document.write( "Question 1196904: Solve each inequality, express the solution set in interval notation, then graph the solution set.\r
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Algebra.Com's Answer #829965 by ikleyn(52908) $\"\"$ $\"About$

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\n" ); document.write( "Solve each inequality, express the solution set in interval notation, then graph the solution set.
\n" ); document.write( "2x/x−7 ≤ 5
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\n" ); document.write( "\n" ); document.write( "        The solution by Alan is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        It is incorrect because multiplying by (x-7) both sides of the given inequality \r
\n" ); document.write( "\n" ); document.write( "        without changing the inequality sign is valid ONLY if x-7 > 0.\r
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\n" ); document.write( "\n" ); document.write( "So, the Alan's solution is valid only if x > 7 and it produces then the solution set x >= 35/3.\r
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\n" ); document.write( "\n" ); document.write( "Thus, when x > 7, the solution set in this domain is { x >= $\"35%2F3\"$ }. \r
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\n" ); document.write( "\n" ); document.write( "                It is the part, that Alan found out.\r
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\n" ); document.write( "\n" ); document.write( "But it is only one part of the work which should be done.\r
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\n" ); document.write( "\n" ); document.write( "            In addition, the case x-7 < 0 should be (and MUST be) analyzed separately.\r
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\n" ); document.write( "\n" ); document.write( "                                    I do it below.\r
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document.write( "So, we assume now that  x-7 < 0,   which means  x < 7.\r\n" );
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document.write( "Multiply both sides of the given inequality by (x-7).\r\n" );
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document.write( "Since this factor  (x-7)  is NEGATIVE  now, we should change the sign of the inequality.\r\n" );
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document.write( "Thus after multiplying we have\r\n" );
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document.write( "    2x >= 5*(x-7).\r\n" );
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document.write( "Simplify and solve for x\r\n" );
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document.write( "    2x >= 5x - 35\r\n" );
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document.write( "    35 >= 5x - 2x\r\n" );
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document.write( "    35 >= 3x\r\n" );
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document.write( "    x <= \r\n" );
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document.write( "Thus in the case when x < 7, we have second inequality  x <= .\r\n" );
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document.write( "Thus, the second solution subset is {x < 7}.\r\n" );
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document.write( "    \r\n" );
document.write( "        Finally, from two basic cases, the solution set        \r\n" );
document.write( "        consists of two subsets  x >=  and  x < 7.   \r\n" );
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document.write( "In other words, the solution set is the union  (-oo,7) U [,oo).\r\n" );
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document.write( "This solution set is plotted below\r\n" );
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document.write( "      ==============|============)--------------|==================\r\n" );
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document.write( "     -oo            0            7                            oo\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, and fully explained.\r
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\n" ); document.write( "\n" ); document.write( "That mistake that Alan did, the beginner students make every day, \r
\n" ); document.write( "\n" ); document.write( "until a teacher or an expert will show them a right way solving such inequalities.\r
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\n" ); document.write( "\n" ); document.write( "So, I am glad to have this opportunity to explain the things to you.\r
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