; \u22123\/2 ou \ud835\udc65 = 1 ou \ud835\udc65 &gt; 3}.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-inequacoes-quociente-nbsp\"><span class=\"ez-toc-section\" id=\"Inequacoes-quociente\"><\/span>Inequa\u00e7\u00f5es-quociente&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Existem tamb\u00e9m as possibilidades de<strong> inequa\u00e7\u00e3o-quociente<\/strong>, sendo \ud835\udc53, \ud835\udc54: \ud835\udc34 \u2192 \u211d, duas fun\u00e7\u00f5es na vari\u00e1vel \ud835\udc65. As possibilidades s\u00e3o:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/KnAZjpIPXsLYMgtMdzXkwBaCzfD1FxwtI7CBJDF1OFrcudHiCRUuFm7WdVxxXvq5kUSYgF3VbtenKFlgmLWTdg7Z_eXuJxXrh73HbIvWrGHInHgehMz5Hsmi9ZE5dKYNEdZ8DPGkQGr-29b_ER2hcFVznGv6e1LpOsDDv431TxZp0C64YpVRxagSk5btrg\" alt=\"\" style=\"width:80px;height:260px\"\/><\/figure><\/div>\n\n\n<p>Assim como as inequa\u00e7\u00f5es-produto, <strong>as inequa\u00e7\u00f5es-quocientes tamb\u00e9m s\u00e3o resolvidas com o quadro de sinais das fun\u00e7\u00f5es envolvidas. <\/strong>No entanto, \u00e9 preciso se atentar \u00e0 <strong>fun\u00e7\u00e3o do denominador, que conta com uma condi\u00e7\u00e3o de exist\u00eancia: ser diferente de zero.&nbsp;<\/strong><\/p>\n\n\n\n<p>Como exemplo, resolva a seguinte inequa\u00e7\u00e3o definida em \u211d:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/06uymM4Q0yZu4MpZgOIXzV8t5f4LpkskzuZI2GTJLZAZ2RUuOPIAMxrYAInQhx8l_KcaIHPxWifSMgL-fbcWmzLxXrR_ilw9zikzgdVrK9xT0cGivaYtzFmgdmFWRuD1SVBt3ZxNX927k1JLwxN9-FEJfYHRe_iFT7wbZy2fdILUUCVwFJp3jrTg50ylgQ\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Para encontrar a solu\u00e7\u00e3o <strong>\u00e9 necess\u00e1rio zerar o outro lado da inequa\u00e7\u00e3o. <\/strong>Assim, \u00e9 preciso 2 em ambos os lados:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/GW33SGYFnkdaeeSAoQFVkWmvGJzTGjrKhrZQv85l6JkCHk1YYP-621hdxmFxnA36y4Cgux944gTitXtg8Y-oRV8jdM6RMC6szVN8bnTpjuLKOQI_p79G0hNZnkr9vknkzrtJDnq3UniJD5g0XxorVyBgO_B9oGTLhTqsN7b4ydurTNc8zHV_N-jld5Pe_g\" alt=\"\" style=\"width:174px;height:191px\"\/><\/figure><\/div>\n\n\n<p>Agora, antes de resolv\u00ea-la, \u00e9 preciso<strong> encontrar a condi\u00e7\u00e3o de exist\u00eancia da inequa\u00e7\u00e3o &#8211; <\/strong>lembrando que o denominador deve ser diferente de zero:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2212 1 \u2260 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 \u2260 1<\/p>\n\n\n\n<p>Com a condi\u00e7\u00e3o em m\u00e3os, \u00e9 hora de proceder para o estudo do sinal:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65) = \ud835\udc65 + 11 = 0 \u21d2 \ud835\udc65 = \u221211<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53 \u00e9 crescente<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc54(\ud835\udc65) = \ud835\udc65 \u2212 1 = 0 \u21d2 \ud835\udc65 = 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc54 \u00e9 crescente&nbsp;<\/p>\n\n\n\n<p>O estudo do sinal \u00e9 o seguinte:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Zh-BfsTjNW7bETlWaX6fNSiQ_4BZ7vSHfk0H_VnEKZ6rVINk-lR1jtf2u7oyThodv45iPmSXkKjCzqG01GFKvGyDxPcpODju-PwfYTGSjjfyXSs09naoOmyD2xhByJUB8EPxP6ndT7bWUoi9gdxc0obVI8Ydtx1wjaqtB0XbwR1PhUyhWjSh6m_K5Ey_iQ\" alt=\"\" style=\"width:-128px;height:-66px\"\/><\/figure><\/div>\n\n\n<p>Note que o c\u00edrculo aberto no ponto \ud835\udc65 = 1 para indicar que ela n\u00e3o pertence \u00e0 solu\u00e7\u00e3o do problema. Logo, a solu\u00e7\u00e3o do problema \u00e9 \ud835\udc46 = {\ud835\udc65 \u2208 \u211d|\ud835\udc65 \u2264 \u22121 \ud835\udc5c\ud835\udc62 \ud835\udc65 &gt; 1}.