document.write( "Question 1047889: in a triangle ABC, AB=x+2, BC=2x+3,AC=3x-5. Find all possible values of x \n" ); document.write( "

Algebra.Com's Answer #663741 by ikleyn(52782) $\"\"$ $\"About$

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\n" ); document.write( "in a triangle ABC, AB=x+2, BC=2x+3,AC=3x-5. Find all possible values of x
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document.write( "To solve the problem you need to solve all three \"triangle inequalities\":\r\n" );
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document.write( "AB + BC > AC,   i.e.  (x+2)  + (2x+3) > 3x-5,    (1)\r\n" );
document.write( "AB + AC > BC,   i.e.  (x+2)  + (3x-5) > 2x+3,    (2)   and\r\n" );
document.write( "BC + AC > AB,   i.e.  (2x+3) + (3x-5) > x+2.     (3)\r\n" );
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document.write( "(1) is equivalent to 5 > -5, which is always true and doesn't carry useful information;\r\n" );
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document.write( "(2) is equivalent to 4x-3 > 2x+3, i.e. 2x > 6, which means x > 3.\r\n" );
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document.write( "(3) is equivalent to 5x-2 > x+2, i.e. 4x > 4, which means x > 1.\r\n" );
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document.write( "So, you solved the problem. All possible values of x are x > 3.\r\n" );
document.write( "It is the necessary and sufficient condition.\r\n" );
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document.write( "Answer. All possible values of \"x\" are determined by only one condition  x > 3.\r\n" );
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