document.write( "Question 828701: The sides of a triangle all have integral measure and can be represented by the expressions x + 10, 2x -3, and 4x. For how many values of x will the sum of the two longer sides be divisible by the shortest side? \n" ); document.write( "
Algebra.Com's Answer #499505 by math-vortex(648)\"\" \"About 
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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "The sides of a triangle all have integral measure and can be represented by the expressions x \r\n" );
document.write( "+ 10, 2x -3, and 4x. For how many values of x will the sum of the two longer sides be \r\n" );
document.write( "divisible by the shortest side?\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.\r\n" );
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document.write( "We have 3 situations that must all be true.\r\n" );
document.write( "I. x + 10 < (2x - 3) + (4x)\r\n" );
document.write( "II. 2x-3 < (x + 10) + (4x)\r\n" );
document.write( "III. 4x < (x + 10) + (2x - 3)\r\n" );
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document.write( "CASE I:\r\n" );
document.write( "x + 10 < 2x - 3 + 4x\r\n" );
document.write( "x + 10 < 6x - 3\r\n" );
document.write( "x < 6x - 13\r\n" );
document.write( "-5x < -13\r\n" );
document.write( "x > 13/5 (Remember to reverse the direction of the inequality.)\r\n" );
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document.write( "CASE II:\r\n" );
document.write( "2x - 3 < x + 10 + 4x\r\n" );
document.write( "2x - 3 < 5x + 10\r\n" );
document.write( "-3x - 3 < 10\r\n" );
document.write( "-3x < 10 + 3\r\n" );
document.write( "-3x < 13\r\n" );
document.write( "x > 13/-3 (Reverse inequality.)\r\n" );
document.write( "x > -13/3\r\n" );
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document.write( "CASE III: \r\n" );
document.write( "4x < 2x - 3 + 10 + 4x\r\n" );
document.write( "4x < 3x + 7\r\n" );
document.write( "x < 7\r\n" );
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document.write( "Now we combine the information from each inequality Recall that the sides of this triangle \r\n" );
document.write( "have integral measure. Also, because it is the side length of the triangle.\r\n" );
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document.write( "x > 5/13 AND x > -13/3 ANG x < 7\r\n" );
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document.write( "x must be an integer such that x>13/5, so x = 3, 4, 5…\r\n" );
document.write( "AND\r\n" );
document.write( "x is an integer such that x>-13/3, so x = 1, 2, 3,…\r\n" );
document.write( "AND\r\n" );
document.write( "x is an integer such that x<7, so x = 1, 2, 3, 4, 5, 6, 7\r\n" );
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document.write( "Taken together, we see that  must equal 3, 4, 5, or 6 in order for the triangle to exist. \r\n" );
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document.write( "For which values of x will the sum of the longer sides be divisible by the shortest side?\r\n" );
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document.write( "When x = 3,\r\n" );
document.write( "x + 10 = 13\r\n" );
document.write( "2x - 3 = 3\r\n" );
document.write( "4x = 12\r\n" );
document.write( "The sum of the longer sides is 25 which is not divisible by 3.\r\n" );
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document.write( "When x = 4,\r\n" );
document.write( "x + 10 = 14\r\n" );
document.write( "2x - 3 = 5\r\n" );
document.write( "4x = 16\r\n" );
document.write( "The sum of the longer sides is 30 which is divisible by 5.\r\n" );
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document.write( "When x = 5,\r\n" );
document.write( "x + 10 = 15\r\n" );
document.write( "2x - 3 = 7\r\n" );
document.write( "4x = 20\r\n" );
document.write( "The sum of the longer sides is 35 which is divisible by 7\r\n" );
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document.write( "When x = 6,\r\n" );
document.write( "x + 10 = 16\r\n" );
document.write( "2x - 3 = 9\r\n" );
document.write( "4x = 24\r\n" );
document.write( "The sum of the longer sides is 40 which is not divisible by 9.\r\n" );
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document.write( "For two values, x=4 and x=5, the sum of the longer sides is divisible by the shorter side.\r\n" );
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document.write( "Interesting problem. I hope this helps! Feel free to email if you have any questions about the \r\n" );
document.write( "solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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