&nbsp;<\/p>\n\n\n\n<p><strong>Aten\u00e7\u00e3o! <\/strong>Podemos ficar tentados a usar a<strong> regra do cruzado<\/strong> para simplificar a inequa\u00e7\u00e3o; desse modo, no entanto, <strong>perde-se o denominador e consequentemente chega-se a um<\/strong><\/p>\n\n\n\n<p><strong>resultado incorreto<\/strong>. Como no exemplo abaixo:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/bQFrujrcbddg5InqYgrF0TXtSsqsynRQQoSRmyp2Yx4BRFvBsyC3VZ_3StW2Q0JRWfOJfx1b7pi9aHWXLUSbtwlo-7imKVWJDBmDePgqrMT4PvB4jsvDEmutN4iPox2QAxVhJzMh8IkS5y9_JG7MAlEr9LUFlEwD-bwb657Gln_c9dAoabi56EuEjJRdRQ\" alt=\"\" style=\"width:-28px;height:-13px\"\/><\/figure><\/div>\n\n\n<p class=\"has-text-align-center\">(3\ud835\udc65 + 9) \u2265 2(\ud835\udc65 \u2212 1)<\/p>\n\n\n\n<p class=\"has-text-align-center\">(3\ud835\udc65 + 9) \u2212 2(\ud835\udc65 \u2212 1) \u2265 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">3\ud835\udc65 + 9 \u2212 2\ud835\udc65 + 2 \u2265 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc65 + 11 \u2265 0&nbsp;<\/p>\n\n\n\n<p>Tenha isso em mente ao resolver as quest\u00f5es do seu seletivo, j\u00e1 que as inequa\u00e7\u00e3o geralmente n\u00e3o estar\u00e3o na forma simplificada como:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/9wUQPM1oRPrstyVNicgd-JDiz_V2TVFdX9YfaO2_SGbT-WdiqbycD7qN9xc1ANeiEhLhTk0Z9VGFba_kuJhQvjFYQHC9iNR_YkijNiCeQS8wrB_YtPrjJier8u9fpRETTliZSa8W3CE1H2hvZnauYNFG8R5Bo5Axaglyt0reW7u7Dto0z3ik8RTFCJdWrg\" alt=\"\" style=\"width:77px;height:49px\"\/><\/figure><\/div>\n\n\n<p>Elas ser\u00e3o da seguinte forma:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/Afjy-ZyNbkKT91Z2ydQ-ceaD_iVFHt8Qu35gliY2MnB71sxm_Eo84fP3K6rBpyLVlA7Ph9syhXDQ71-Y7p9LcGjM3tXABcueUiRLMfRuSlyPp0HTkWiaFAip4nBdcMs_NzHjWEdY-qt6GCnYL9k1V9H4v3XV_YZ1o48vitbL98aKLbFRQkH3CpIbtYgIkA\" alt=\"\" style=\"width:106px;height:56px\"\/><\/figure><\/div>\n\n\n<p>Como mencionamos, para resolv\u00ea-la, basta subtrair \u210e(\ud835\udc65) nos dois lados da inequa\u00e7\u00e3o para proceder com o estudo do sinal:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Eih_pSpdUNau05XUFWML-e22oHbL4bpbTdl6l-PeFWbyePiBW6v8X6dUMCGt2cfiFWR8Aa86YiDWufIPB-vzfJ3kJj_uvI7qpcH1wrQNmOX12UZXZgelTsUuQWnJi98OrzRfwEHjQfScgNBGaleIEk9lBAaiUBRvcV5VqQYBVy1E1AXe89cWVlewpSb2jw\" alt=\"\" style=\"width:-41px;height:-28px\"\/><\/figure><\/div>\n\n\n<p>O estudo do sinal envolver\u00e1 as fun\u00e7\u00f5es \ud835\udc61(\ud835\udc65) = \ud835\udc53(\ud835\udc65) \u2212 \u210e(\ud835\udc65)\ud835\udc54(\ud835\udc65) e \ud835\udc54(\ud835\udc65). O resultado ser\u00e1 a an\u00e1lise do sinal de \ud835\udc61(\ud835\udc65)\/\ud835\udc54(\ud835\udc65).