{"id":11777,"date":"2014-02-07T02:45:29","date_gmt":"2014-02-07T02:45:29","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11777"},"modified":"2022-01-14T19:05:53","modified_gmt":"2022-01-14T19:05:53","slug":"determine-o-conjunto-solucao-de-cada-uma-das-condicoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11777","title":{"rendered":"Determine o conjunto solu\u00e7\u00e3o de cada uma das condi\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_11777' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11777' class='GTTabs_curr'><a  id=\"11777_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11777' ><a  id=\"11777_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11777'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a fun\u00e7\u00e3o $f$ definida\u00a0por: \\[f\\left( x \\right) = \\frac{x}{{{x^2} &#8211; 3x + 2}}\\]<\/p>\n<p>Determine o conjunto solu\u00e7\u00e3o de cada uma das inequa\u00e7\u00f5es:<\/p>\n<ol>\n<li>$f\\left( x \\right) &gt; 0$<\/li>\n<li>$f\\left( {x &#8211; 2} \\right) &gt; 0$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11777' onClick='GTTabs_show(1,11777)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11777'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, tem-se sucessivamente:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) &gt; 0}&amp; \\Leftrightarrow &amp;{\\frac{x}{{{x^2} &#8211; 3x + 2}} &gt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{x}{{\\left( {x &#8211; 1} \\right)\\left( {x &#8211; 2} \\right)}} &gt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x &lt; 0} \\\\<br \/>\n{\\left( {x &#8211; 1} \\right)\\left( {x &#8211; 2} \\right) &lt; 0}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x &gt; 0} \\\\<br \/>\n{\\left( {x &#8211; 1} \\right)\\left( {x &#8211; 2} \\right) &gt; 0}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x &lt; 0} \\\\<br \/>\n{x \\in \\left] {1,2} \\right[}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x &gt; 0} \\\\<br \/>\n{x \\in \\left] { &#8211; \\infty ,1} \\right[ \\cup \\left] {2, + \\infty } \\right[}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x \\in \\emptyset }&amp; \\vee &amp;{x \\in \\left] {0,1} \\right[ \\cup \\left] {2, + \\infty } \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] {0,1} \\right[ \\cup \\left] {2, + \\infty } \\right[}<br \/>\n\\end{array}\\]<\/li>\n<li>Seja $g\\left( x \\right) = f\\left( {x &#8211; 2} \\right)$.<br \/>\nOra, o gr\u00e1fico da fun\u00e7\u00e3o $g$ pode obter-se a partir do gr\u00e1fico de $f$ por uma transla\u00e7\u00e3o associada ao vetor de coordenadas $\\left( {2,0} \\right)$.<\/p>\n<p>Assim, como o conjunto solu\u00e7\u00e3o da condi\u00e7\u00e3o ${f\\left( x \\right) &gt; 0}$ \u00e9 ${S_1} = \\left] {0,1} \\right[ \\cup \\left] {2, + \\infty } \\right[$, ent\u00e3o o conjunto solu\u00e7\u00e3o da condi\u00e7\u00e3o $f\\left( {x &#8211; 2} \\right) &gt; 0 \\Leftrightarrow g\\left( x \\right) &gt; 0$ ser\u00e1 ${S_2} = \\left] {2,3} \\right[ \\cup \\left] {4, + \\infty } \\right[$.<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":928,\r\n\"height\":467,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of 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