document.write( "Question 337862: A farmer plans to build a rectangular pen taht is 16 feet long. The area of the pen is 80 square feet. Is 40 feet of fencing enough to fence in the pen? \n" ); document.write( "
Algebra.Com's Answer #242300 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the length be x\r\n" );
document.write( "Let the width by y\r\n" );
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document.write( "Area = xy = 80\r\n" );
document.write( "Perimeter = 2x + 2y < 40 \r\n" );
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document.write( "So we have \r\n" );
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document.write( "\"system%28xy=80%2C2x%2B2y%3C=40%29\"\r\n" );
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document.write( "Solve the equation for y \"y=80%2Fx\"\r\n" );
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document.write( "Substitute in the inequality\r\n" );
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document.write( "\"2x+%2B+2%2880%2Fx%29+%3C=40\"\r\n" );
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document.write( "\"2x+%2B+160%2Fx+%3C=+40\"\r\n" );
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document.write( "Since x is positive we can multiply through by x without\r\n" );
document.write( "reversing the inequality symbol\r\n" );
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document.write( "\"2x%5E2%2B160+%3C=+40x\"\r\n" );
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document.write( "\"2x%5E2-40x%2B160%3C=0\"\r\n" );
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document.write( "Divide through by 2\r\n" );
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document.write( "\"x%5E2-20x%2B80%3C=0\"\r\n" );
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document.write( "Find the zeros of the expression on the left.\r\n" );
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document.write( "It doesn't factor, so:\r\n" );
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document.write( " \"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"\r\n" );
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document.write( " \"x+=+%28-%28-20%29+%2B-+sqrt%28%28-20%29%5E2-4%2A1%2A80+%29%29%2F%282%2A1%29+\"\r\n" );
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document.write( " \"x+=+%2820+%2B-+sqrt%28400-320%29%29%2F2+\"\r\n" );
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document.write( " \"x+=+%2820+%2B-+sqrt%2880%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%2820+%2B-+sqrt%2816%2A5%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%2820+%2B-+4sqrt%285%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%284%285+%2B-+sqrt%285%29%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+2%285+%2B-+sqrt%285%29%29+\"\r\n" );
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document.write( "The two zeros are approximately 5.5 and 14.5\r\n" );
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document.write( "So we mark those two points on a number line:\r\n" );
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document.write( "\"number_line%28+700%2C+-1%2C20%2C+2%285%2Bsqrt%285%29%29%2C+2%285-sqrt%285%29%29+%29\"\r\n" );
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document.write( "So we test a point is the region left of 5.5, say 5, in\r\n" );
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document.write( "the region left of the 5 in the inequality:\r\n" );
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document.write( "\"x%5E2-20x%2B80%3C=0\"\r\n" );
document.write( "\"5%5E2-20%285%29%2B80%3C=0\"\r\n" );
document.write( "\"25-100%2B80%3C=0\"\r\n" );
document.write( "\"5%3C=0\"\r\n" );
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document.write( "That's false so we do not shade the part of the number line\r\n" );
document.write( "left of the green point.\r\n" );
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document.write( "Next we test a point is the region between 5.5 and 14.5, say 6, in\r\n" );
document.write( "the inequality:\r\n" );
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document.write( "\"x%5E2-20x%2B80%3C=0\"\r\n" );
document.write( "\"6%5E2-20%286%29%2B80%3C=0\"\r\n" );
document.write( "\"36-120%2B80%3C=0\"\r\n" );
document.write( "\"-4%3C=0\"\r\n" );
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document.write( "That's true so we shade the part of the number line between the\r\n" );
document.write( "green and red points:\r\n" );
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document.write( "Now we test a point is the region right of 14.5, say 15, in\r\n" );
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document.write( "the region left of the 5 in the inequality:\r\n" );
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document.write( "\"x%5E2-20x%2B80%3C=0\"\r\n" );
document.write( "\"15%5E2-20%2815%29%2B80%3C=0\"\r\n" );
document.write( "\"225-300%2B80%3C=0\"\r\n" );
document.write( "\"5%3C=0\"\r\n" );
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document.write( "That's false so we do not shade the part of the number line\r\n" );
document.write( "right of the red point.\r\n" );
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document.write( "So the solution set for the inequality in interval notation is\r\n" );
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document.write( "\"solution_set%282%285%2Bsqrt%285%29%29%2C+2%285-sqrt%285%29%29+%29\"\r\n" );
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document.write( "So yes we can make the length any value  of x between those two\r\n" );
document.write( "values, and the width \"y=80%2Fx\", and 40 will be enough fencing.\r\n" );
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document.write( "Edwin
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