<\/p>\n\n\n\n<p>E a\u00ed, guerreiro? Quer mais resumos dos assuntos que caem nos seletivos das For\u00e7as Armadas? Ent\u00e3o, acompanhe o Portal do Estrat\u00e9gia Militares e assine nossa Newsletter semanal! Al\u00e9m disso, clique no banner e conhe\u00e7a nossa plataforma de cursos on-line preparat\u00f3rios!&nbsp;<\/p>\n\n\n\n<p><strong><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/militares.estrategia.com\/curso\/banco-de-questes-anual\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/militares.estrategia.com\/portal\/wp-content\/uploads\/2023\/02\/EM-Banco-de-Questoes.jpg\" alt=\"EM - Banco de Questo\u0303es\" title=\"EM - Banco de Questo\u0303es\">\n        <\/a><\/strong><\/p>\n\n\n\n\n\n<p><em>Texto escrito com base no conte\u00fado elaborado pelo Prof. Victor So<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\">Progress\u00e3o Aritm\u00e9tica (PA): o que \u00e9, tipos e propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/\">Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\">Par ordenado e produto cartesiano: o que s\u00e3o e suas propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/\">O que \u00e9 uma Rela\u00e7\u00e3o Bin\u00e1ria na Matem\u00e1tica<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/\">Entenda a equa\u00e7\u00e3o geral da reta na Geometria Anal\u00edtica<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/estatistica-descritiva-conheca-tecnicas-e-aplicacoes\/\">Estat\u00edstica descritiva: conhe\u00e7a t\u00e9cnicas e aplica\u00e7\u00f5es<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/polinomios-definicoes-operacoes-fatoracao-e-mais\/\">Polin\u00f4mios: defini\u00e7\u00f5es, opera\u00e7\u00f5es, fatora\u00e7\u00e3o e mais!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/conheca-os-poligonos-regulares-e-suas-propriedades\/\">Conhe\u00e7a os pol\u00edgonos regulares e suas propriedades!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/matriz-entenda-os-conceitos-basicos\/\">Matriz: entenda os conceitos b\u00e1sicos!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/regra-de-tres-domine-facilmente-essa-ferramenta\/\">Regra de tr\u00eas: domine facilmente essa ferramenta!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entendendo-os-numeros-primos-e-sua-importancia-na-matematica\/\">Entendendo os n\u00fameros primos e sua import\u00e2ncia na matem\u00e1tica<\/a><\/li>\n<\/ul>\n\n\n","protected":false},"excerpt":{"rendered":"Est\u00e1 estudando para os seletivos das For\u00e7as Armadas? Ent\u00e3o confira os resumos dos principais conte\u00fados de Matem\u00e1tica do&hellip;\n","protected":false},"author":9,"featured_media":62629,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[307],"tags":[],"wl_entity_type":[187],"class_list":{"0":"post-62628","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Inequa\u00e7\u00e3o e Fun\u00e7\u00e3o de 1\u00ba grau: como resolver?<\/title>\n<meta name=\"description\" content=\"Neste resumo de Matem\u00e1tica, trazemos para voc\u00ea como resolver inequa\u00e7\u00f5es envolvendo fun\u00e7\u00f5es de 1\u00ba grau, n\u00e3o perca!\" \/>\n<meta name=\"robots\" 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