document.write( "Question 1194429: The numerator of the fraction is x, while the denominator is 5 more the numerator if the value of the fraction is less than 2/5, find the value of x \n" ); document.write( "

Algebra.Com's Answer #826630 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The numerator of the fraction is x, while the denominator is 5 more the numerator.
\n" ); document.write( "if the value of the fraction is less than 2/5, find the value of x
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document.write( "Let x be the numerator;  then the denominator is (x+5) and the fraction is  .\r\n" );
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document.write( "They want you solve this inequality\r\n" );
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document.write( "     < .     (1)\r\n" );
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document.write( "                        +---------------------+\r\n" );
document.write( "                        |     Attention (!)   |\r\n" );
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document.write( "     Do not multiply both sides by (x+5), because you don't know in advance, \r\n" );
document.write( "                  if this value is positive or negative.\r\n" );
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document.write( "    +----------------------------------------------------------------------+\r\n" );
document.write( "    |   THERE IS ANOTHER METHOD TO SOLVE IT without making wrong steps !   |\r\n" );
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document.write( "Re-write equation (1)  equivalently  in this form\r\n" );
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document.write( "     -  < 0.\r\n" );
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document.write( "Write it with the common denominator, which is  5*(x+5).\r\n" );
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document.write( "     -  < 0.\r\n" );
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document.write( "Simplify\r\n" );
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document.write( "     < 0,\r\n" );
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document.write( "     < 0,\r\n" );
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document.write( "     < 0.\r\n" );
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document.write( "This inequality is equivalent to\r\n" );
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document.write( "     < 0.\r\n" );
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document.write( "The last inequality is hold in these two cases:\r\n" );
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document.write( "    (a)  3x-10 < 0  and  x+5 > 0\r\n" );
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document.write( "or \r\n" );
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document.write( "    (b)  3x-10 > 0  and  x+5 < 0.\r\n" );
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document.write( "In case (a),  it implies  3x < 10  and  x > -5,  which is the same as  x < 10/3  and  x > -5,\r\n" );
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document.write( "              which gives the answer -5 < x < 10/3.\r\n" );
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document.write( "In case (b),  it implies  3x > 10  and  x < -5,  which  has no solutions.\r\n" );
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document.write( "ANSWER.  The solution set is  -5 < x < 10/3,  or,  in interval form,  (-5, 10/3).\r\n" );
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